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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(f:C\to X\) be a multiple covering of smooth projective curves. Here we discuss the non-existence of base point free linear systems on \(C\). multiple coverings of curves; double coverings; base point free pencils Special divisors on curves (gonality, Brill-Noether theory), Coverings of curves, fundamental group Linear systems on multiple coverings of a smooth curve | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(Z\) be a Fano manifold with Picard number \(1\) and index \(i(Z) \geq 3\). For a generic point \(y \in Z\) one defines the \textit{normalized space \(\mathcal{K}_y\) of minimal rational curves through \(y\)} (see [\textit{J. Kollár}, Rational curves on algebraic varieties. Berlin: Springer-Verlag (1995; Zbl 0877.14012)]). It parameterizes all rational curves on \(Z\) which pass through \(y\) and have minimal intersection index with \(-K_Z\). General such curve is smooth at \(y\) and one obtains a rational map \(\tau_y: \mathcal{K}_y \dashrightarrow \mathbb{P}T_y(Z)\) sending a given rational curve to its tangent at \(y\). The map \(\tau_y\) is known to be birational (see e.g. [\textit{J.-M. Hwang} and \textit{N. Mok}, Asian J. Math. 8, No. 1, 51--64 (2004; Zbl 1072.14015)]) and the closure \(\mathcal{C}_y\) of \(\tau_y\) is called the \textit{variety of minimal rational tangents} at \(y\). Let further the generator of \(\mathrm{Pic}(Z)\) be very ample and for the corresponding embedding \(Z \subset \mathbb{P}^N\) variety \(Z\) be covered by lines (so that in particular \(\mathcal{C}_y \neq \emptyset\) for all \(y\)).
In the paper under review, the author studies double covers \(\phi: X^Z \longrightarrow Z\) ramified in smooth section of \(Z\) by a hypersurface \(Y\) of degree \(2m, 1 \leq m \leq i(Z) - 2\), for which she proves the following (see Theorem 1.1 in the text). If \(x \in X^Z\) is a generic point, then \(\mathcal{C}_x\) is smooth of dimension \(i(Z) - m - 2\), and the differential \(d\phi_x: \mathbb{P}T_x(X^Z) \longrightarrow \mathbb{P}T_{\phi(x)}(Z)\) maps \(\mathcal{C}_x\) isomorphically onto an intersection of \(\mathcal{C}_{\phi(x)}\) and \(m\) hypersurfaces of degrees \(m+1,\ldots,2m\).
The main result is proved in Sections \(3\) and \(4\) (after some preliminaries in Section \(2\)). There the author proves that the image \(d\phi_x(\mathcal{C}_x)\) parameterizes such lines on \(Z\) which have all even local intersection indices with ramification divisor of \(\phi\) (cf. Proposition 4.2). The proof then follows, for generic \(Y\), essentially from \textit{J.-M. Hwang} and \textit{H. Kim} [Math. Z. 275, No. 1--2, 109--125 (2013; Zbl 1282.14070)], whereas the case of arbitrary \(Y\) as above is deduced by simple deformation argument.
In conclusion, the author also considers the so-called \textit{CR-rigidity} of \(\phi\), after [\textit{J.-M. Hwang} and \textit{N. Mok}, J. Math. Pures Appl. (9) 80, No. 6, 563--575 (2001; Zbl 1033.32013)], and establishes it in some special cases (cf. Theorem 1.4). double covers of Fano manifolds; varieties of minimal rational tangents; Cartan-Fubini type rigidity Fano varieties Cartan-Fubini type rigidity of double covering morphisms of quadratic manifolds | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Given a line bundle \(L\) on a smooth projective variety \(X\), the Gauss map
\[
\bigwedge^ 2 H^ 0(X,L) \to H^ 0 (X,L^{\otimes 2})
\]
is defined by sending \(\sum s_ i \otimes t_ i\) to \(\sum(ds_ i\otimes t_ i- s_ i\otimes dt_ i)\). If \(X \to Y\) is a cyclic covering, with \(Y\) and \(X\) smooth and projective, and \(L\) is a pull-back from \(Y\), then, using projection formulas, it is possible to express the Gauss map for \(L\) in terms of similar maps on \(Y\).
After having described this general setting, the author focuses on the case of double covers of \(\mathbb{P}^ m\) \((m>1)\) and computes the cokernel of the Gauss map for the pull-back of any effective line bundle on \(\mathbb{P}^ m\). The paper also contains several references on the general theory and applications of the Gauss maps. projection; Gauss map; double covers Duflot, J.: Gaussian maps for double coverings. Manuscript Math. 82, 71--87 (1994) Coverings in algebraic geometry, Projective techniques in algebraic geometry Gaussian maps for double coverings | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In 1901 Severi proved that the only surface in \(\mathbb{P}^5\) that can be projected isomorphically in \(\mathbb{P}^4\) is the Veronese surface in \(\mathbb{P}^5\). In the same paper, Severi claims that the only smooth surfaces in \(\mathbb{P}^5\) having one apparent double point are the rational scroll(s) of degree 4 and the Del Pezzo surface of degree 5. But the Severi's proof has a gap, which was remarked first by Ciliberto and Sernesi. In this paper, the author fills the gap and proves that the only surfaces in \(\mathbb{P}^5\) having one apparent double point are the rational normal scrolls \(S(2,2)\) and \(S(1,3)\) of degree 4 and the Del Pezzo surface of degree 5. Moreover, the author shows that the rational normal scrolls \(S(\underbrace{1,\dots,1}_{r-1},3)\), \(r\geq 1\), and \(S(\underbrace{1,\dots,1}_{r-2},2,2)\), \(r\geq 2\), in \(\mathbb{P}^{2r+1}\) of degree \(r+2\) have one apparent double point. A consequence is that the smooth \(r\)-dimensional subvarieties of \(\mathbb{P}^{2r+1}\) of minimal degree, with the exception of the Veronese surface, have one apparent double point. Veronese surface; rational normal scroll; Del Pezzo surface; double point Russo, F, On a theorem of Severi, Math. Ann., 316, 1-17, (2000) Projective techniques in algebraic geometry, Rational and ruled surfaces On a theorem of Severi | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper, we construct stable Bott-Samelson classes in the projective limit of the algebraic cobordism rings of full flag varieties, upon an initial choice of a reduced word in a given dimension. Each stable Bott-Samelson class is represented by a bounded formal power series modulo symmetric functions in positive degree. We make some explicit computations for those power series in the case of infinitesimal cohomology. We also obtain a formula of the restriction of Bott-Samelson classes to smaller flag varieties. Schubert calculus; cobordism; flag variety; Bott-Samelson resolution Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Grassmannians, Schubert varieties, flag manifolds, Generalized (extraordinary) homology and cohomology theories in algebraic topology, Bordism and cobordism theories and formal group laws in algebraic topology, Symmetric functions and generalizations, Classical problems, Schubert calculus, Combinatorial aspects of algebraic geometry Stability of Bott-Samelson classes in algebraic cobordism | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper provides an example of a simple highest weight module over \(\mathfrak{sl}_{12}(\mathbb{C})\) whose characteristic variety is reducible. The proof of reducibility is rather indirect, it uses the theories of \(p\)-canonical bases, \(W\)-graphs and perverse sheaves. More precisely, the paper gives two permutations \(x\) and \(y\) in \(S_{12}\) which lie in the same right Kazhdan-Lusztig cell and such that a normal slice to the Schubert variety corresponding to \(y\) along the Schubert cell corresponding to \(x\) is isomorphic to the Kashiwara-Saito singularity. This is equivalent to the reducibility of a certain characteristic variety. That characteristic varieties in other types can be reducible was already known. characteristic variety; Schubert variety; Kazhdan-Lusztig cell; highest weight module; Kashiwara-Saito singularity Williamson, Geordie, A reducible characteristic variety in type \(A\).Representations of reductive groups, Prog. Math. Phys. 312, 517-532, (2015), Birkhäuser/Springer, Cham Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.), Intersection homology and cohomology in algebraic topology, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Grassmannians, Schubert varieties, flag manifolds A reducible characteristic variety in type \(A\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The aim of the article is to give resolution of the singularities of a Schubert cycle in the Grassmannian \(G(r,n)\) by blowing-up certain sub- Schubert cycles.
Let \(K\) a field, \(r\) and \(n\) positive integers with \(r<n\), and \(G(r,n)\) the Grassmannian of \(r\)-dimensional subspaces of \(K^ n\). We call \(T\) the set of \(r\)-uples \((t) = (t_ 1, \dots, t_ r)\) of integers satisfying \(1 \leq t_ 1 < \cdots < t_ r \leq n\), and for each \((t)\) in \(T\) we introduce \(r(n - r)\) variables \(X^{(t)}_{i,j}\), \(1 \leq i \leq r\), \(1 \leq j \leq n\) and \(j \neq t_ 1, \dots, t_ r\), and we define an affine scheme \(U^{(t)} = \text{Spec} K[X^{(t)}_{i,j}]\). The scheme structure on the Grassmannian \(G(r,n)\) is obtained by gluing the affine sets \(U^{(t)}\). For any \((l)\) in \(T\), we define the Schubert cycle \(S_{(l)}\). For any \((t)\) and \((l)\) in \(T\), we define a closed subscheme \(S^{(t)}_{(l)}\) of \(U^{(t)}\) whose equations are the \((r - i + 1)\) minors of some matrix. The structure scheme on the Schubert cycle \(S_{(l)}\) is obtained by gluing the sets \(S^{(t)}_{(l)}\). The scheme \(S_{(l)}\) is not necessarily smooth and its dimension is equal to \(\sum^ r_{i = 1} (l_ i - i)\). We want to study the blowing-up of a Schubert cycle \(S_{(l)}\) in \(G(r,n)\), more precisely if we order the sub-Schubert cycles \(\sigma_ 1 \leq \cdots \leq \sigma_ k\) of \(S_{(l)}\) by dimension, we want to determine their strict transforms by this blowing-up. We put an order on \(T\) (inverse lexicographic): \((k) > (l)\) if and only if there exists \(s \geq 0\) so that \(k_ i = l_ i\) for \(i \leq s\) and \(k_{s + 1} < l_{s + 1}\).
Proposition: For any \((l)\) in \(T\), \(S^{(l)}_{(l)}\) is the first nontrivial Schubert cycle in \(S^{(l)}\), i.e. \(S^{(l)}_{(l)} \neq \emptyset\) and \(S^{(l)}_{(t)} = \emptyset\) for \((t) < (l)\).
Blowing-up \(S_{(l)}\), the strict transforms of the Schubert cycles in \(U^{(l)} = \text{Spec} K [X_{i,j}^{(l)}]\) are isomorphic to the Schubert cycles in \(U^{(l')}\), where \((l) > (l')\). More precisely, we can cover the blowing-up \(\widetilde {U^{(l)}}\) of \(U^{(l)}\) by affine chart \(B_{(l)} (s,t)\), with \(1 \leq s \leq r\), \(t \geq l_ s + 1\) and \(t \neq l_ 1, \dots, l_ r\). For any \((s,t)\), we can construct \((l')\) such that \((l) > (l')\) and the chart \(B_{(l)} (s,t)\) is isomorphic to \(U^{(l')}\), then the strict transform of a Schubert cycle in this chart is isomorphic to the Schubert cycle in \(U^{(l')}\). -- From the proposition, the author deduces the principal result:
Corollary: Let \(\sigma\) be a Schubert cycle in \(G(r,n)\), let \(\{\sigma_ i\}\) be the Schubert cycles contained in \(\sigma\) ordered so that \(\dim \sigma_ i \leq \dim \sigma_{i + 1}\). Then \(\sigma\) can be desingularized by blowing up \(\sigma_ 1\), then by blowing up the strict transform of \(\sigma_ 2\), and so on. resolution of the singularities of a Schubert cycle; Grassmannian; strict transforms of the Schubert cycles Boudhraa, Z.: Resolution of singularities of Schubert cycles. J. pure appl. Algebra 90, No. 2, 105-113 (1993) Global theory and resolution of singularities (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds Resolution of singularities of Schubert cycles | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper is an outstanding survey of Thomason's theory of hypercohomology spectra associated to presheaves of spectra on a Grothendieck site. This theory is explained along with celebrated applications to \(K\)-theory, and compared to Jardine's approach to simplicial presheaves. hypercohomology spectra; presheaves of spectra; Grothendieck site; simplicial presheaves Stephen A. Mitchell, Hypercohomology spectra and Thomason's descent theorem, Algebraic \?-theory (Toronto, ON, 1996) Fields Inst. Commun., vol. 16, Amer. Math. Soc., Providence, RI, 1997, pp. 221 -- 277. \(K\)-theory of schemes, Étale and other Grothendieck topologies and (co)homologies, Stable homotopy theory, spectra, Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) Hypercohomology spectra and Thomason's descent theorem | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials [For the entire collection see Zbl 0707.00010.]
The authors give a view of the status of art of the theory of de Rham- Grothendieck coefficients and the methods of proofs. differential operator; de Rham-Grothendieck coefficients Mebkhout, Z.; Narvaez, L., Sur LES coefficients de de Rham-Grothendieck des variétés algébriques, (\textit{p}-adic Analysis, Lect. Notes Math., vol. 1454, (1990), Springer-Verlag Heidelberg), 267-309 de Rham cohomology and algebraic geometry, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials Sur les coefficients de De Rham-Grothendieck des variétés algébriques. (On the De Rham-Grothendieck coefficients of algebraic varieties) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In the paper under review, the author introduces concepts which are weaker than coordinates and use them to study coordinates in the polynomial ring \(k^{[2]}=k[x,y]\) in two variables over an algebraically closed field of characteristic 0. A polynomial \(f\) is a coordinate if \(k^{[2]}\) if there exists another polynomial \(f_2\) such that \(k[f,f_2]=k^{[2]}\). Let \(f\in k^{[2]}\setminus k\) be a nonconstant polynomial and let \(\Phi_f:\text{Spec } k^{[2]}\to\text{Spec }k[f]\) be the morphism associated to the inclusion \(k[f]\hookrightarrow k^{[2]}\). Let \(\Phi_f^{-1}(P)\) by a fiber of \(\Phi_f\) at a close point \(P\in\text{Spec }k[f]\). Then \(\Phi_f^{-1}(P)=\text{Spec}(k[x,y]/(f-\lambda))\), \(\lambda\in k\). The first main result of the paper gives equivalent conditions in terms of \(\Phi_f\) and \(\Phi_f^{-1}(P)\) for \(f\) to be factorially closed (i.e., \(k[f]\) is factorially closed in \(k^{[2]}\)) and univariate (i.e., \(f\in k[g]\) for some coordinate \(g\)).
Then the author considers an integral domain \(R\) and an \(R\)-domain \(A\). Let \(S(r,A)\) be the set of all \(R\)-subalgebras of \(A\) of transcendence degree \(r\) over \(R\). The main result characterizes the maximal elements of \(S(r,A)\) in terms of algebraic closeness, integral closeness and constants of derivations. This is a generalization of a result in [\textit{H. Kojima} and \textit{T. Nagamine}, J. Pure Appl. Algebra 219, No. 12, 5493--5499 (2015; Zbl 1320.13010)]. closed polynomials; coordinates; derivations Polynomials over commutative rings, Integral closure of commutative rings and ideals, Polynomial rings and ideals; rings of integer-valued polynomials, Derivations and commutative rings, Affine fibrations On some properties of coordinates in polynomial rings | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(R\) be an integral domain and \(K\) its quotient field. The main purpose of the first part of the book is to study the ring \(\text{Int} (R) = \{f \in K [X]\); \(f(R) \subset R\}\). It is well known (G. Pólya, 1915) that, when \(R = \mathbb{Z}\), \(\text{Int} (R)\) is a free \(R\)-module generated by a family \(\{h_n\}_{n \in \mathbb{N}}\) of polynomials with \(\deg h_n = n\) (these are in fact the binomial polynomials). An open problem is the determination of all rings \(R\) with this property.
The author discusses this problem, in particular in the case when \(R = I_K\) is the ring of integers of a number field \(K\). Another interesting question is about the algebraic properties of \(\text{Int} (R)\): noetherianity, Skolem property, maximal and prime ideals, Krull dimension, Prüfer property, etc. This first part also deals with the values of the successive derivatives of polynomials or of rational functions.
The second part is devoted to the study of fully invariant subsets of a field by polynomial mappings: if \(f \in \mathbb{Q} [X]\) and \(S \subset \mathbb{Q}\) satisfy \(f(S) = S\), then either \(S\) is finite or \(\deg f = 1\). The aim of study is to determine the fields with this property or its analogue in the case of several variables. In particular, is this property stable by purely transcendental extension (yes) or by finite extension? The last chapter deals with polynomial cycles: by a theorem of I. N. Baker (1960) every polynomial of degree \(\geq 2\) in \(\mathbb{C} [X]\) has cycles of every order with at most one exception. The author considers this question in algebraic number fields.
This nice, short (130 pages) but dense book makes a sound review of the question. As often as possible, concise proofs are given. More technical results or related questions are described and references are given; the text is well supplemented by many exercises given at the end of each chapter. The appendix states a list of 21 open problems and the book contains 11 pages of bibliographical references (from 1895 to 1994). It is interesting to have such a synthesis on questions which are often studied but scattered in the literature. binomial polynomials; values of the successive derivatives of polynomials or rational functions; polynomial functions; integral domain; ring of integers of a number field; fully invariant subsets; polynomial mappings; several variables; transcendental extension; finite extension; polynomial cycles; many exercises; open problems; bibliographical references Narkiewicz, Władysław, Polynomial mappings, Lecture Notes in Mathematics 1600, viii+130 pp., (1995), Springer-Verlag, Berlin Research exposition (monographs, survey articles) pertaining to number theory, Polynomials in number theory, Polynomials in general fields (irreducibility, etc.), Research exposition (monographs, survey articles) pertaining to commutative algebra, Polynomials over commutative rings, Polynomials over finite fields, Polynomial rings and ideals; rings of integer-valued polynomials, Rational and birational maps, Polynomials (irreducibility, etc.) Polynomial mappings | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials A collection \(A=\{D_1,\dots,D_n\}\) of divisors on a smooth variety \(X\) is an \textit{arrangement} if the intersection of every subset of \(A\) is smooth. We show that, if \(X\) is defined over a field of characteristic not equal to 2, a double cover of \(X\) ramified on an arrangement has a crepant resolution under additional hypotheses. Namely, we assume that all intersection components that change the canonical divisor when blown up are \textit{splayed}, a property of the tangent spaces of the components first studied by Faber. This strengthens a result of Cynk and Hulek, which requires a stronger hypothesis on the intersection components. Further, we study the singular subscheme of the union of the divisors in \(A\) and prove that it has a primary decomposition where the primary components are supported on exactly the subvarieties which are blown up in the course of constructing the crepant resolution of the double cover. double covers; crepant resolution; primary decomposition; Calabi-Yau varieties Calabi-Yau manifolds (algebro-geometric aspects), Singularities in algebraic geometry, Coverings in algebraic geometry, Polynomial rings and ideals; rings of integer-valued polynomials On the Cynk-Hulek criterion for crepant resolutions of double covers | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors study \textit{symmetric} nonnegative homogeneous polynomials and relationships between the cone of symmetric sums of squares and the cone of symmetric nonnegative forms of fixed degree \(2d\) (in arbitrary numbers of variables).
They provided a uniform representation of the cone of symmetric sums of squares and its dual cone in terms of linear matrix polynomials. In particular, by using the representation, they completely characterize the sums of squares cone \(\Sigma_{n,4}\) of degree \(4\) in \(n\) variable and its boundary, and therefore certify the difference between symmetric sums of squares and symmetric non-negative quartics.
Also, they investigated the asymptotic behavior of the cone of sums of squares and nonnegative forms of fixed degree \(2d\) as the number \(n\) of variables grows. In detail, they showed that the difference between the cone of symmetric nonnegative forms and sums of squares does not grow arbitrarily large for any fixed degree \(2d\) (even though the (volume) difference between the two cones increases exponentially for any even degree \(2d\) as the number \(n\) of variables grows). In particular, they show that the cone of symmetric non-negative quartics and the cone of quartic symmetric sums of squares asymptotically become closer as the number of variables grows by proving the two cones approach the same limit (in degree \(4\)). They conjectured that the limits agree in any degree \(2d\) for \(d>2\). non-negative polynomials; sums of squares; symmetric polynomials; symmetric inequalities; symmetric group Semialgebraic sets and related spaces, Representations of finite symmetric groups Symmetric non-negative forms and sums of squares | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials At the time when A. Grothendieck established his ingenious ideas about schemes as the foundation of algebraic geometry there grew out a deep interest in the study of section functors and its right derived functors. Motivated by questions about fundamental groups, Lefschetz theorems a.o., \textit{A.Grothendieck} [see ``Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux'', Sémin. géométrie algébrique 2 (SGA2) (1962; Zbl 0159.50402), see also the enlarged edition (Amsterdam 1968; Zbl 0197.47202)] developed the local cohomology theory in algebraic geometry. In his work as well as in R. Hartshorne's notes about A. Grothendieck's ideas [see \textit{R. Hartshorne}, Notes in the book by \textit{A. Grothendieck}: ``Local cohomology'', Lect. Notes Math. 41 (1967; Zbl 0185.49202)] it turned out that the power of these techniques is -- at least -- two-fold. Firstly it allows to transform the homological techniques invented by J.-P. Serre in algebraic geometry [see \textit{J.-P. Serre}, ``Faisceaux algébriques cohérents'', Ann. Math., II. Ser. 61, 197-278 (1955; Zbl 0067.16201)] to commutative ring theory. Secondly it opens a wide field for research activities on local cohomology in commutative algebra, as one might see by the literature over the last decades. In the mean time there are several approaches to local cohomology, among them those by \textit{W. Bruns} and \textit{J. Herzog} [``Cohen-Macaulay rings'' (revised edition 1998; see also the first edition 1993; Zbl 0788.13005)], \textit{D. Eisenbud} [``Commutative algebra. With a view toward algebraic geometry'' (1995; Zbl 0819.13001)], \textit{M. Herrmann, S. Ikeda} and \textit{U. Orbanz} [``Equimultiplicity and blowing up. An algebraic study'' (1988; Zbl 0649.13011)], \textit{M. Hochster} [``Notes on local cohomology'', Lect. Univ. Michigan, Ann Arbor], \textit{P. Schenzel} [``On the use of local cohomology in algebra and geometry'', Lect. Summer School Commutative Algebra and Algebraic Geometry, Bellaterra 1996 (Birkhäuser 1998)], and \textit{C. Weibel} [``An introduction to homological algebra'' (1994; Zbl 0797.18001)].
The basic motivations for the introduction under review are the following: 1. The authors feel a challenge for an algebraic introduction to Grothendieck's local cohomology theory originally invented by the aid of scheme theory and in that form not available yet. -- 2. The introduction is designed primarily to graduate students who have some experience of basic commutative and homological algebra. So the approach is homologically based on the fundamental `\(\delta\)-functor' technique. -- 3. A large part of the investigations follows algebraic properties of local cohomology, most of them for the first time available in a textbook, e.g., local duality, secondary representations of local cohomology modules, annihilator and finiteness results, graded local cohomology, Hilbert polynomials etc. -- 4. From an algebraic point of view the authors illustrate the geometric significance of various aspects of local cohomology, in particular by applications of local cohomology to connectivity, Castelnuovo-Mumford regularity, sheaf cohomology.
The authors expect that the interested reader should be familiar with the basic sections of the books by \textit{H. Matsumura} [``Commutative ring theory'' (1986; Zbl 0603.13001)] and \textit{J. J. Rotman} [``An introdution to homological algebra'' (1979; Zbl 0441.18018)]. Consequently they included expositions about Matlis duality, the indecomposable injective modules, Hilbert polynomials, foundations about \(\mathbb Z\)-graded rings and modules for the use in graded local cohomology theory. Besides of the authors' approach to the fundamental vanishing theorems on local cohomology, the Lichtenbaum-Hartshorne vanishing theorem a.o. there are very interesting chapters about the annihilation and finiteness theorems on local cohomology, Castelnuovo-Mumford regularity in geometry, connectivity in algebraic geometry, where research results are provided in a textbook form for the first time.
The text is carefully and clearly written. Very often it is completed by examples and easy exercises. They make it easier to a beginner to learn the subject. The connectivity results are -- at least for the reviewer -- the highlights of the book. The authors' use of local cohomology leads to proofs of major results involving connectivity, such as Grothendieck's connectedness theorem, the Bertini-Grothendieck connectivity theorem, the connectedness theorem for projective varieties due to W. Barth, to W. Fulton and W. Hansen, and to G. Faltings, as well as a ring theoretic version of Zariski's main theorem. By the authors' intention the characteristic \(p\) methods in local cohomology -- introduced by \textit{C. Peskine} and \textit{L. Szpiro} [Inst. Hautes Étud. Sci., Publ. Math. 42 (1972), 47-119 (1973; Zbl 0268.13008)] and further developed by M. Hochster and C. Huneke in the notion of tight closure and related research as well as results about big Cohen-Macaulay modules and the rôle of local cohomology in intersection theorems are beyond the scope of the book. For an introduction to this subject the interested reader, prepared by the basics of that book, might and should consult the third part of the book by \textit{W. Bruns} and \textit{J. Herzog}, ``Cohen-Macaulay rings'', cited above.
The value of the book under review consists in its consequent introductory nature, which may be welcome to the beginners of the subject. By the aid of illuminating examples and exercises the authors shead different colours on several subjects of commutative algebra and algebraic geometry. The interested reader will be guided to research problems connected to local cohomology as well as the power of the methods in algebraic geometry. ideal transform; local duality; Hilbert polynomials; reduction of ideals; connectivity; sheaf cohomology; vanishing theorems; annihilation; finiteness theorems on local cohomology; Castelnuovo-Mumford regularity M.P. Brodmann, R.Y. Sharp, \(Local Cohomology: An Algebraic Introduction with Geometric Applications\). Cambridge Studies in Advanced Mathematics, vol. 60 (Cambridge University Press, Cambridge, 1998). MR 1613627 (99h:13020) Local cohomology and commutative rings, Local cohomology and algebraic geometry, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra, Homological methods in commutative ring theory Local cohomology. An algebraic introduction with geometric applications | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper extends results on structure sheaves associated to a module over a commutative Noetherian ring, which the author has extended to the non-commutative case, to additive categories. Two results indicate the nature of this extension:
Theorem 1: For any quasi-spectrum \(S\) of an additive category \({\mathcal C}\), if, for each \(F:{\mathcal C}^{op}\rightarrow{\mathcal A}b\) which is an additive presheaf of abelian groups and each \(\tau\in S\), \(\widetilde{Q}_{F}(\tau)\) is a \(\tau\)-sheaf, then the sheaf \(\widetilde{Q}\) has a right adjoint. Here \(\widetilde{Q}_{F}(\tau)\) is the sheafification of \(Q_{F}\), and \(Q_{F}\) is the structure presheaf associated to \(F\).
Theorem 2: Let \({\mathcal C}\) and \(F\) be as above. Let \(\mathcal{B}\) be a basis of \(\operatorname {Spec}{\mathcal C}\) consisting of quasi-compact open sets. Then, the associated presheaf \(Q_{F}\) is a sheaf over \(\mathcal{B}\) if and only if, for \(D(I), D(J)\in\mathcal{B}\), the sequence
\[
0\rightarrow Q_{I+J}\to Q_{I}\oplus Q_{J}\to Q_{IJ}
\]
is exact. structure sheaf; additive category; Grothendieck topology; exactness; adjoint Preadditive, additive categories, Étale and other Grothendieck topologies and (co)homologies, Torsion theories, radicals, Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) Structure sheaves for additive categories | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author gives a definition of the category \(M(X,{\mathcal D})\) of ``\(\mathcal D\)-modules'' on a complex analytic space \(X\), without using any structure sheaf on \(X\). In the smooth case, this category agrees with the category of usual \({\mathcal D}_X\)-modules. He extends the notions and results of the smooth case to the singular case: coherence, holonomy, regularity, local algebraic cohomology, \(\otimes\), duality, direct and inverse images. De Rham function, Riemann-Hilbert correspondence. The key points are: 1) Kashiwara's theorem, giving the equivalence between the category of \(\mathcal D\)-modules over a complex submanifold \(M\) over \(Z\) satisfying \(\Gamma_{[Y]}(M)\simeq M\). 2) The fact that every complex analytic space is locally embeddable in a complex manifold. 3) The construction of a Grothendieck topology \({\mathcal C}(X)\) over \(X\) [cf. `Grothendieck topologies' by \textit{M. Artin}, Cambridge, Mass.: Harvard University (1962; Zbl 0208.48701) and `Notes on crystalline cohomology' by \textit{P. Berthelot} and \textit{A. Ogus}, Princeton, N.J.: Princeton University Press (1978; Zbl 0383.14010)], whose objects are the closed embeddings \(U\hookrightarrow V\) such that \(U\subset X\) is an open set and \(V\) is smooth, and whose morphisms are the morphisms of \(V\) such that their restrictions to \(U\) are the inclusions as subsets of \(X\).
The objects of \(M(X,{\mathcal D})\) are defined locally for \({\mathcal C}(X)\), and the glueing is accomplished by the !-direct image functor. The proofs of the results in the paper depend only on the corresponding results in the smooth case, and the compatibility with the glueing data. Therefore, the main author's contribution is a convenient technical manipulation of these compatibilities.
It seems possible to take the point of view of `Le théorème de comparaison entre cohomologies de De Rham d'une variété algébrique complexe et le théorème d'existence de Riemann' by \textit{Z. Mebkhout}, Publ. Math., Inst. Hautes Etud. Sci. 69, 47-89 (1989; Zbl 0709.14015), to be able to define the ``irregularity complexes'', and to avoid the use of the general theorem of desingularization of Hironaka. \(\mathcal D\)-module; local algebraic cohomology; de Rham function; Grothendieck topology DOI: 10.2977/prims/1195169840 Sheaves of differential operators and their modules, \(D\)-modules, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, de Rham cohomology and algebraic geometry \(\mathcal D\)-modules on analytic spaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials [For the entire collection see Zbl 0695.00010.]
The purpose of the paper under review is to give a proof of six formulas by Schubert (two of which he proved and four of which he only conjectured) concerning the number of double contacts among the curves of two families of plane curves. The method consists in finding bases of the Chow groups of the Hilbert scheme of length 2 subschemes of the point- line incidence variety. This approach turns out to be much simpler than the one using the space of triangles as suggested by Schubert.
As a byproduct, the authors obtain proofs of the classical formulas on triple contacts (i.e., single contacts of third order) between two such families of curves. flag variety; number of double contacts; families of plane curves; Chow groups; triple contacts Enumerative problems (combinatorial problems) in algebraic geometry, Parametrization (Chow and Hilbert schemes), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Proof of Schubert's conjectures on double contacts | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper describes the interaction between classical algebraic geometry, general algebra, and formal concept analysis. Its goal is to elaborate the general core of the basic results of algebraic geometry using this interaction. We start from a general polynomial context of the form \(\mathbb{K}_{n,A}: =(A^n,F_n (X,A)\times F_n(X,A), \perp)\). Here \(A\) is a general algebra, \(F_n(X,A)\) is the free algebra in \(n\) variables in the variety (in the sense of general algebra) \(\text{Var} A\) generated by \(A\), and we have \(\vec a\perp(p,q): \Leftrightarrow p(\vec a)=q (\vec a)\) for \(\vec a\in A^n\) and \(p,q\in F_n(X,A)\). Extents of this formal context will be called \(A\)-algebraic sets. We find that the intents of \(\mathbb{K}\) are certain congruence relations on \(F_n(X,A)\), which we will call radical congruences. We conclude that the lattice of \(A\)-algebraic sets in \(A^n\) and the lattice of radical congruences on \(F_n(X,A)\) are dually isomorphic. When we choose a general algebra such that \(F_n(X,A)\) is the ring of polynomials \(K[x_1, \dots,x_n]\) over an algebraically closed field, we obtain the classical correspondence between algebraic varieties in \(K^n\) and reduced ideals in \(K[x_1, \dots,x_n]\). In algebraic geometry we have a functorial correspondence between algebraic varieties and coordinate algebras \(K[V]:=K[x_1, \dots, x_n]/V^\perp\). (Here \(V^\perp\) is the ideal of polynomials that vanish on \(V)\). For \(A\)-algebraic sets \(V\), we define a coordinate algebra \(\Gamma(V)\) by \(\Gamma(V):= F_n(X,A)/ \Phi\), where \(\Phi:= V^\perp\) is the congruence relation corresponding to \(V\). Since \(A\)-algebraic sets can be understood as homomorphisms from \(F_n(X)/ \Phi\) to \(A\) and since coordinate algebras can be understood as finitely generated subalgebras of a power of \(A\), we get a dual equivalence between the category of \(A\)-algebraic sets with polynomial morphisms -- yet to be defined -- and the category of finitely generated subalgebras of a power of \(A\) with homomorphisms. This result is due to \textit{H. Bauer} [About Hilbert's and Rückert's Nullstellensatz. (FB4-Preprint No. 722, TU Darmstadt) (1983)]. In the classical case we get the dual equivalence mentioned above. We will use the general results to deduce the classical results of algebraic geometry. Along with basic results from formal concept analysis we determine which classical results follow from the general treatment and which results require commutative algebra. \(A\)-algebraic sets; classical algebraic geometry; formal concept analysis; polynomial context; free algebra; formal context; congruence relations; radical congruences; ring of polynomials; algebraically closed field; algebraic varieties; reduced ideals; functorial correspondence; coordinate algebras; dual equivalence; category Lattices, Foundations of algebraic geometry, Galois correspondences, closure operators (in relation to ordered sets), Complete lattices, completions, Varieties and morphisms, Free algebras, Subalgebras, congruence relations General algebraic geometry and formal concept analysis | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We give a description of the singular locus of a Schubert variety X, in the flag variety G/B, where G is a classical group and B is a Borel subgroup. The singular locus is determined by using standard monomial theory as developed in ''Geometry of G/B'' [by the authors and \textit{C. Musieli}; part I-IV in Collect. Publ. C. P. Ramanujan and Papers in his Mem., Tata Inst. Fundam. Res., Stud. Math. 8, 207-239 (1978); Proc. Indian Acad. Sci., Sect. A 87, No.2, 1-54 (1978); ibid. 88, No.2, 93-177 (1979); ibid. 88, No.4, 279-362 (1979; Zbl 0447.14010-14013); part V (to appear)]. A consequence of this theory is the determination of the ideal defining X in G/B, using which, we are able to write the Jacobian matrix Jw,\(\tau\) (here w is given by \(X=X(w)\) and \(e_{\tau}\) is the T-fixed point in G/B corresponding to \(\tau\), \(\tau\leq w\), T being a maximal torus contained in B) in the affine neighborhood \(U^-_{\tau}\cdot\tau \) of \(e_{\tau}\), where \(U^-_{\tau}=\tau U^-\tau^{-1}\), \(U^-\) being the unipotent part of the Borel subgroup of G opposite to B. Evaluating Jw,\(\tau\) at \(e_{\tau}\), we obtain the dimension of \(Z_{w,\tau}\), the Zariski tangent space to X(w) at \(e_{\tau}\). Denoting by \(\{\) \(p(\lambda\),\(\mu)\}\) the weight vectors as given by standard monomial theory, let \(R(w,\tau)=\{\beta\in \tau (\Delta^+)\quad there\quad exists\quad a\quad p(\lambda,\mu),\quad such\quad that\quad\quad w\ngeq\lambda \quad and\quad X_{- \beta}p(\lambda,\mu)=cp(\tau),c\in k^*\}.\) Then we have the main theorem \(\dim Z_{w,\tau}=N-\# R(w,\tau)\) where \(N=\# \Delta^+=\# \{positive\quad roots\}).\) In particular we have X(w) is smooth at \(e_{\tau}\) if and only if \(N-\# R(w,\tau)=\upharpoonright (w),\) the length of w. dimension of Zariski tangent space; singular locus of a Schubert variety; flag variety; standard monomial theory; Jacobian matrix; weight vectors C.S. Seshadri : Normality of Schubert variety . Proceeding de ''Algebraic Geometry'' (Bombay, Avril 1984). Grassmannians, Schubert varieties, flag manifolds, Singularities in algebraic geometry Singular locus of a Schubert variety | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The superb paper under review is quite (and probably unavoidably) technical. That is why the reviewer will not put a special effort to simplify the description of its content. Otherwise this short summary may turn useless for both non experts and specialists: the former because in any case would lack background, and the latter because the essential mathematical content of the paper would be lost behind excess of simplification. In any case, to save a minimum of friendly shape, let us begin this short summary as if it were a tale.
Schubert calculus. At the very beginning was the intersection theory of the complex Grassmann manifolds \(G(k,n)\) parametrizing \(k\)-dimensional subspaces of \({\mathbb C}^n\). Grassmannians, however, are a special kind of flag varieties, which are in turn special kind of homogeneous projective varieties, i.e. quotient \(G/B\) of a complex connected semi-simple Lie group modulo the action of a Borel subgroup \(B\). Thus, nowadays, the location Schubert calculus has acquired a much broader meaning. Not only for Grassmannians, but on general flag varieties, and not only classical cohomology, but also quantum, equivariant or quantum-equivariant, up to its \(K\)-theory and its connective \(K\)-theory (a theory interpolating, in a suitable sense, the \(K\)-theory and the intersection theory of a homogeneous space).
The paper under review puts itself in this very general framework using in a creative manner a new algebraic tool, what the authors call \textsl{formal root polynomials}, with the purpose of studying the elliptic cohomology of the homogeneous space \(G/B\): this means a cohomology theory where all the odd parts vanish and there is an invertible element \(h\in H^2\) inducing a complex orientation with the same formal group law as that of an elliptic curve. There is a correspondence between generalized cohomology theories and formal group laws, and in particular the authors investigate the \textsl{hyperbolic group law} introduced in Section 2.2. The corresponding Schubert calculus is so called by the authors \textsl{hyperbolic Schubert calculus.} and enables to study the elliptic cohomology of homogeneous spaces, extending previous work by Billey and Graham-Willems letting it to work uniformly in all Lie type. The definition of formal root polynomial is quite technical and is not worth to be recalled in the present review. However the idea is that of replacing, or rather extend, the notion of root polynomials heavily used by \textit{S. C. Billey} [Duke Math. J. 96, No. 1, 205--224 (1999; Zbl 0980.22018)] and by \textit{M. Willems} [Bull. Soc. Math. Fr. 132, No. 4, 569--589 (2004; Zbl 1087.19004)].
After introducing the formal root polynomial, whose definition depends on a reduced word for a Weyl group element, the main Theorem 3.10 states that indeed it does not depend on such a word provided that the formal group law is the hyperbolic one. Section 4 is devoted to applications: in particular the authors show how their techniques provide an efficient method to compute the transition matrix between two natural bases of the formal \textsl{Demazure algebra}, another gadget introduced and explained in Section 2. Section 5 is concerned with localization formulas in cohomology and \(K\)-theory, while section 6 is not only devoted to show further applications of root polynomials to compute Bott-Samelson classes, but also to propose a couple of conjectures in the hyperbolic Schubert calculus, based mainly on analogies and experimental evidence. The paper ends with a comprehensive reference list: among the key ones, the paper by Goresky, Kottwitz and MacPherson [\textit{M. Goresky} et al., Invent. Math. 131, No. 1, 25--83 (1998; Zbl 0897.22009)], the important 1974 paper by \textit{M. Demazure} [Ann. Sci. Éc. Norm. Supér. (4) 7, 53--88 (1974; Zbl 0312.14009)] on desingularization of generalized Schubert varieties, a couple of papers by Graham and Graham-Kumar on equivariant \(K\)-theory, and the papers by Billey and Willems, that inspired the research developed in this amazing step forward a generalized cohomology Schubert calculus. Schubert calculus; equivariant oriented cohomology; flag variety; root polynomial; hyperbolic formal group law ] C. Lenart and K. Zainoulline, Towards generalized cohomology Schubert calculus via formal root polynomials, arXiv:1408.5952. Grassmannians, Schubert varieties, flag manifolds, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Equivariant \(K\)-theory, Generalized (extraordinary) homology and cohomology theories in algebraic topology, Bordism and cobordism theories and formal group laws in algebraic topology, Algebraic combinatorics Towards generalized cohomology Schubert calculus via formal root polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The Real Nullstellensatz, a classic fundamental result in Real Algebraic Geometry, was generalized in [\textit{G. Alon} and \textit{E. Paran}, J. Pure Appl. Algebra 225, No. 4, Article ID 106572, 9 p. (2021; Zbl 1483.16027)] to quaternionic polynomials. The main results of this paper (Theorems E and F) extend the Quaternionic Nullstellensatz to quaternionic matrix polynomials. The author also proves a matrix version of the Real Nullstellensatz (Theorem D) that improves, thanks to a simpler definition of real left ideal (Sect.~1.4) the main theorem in [\textit{J. Cimprič}, J. Algebra 396, 143--150 (2013; Zbl 1423.13139)]. The proof of the Matricial Real Nullstellensatz makes use of the main results of [\textit{J. Cimprič}, J. Algebra Appl. 21, No. 11, Article ID 2250217, 11 p. (2022; Zbl 07596140)], which are recalled in Sect.~2. Real and mildly non-commutative versions of these results are investigated in Sect.~3. real algebraic geometry; nullstellensatz; polynomials over division algebras; matrix rings; quaternions Projective and free modules and ideals in commutative rings, Ideals in associative algebras, Noncommutative algebraic geometry, Real algebraic and real-analytic geometry Matrix versions of real and quaternionic nullstellensätze | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper, we define generalized Hardy-Berndt sums and elliptic analogue of the generalized Hardy-Berndt sums related to elliptic Bernoulli polynomials. We give relations between the Weierstrass \(\wp (z)\)-function, Hardy-Berndt sums, theta functions and generalized Dedekind eta function. Bernoulli polynomials and Bernoulli functions; Dedekind sums; Dedekind-Rademacher sums; Hardy sums; Theta functions Simsek, Y, Elliptic analogue of the Hardy sums related to elliptic Bernoulli functions, Gen. Math, 15, 3-23, (2007) Dedekind eta function, Dedekind sums, Bernoulli and Euler numbers and polynomials, Theta functions and abelian varieties, Theta functions and curves; Schottky problem Elliptic analogue of the Hardy sums related to elliptic Bernoulli functions | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We present a new numerical homotopy continuation algorithm for finding all solutions to Schubert problems on Grassmannians. This Littlewood-Richardson homotopy is based on Vakil's geometric proof of the Littlewood-Richardson rule. Its start solutions are given by linear equations and they are tracked through a sequence of homotopies encoded by certain checker configurations to find the solutions to a given Schubert problem. For generic Schubert problems the number of paths tracked is optimal. The Littlewood-Richardson homotopy algorithm is implemented using the path trackers of the software package PHCpack. Grassmannian; Schubert problems; continuation; geometric Littlewood-Richardson rule; homotopies; numerical Schubert calculus; path following; polynomial system F. Sottile, R. Vakil, and J. Verschelde, \textit{Solving Schubert problems with Littlewood-Richardson homotopies}, Proc. ISSAC 2010 (Stephen M. Watt, ed.), ACM, 2010, pp. 179-186. Symbolic computation and algebraic computation, Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus Solving Schubert problems with Littlewood-Richardson homotopies | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We provide an explicit formula for the following enumerative problem: how many (absolutely) indecomposable vector bundles of a given rank \(r\) and degree \(d\) are there on a smooth projective curve \(X\) of genus \(g\) defined over a finite field \(\mathbb{F}_{q}\)? The answer turns out to only depend on the genus \(g\), the rank \(r\) and the Weil numbers of the curve \(X\). We then provide several interpretations of these numbers, either as the Betti numbers or counting polynomial of the moduli space of stable Higgs bundles (of same rank \(r\) and degree \(d\)) over \(X\), or as the character of some infinite dimensional graded Lie algebra. We also relate this to the (cohomological) Hall algebras of Higgs bundles on curves and to the dimension of the space of absolutely cuspidal functions on \(X\). Kac polynomials; dimension of graded vector spaces; representations of Kac-Moody algebras; graded Borcherds algebras; character of infinite dimensional graded Lie algebra Infinite-dimensional Lie (super)algebras, Graded Lie (super)algebras, Finite ground fields in algebraic geometry, Vector bundles on curves and their moduli Kac polynomials and Lie algebras associated to quivers and curves | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper discusses an equivariant version of the Riemann-Roch theorem. Let \(k\) be a field and \(G\) an algebraic group over \(k\). Let \(X\) be a good \(G\)-scheme -- that is, the action of \(G\) is proper and the stabilizer of any geometric point of \(X\) is finite and reduced. Then (proposition 2.1) there is a unique ring homomorphism \(ch:K_ 0 \bigl(X \| G \bigr) \to A \bigl( X \| G \bigr)\) compatible with pullbacks, and (2.2) a group homomorphism \(\tau_ X:K_ 0' \bigl( X \| G \bigr) \to\)CH\(\bigl( X \| G \bigr)\) which commutes with proper pushforwards and is onto. The homomorphisms \(ch\) and \(\tau\) satisfy several other properties which I will not list. The concepts mentioned above are all defined in this paper or in a previous paper of the author [Invent. Math. 97, No. 3, 613-670 (1989; Zbl 0694.14001)]. For example,
\[
K_ 0'\bigl( X \| G \bigr)=\text{(Grothendieck group of } G\text{-equivariant coherent sheaves on} X) \otimes \mathbb{Q},
\]
and CH\(\bigl( X \| G)\) is the Chow group with rational coefficients of the quotient stack \(X \| G\). The theorems mentioned above are not proved in this paper. Proofs will appear in a forthcoming paper of the author. Note that if the action of \(G\) is trivial then \(\tau_ X\) is an isomorphism [\textit{W. Fulton}, ``Intersection theory'' (1984; Zbl 0541.14005); corollary 18.3.2]. If the \(G\)-action is not trivial then this need not be true. The kernel is described if \(X\) is smooth, and conjectured in more general cases. The paper concludes with a discussion of a totally split torus acting linearly on a finite dimensional vector space. equivariant Grothendieck groups; equivariant Chow groups; good scheme; Riemann-Roch theorem; Chow group; quotient stack; totally split torus --. --. --. --., ``Equivariant Grothendieck groups and equivariant Chow groups'' in Classification of Irregular Varieties (Trento, Italy, 1990) , ed. E. Ballico, F. Catanese, and C. Ciliberto, Lecture Notes in Math. 1515 , Springer, Berlin, 1992, 112--133. Applications of methods of algebraic \(K\)-theory in algebraic geometry, Parametrization (Chow and Hilbert schemes), Grothendieck groups (category-theoretic aspects) Equivariant Grothendieck groups and equivariant Chow groups | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(A\) be an abelian group. An \(A\)- valued Euler characteristic with compact support for varieties over field of characteristic zero \(k\) is an invariant of an isomorphism class of a variety \(e(X)\in A\) satisfying \(e(X)=e(X-Y)+e(Y)\) for every closed subvariety \(Y\subset X\). The universal Euler characteristic takes values in the Grothendieck group \(K_0(\text{Var}_k)\) - the free abelian group generated by the isomorphism classes of varieties over \(k\) modulo the relations \([X]=[X-Y]+ [Y]\), where \(Y\subset X\) is a closed subvariety. The author proves that \(K_0(\text{Var}_k)\) is the free abelian group on isomorphism classes of smooth projective varieties modulo the relations \([\emptyset]=0\) and \([X]-[Y]=[\text{Bl}_{Y}X]-[E]\), where \(\text{Bl}_{Y}X\) denotes the blow-up of \(X\) along \(Y\) and \(E\) is the exceptional divisor of the blow-up. In the proof the author uses the factorization theorem proven by \textit{J. Włodarczyk} [Invent. Math. 154, No. 2, 223--331 (2003; Zbl 1130.14014)] and \textit{D. Abramovich, K. Karu, K. Matsuki} and \textit{J. Włodarczyk} [J. Am. Math. Soc. 15, No. 3, 531--572 (2002; Zbl 1032.14003)]. As an application, the author proves that there is a unique Euler characteristic with compact support for \(k\)-varieties , which takes values in the Grothendieck group of Chow motives. This result has been already obtained independently by \textit{H. Gillet} and \textit{C. Soulé} [J. Reine Angew. Math. 478, 127--176 (1996; Zbl 0863.19002)] and \textit{F. Guillen} and \textit{V. Navarro Aznar} [Publ. Math., Inst. Hautes Étud. Sci. 95, 1--91 (2002; Zbl 1075.14012)]. The author also generalizes the presentation of \(K_0(\text{Var}_k)\) to the relative setting. Euler characteristic; Grothendieck group; Chow motives F. Bittner. The universal euler characteristic for varieties of characteristic zero. \(Comp. Math.\), 140:1011-1032, 2004. Motivic cohomology; motivic homotopy theory, Rational and birational maps, Applications of methods of algebraic \(K\)-theory in algebraic geometry The universal Euler characteristic for varieties of characteristic zero | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We investigate the action of the \(\mathbb{Q}\)-cohomology of the compact dual \(\widehat{X}\) of a compact Shimura variety \(S(\Gamma)\) on the \(\mathbb{Q}\)-cohomology of \(S(\Gamma)\) under a cup product. We use this to split the cohomology of \(S(\Gamma)\) into a direct sum of (not necessarily irreducible) \(\mathbb{Q}\)-Hodge structures. As an application, we prove that for the class of arithmetic subgroups of the unitary groups \(U(p,q)\) arising from Hermitian forms over CM fields, the Mumford-Tate groups associated to certain holomorphic cohomology classes on \(S(\Gamma)\) are Abelian. As another application, we show that all classes of Hodge type \((1,1)\) in \(H^2\) of unitary four-folds associated to the group \(U(2,2)\) are algebraic. cohomology of Shimura varieties; Schubert cycles; Hodge structures; Mumford-Tate group T. N. Venkataramana, Abelianness of Mumford-Tate groups associated to some unitary groups, Compositio Math. 122 (2000), 223--242. Cohomology of arithmetic groups, Arithmetic aspects of modular and Shimura varieties, Modular and Shimura varieties Abelianness of Mumford-Tate groups associated to some unitary groups | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This note presents a proper closed model structure on the category of simplicial presheaves on a small Grothendieck site intermediate between the local projective model structure [cf. \textit{B. Blander}, \(K\)-theory 24, 283--301 (2001; Zbl 1073.14517)] and the standard one [cf. \textit{J. F. Jardine}, J. Pure Appi. Algebra 47, 35--87 (1987; Zbl 0624.18007)]. That is determined by any set of cofibrations containing the standard set of generating projective cofibrations. The weak equivalences are the local weak equivalences in the usual sense. The presented structure is shown to be cofibrantly generated. model category; cofibrantly generated model category; Grothendieck site Jardine, J.F.: Intermediate model structures for simplicial presheaves. Can. Math. Bull. 49(3), 407--413 (2006) Simplicial sets, simplicial objects (in a category) [See also 55U10], Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects), Abstract and axiomatic homotopy theory in algebraic topology, Homotopy theory and fundamental groups in algebraic geometry Intermediate model structures for simplicial presheaves | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author gives results on \(1\)-dimensional Schubert problems over the moduli space \(\overline{\mathcal M_{0,r}}\) whose real geometry is described by orbits of Schützenberger promotion and a related operation involving Young and jeu de taquin tableau evacuation. Actually, the author works on Schubert problems with respect to flags osculating the rational normal curve and the results of this paper shows that the real points of solution curves are smooth. Also, the author finds a new identity involving first order \(K\)-theoretic Littlewood-Richardson coefficients.
Now let us explain osculating flags. Let \(\mathbb P^k\) be \(k\)-dimensional real projective space. Let \(f:\mathbb P^1 \rightarrow \mathbb P^{n-1}\) be the Veronese embedding \(t\rightarrow (t, t^2, \ldots, t^{n-1})\). At each point \(f(p)\in \mathbb P^{n-1}\), there is an osculating flag \(\mathcal F (p)\) of planes intersecting \(f(\mathbb P^1)\) at \(f(p)\) with the highest possible multiplicity. So the author considers this type Schubert conditions with resoect to such flags and works in the Grassmannian
\[
G(k,\mathbb C^n) = G(k, H^0 (\mathcal O_{\mathbb P^1}(n-1)),
\]
of linear series \(V\) of rank \(k\) and degree \(n-1\) on \(\mathbb P^1\).
Let \(|\underline{\overline{\text{-}|\text{-}|\text{-}}}|\) denote \(k\times (n-k)\) rectangular partition. For a partition \(\lambda \subset |\underline{\overline{\text{-}|\text{-}|\text{-}}}|\), \(\Omega(\lambda, p)\) denotes Schubert variety for \(\lambda\) with respect to \(\mathcal F (p)\). For a collection of distinct points \(p_{\bullet}=(p_1, p_2, \ldots, p_r)\) and partitions \(\lambda_{\bullet} = (\lambda_1, \lambda_2, \ldots, \lambda_r)\), let
\[
S(\lambda_{\bullet}, p_{\bullet}) = \bigcap_{i=1}^r \Omega (\lambda_i, p_i).
\]
Note that codimension of \(\Omega(\lambda, p_i)\) is \(|\lambda|\). We call the quantity
\[
\rho(\lambda_{\bullet})= k(n-k) - \sum |\lambda_i|
\]
expected dimension of \( S(\lambda_{\bullet}, p_{\bullet})\). Schubert calculus; stable curves; Shapiro-Shapiro conjecture; jeu de taquin; growth diagram; promotion Levinson, J.: One-dimensional Schubert problems with respect to osculating flags. Canadian J. Math. (2016). 10.4153/CJM-2015-061-1 Classical problems, Schubert calculus One-dimensional Schubert problems with respect to osculating flags | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(\gamma\) be a non-negative integer. A pointed curve \((C,P)\) is called \textit{\(\gamma\)-hyperelliptic} if the Weierstrass semigroup \(H(P)\) at \(P\) has exactly \(\gamma\) even gaps; here by a \textit{curve} we mean a projective, non-singular, irreducible algebraic curve defined over an algebraically closed field of characteristic zero. Then the very semigroup property of \(H(P)\), see e.g. [\textit{F. Torres}, Semigroup Forum 55, No. 3, 364--379 (1997; Zbl 0931.14017)], implies \(H(P)=2\tilde H\cup\{u_\gamma<\ldots<u_1\}\cup\{2g+i: i\in{\mathbb N}_0\}\), where \(g=g(C)\) is the genus of \(C\), \(\tilde H\) is a numerical semigroup of genus \(\gamma\), and the \(u_i's\) are odd integers with \(u_1<2g\); in addition \(u_\gamma\geq 2g-4\gamma+1\, (*)\).
This paper deals with the question \(D(C,P,\gamma)\): If \((C,P)\) is \(\gamma\)-hyperelliptic, do exist a double covering of curves \(F:C\to \tilde C\) which is ramified at \(P\)? If the answer is positive, the Weierstrass semigroup at \(F(P)\) equals \(\tilde H\) above so that \(g(\tilde C)=\gamma\); in particular, \(g\geq 2\gamma\) by the Riemann-Hurwitz formula. If \(\gamma\leq 3\), \(D(C,P,g,\gamma)\) is indeed true; see [\textit{J. Komeda}, Semigroup Forum 83, No. 3, 479--488 (2011; Zbl 1244.14025)] and the references therein. From now we let \(\gamma\geq 4\).
If \(g(C)\geq 6\gamma+4\), \(D(C,P,\gamma)\) is true [\textit{F. Torres}, Manuscr. Math. 83, No. 1, 39--58 (1994; Zbl 0838.14025)]. To see this we consider the linear system \(D_{\gamma+1}:=|(6\gamma+2)P|\) which has dimension \(2\gamma+1\) by \((*)\) above (indeed, this follows provided that \(g(C)\geq 5\gamma+1\)). Then the degree \(t\) of the morphism \(F_1: C\to {\mathbb P}^{2\gamma+1}\) associated to \(D_{\gamma+1}\) is at most \(2\). If \(t=2\), the claimed answer follows. On the contrary, Castelnuovo's genus bound gives \(g(C)\leq \pi_0(6\gamma+2,2\gamma+1)=6\gamma+3\), a contradiction.
The present paper proves that \(D(C,P,\gamma)\) is even true whenever \(g(C)= 6\gamma+1, 6\gamma\). As a matter of fact, \(D(C,P,\gamma\) is also true for \(g(C)=6\gamma+3, 6\gamma+2\) which follow from the techniques used by the authors here.
Let \(g(C)=6\gamma+1\) and notation as above. We claim that \(t=2\). Let \(C_0:=F_1(C)\) and assume \(t=1\). Then \(g(C)=g(C_0)\leq g_a(C_0)\leq c_0(6\gamma+2, 2\gamma+1)=6\gamma+3\), where \(g_a\) is the arithmetic genus of \(C_0\). If \(g(C)=g_a(C_0)\), \(C\) is isomorphic to \(C_0\) and hence \((6\gamma+2)P\sim P+D\) with \(D\) a divisor on \(C\) such that \(P\not\in \text{supp}(D)\). Hence \(6\gamma+1\in H(P)\), a contradiction according to \((*)\) above. Now the number \(\pi_1(6\gamma+2,2\gamma+1)\) in Theorem 3.15 [\textit{J. Harris}, Curves in projective space. Montreal, Quebec, Canada: Les Presses de l'Universite de Montreal (1982; Zbl 0511.14014)] equals \(6\gamma+1\); hence \(C_0\subseteq S\subseteq {\mathbb P}^{2\gamma+1}\), being \(S\) a surface of degree \(2\gamma\) [loc. cit.]. Then by considering the minimal resolution of \(S\) and the adjunction formula, \(g_a(C_0)\) can be computed. Finally the proof that \(t=1\) is a contradiction proceeds via a carefully study of the condition \(1\leq g_a(C_0)-g(C_0)\leq 2\).
The case \(g(C)=6\gamma\) is treated in a similar way; however, here the linear system \(|(6\gamma-2)P|\), which is of dimension \(2\gamma-1\), is used. Weierstrass semigroup; double cover of a curve; rational ruled surface Plane curves of degree 4 Riemann surfaces; Weierstrass points; gap sequences, Coverings of curves, fundamental group, Rational and ruled surfaces On \(\gamma \)-hyperelliptic Weierstrass semigroups of genus \(6\gamma +1\) and \(6\gamma \) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(H_n=\text{Hilb}^n(\mathbb{C}^2)\) be the Hilbert scheme which parametrizes the subschemes \(S\) of length \(n\) of \(\mathbb{C}^2\). To each such subscheme \(S\) corresponds a unordered \(n\)-tuple with possible repetitions \(\sigma(S)=[[P_1,...,P_n]]\) of points of \(\mathbb{C}^2\). There exists an algebraic variety \(X_n\) (called the isospectral Hilbert scheme) which is finite over \(H_n\) and which consists of all ordered \(n\)-tuples \((P_1,...,P_n)\in(\mathbb{C}^2)^n\) whose underlying unordered \(n\)-tuple is \(\sigma(S)\). The main aim of the paper under review is to study the geometry of \(X_n\), which is more complicated than the geometry of \(H_n\). For instance, a classical result of J. Fogarty asserts that \(H_n\) is irreducible and non-singular. The main result of the paper under review asserts that \(X_n\) is normal and Gorenstein (in particular, Cohen-Macaulay). Earlier work of the author indicated that there is a far-reaching correspondence between the geometry and sheaf cohomology of \(H_n\) and \(X_n\) on the one hand, and the theory of Macdonald polynomials on the other hand. The link between Macdonald polynomials and Hilbert schemes comes from some recent work [see \textit{A. M. Garsia} and \textit{M. Haiman}, Proc. Nat. Acad. Sci. USA 90, No. 8, 3607-3610 (1993; Zbl 0831.05062)].
The main result proved in this paper is expected to be an important step toward the proof of the so-called \(n!\)-conjecture and Macdonald positivity conjecture. The main theorem is based on a technical result (theorem 4.1) which asserts that the coordinate ring of a certain type of subspace arrangement is a free module over the polynomial ring generated by some of the coordinates. Macdonald polynomials; isospectral Hilbert schemes; \(n!\)-conjecture; Macdonald positivity conjecture; subspace arrangement K.B. Alkalaev and V.A. Belavin, \textit{Conformal blocks of}\( {\mathcal{W}}_n \)\textit{Minimal Models and AGT correspondence}, arXiv:1404.7094 [INSPIRE]. Parametrization (Chow and Hilbert schemes), Symmetric functions and generalizations, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Hilbert schemes, polygraphs and the Macdonald positivity conjecture | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper, we study the \(T_{\mathrm w}\)-equivariant cohomology of the weighted Grassmannians \(\mathrm{wGr}(d,n)\) introduced by \textit{A. Corti} and \textit{M. Reid} [in: Algebraic geometry. A volume in memory of Paolo Francia. Berlin: de Gruyter. 141--163 (2002; Zbl 1060.14071)], where \(T_{\mathrm w}\) is the \(n\)-dimensional torus that naturally acts on \(\mathrm{wGr}(d,n)\). We introduce the equivariant weighted Schubert classes and, after we show that they form a basis of the equivariant cohomology, we give an explicit formula for the structure constants with respect to this Schubert basis. We also find a linearly independent subset \(\{\mathrm wu_1,\dots,\mathrm wu_{n-1}\}\) of \(\mathrm{Lie}(T_w)^\ast\) such that those structure constants are polynomials in \(\mathrm wu_i\)'s with nonnegative coefficients, up to a permutation on the weights. weighted Grassmannians; equivariant weighted Schubert classes; equivariant cohomology; structure constants Abe, H; Matsumura, T, Equivariant cohomology of weighted Grassmannians and weighted Schubert classes, Int. Math. Res. Not. IMRN, 9, 2499-2524, (2015) Grassmannians, Schubert varieties, flag manifolds, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Equivariant cohomology of weighted Grassmannians and weighted Schubert classes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper, we study the structure of the quantum cohomology ring of a projective hypersurface with nonpositive first Chern class. We prove a theorem which suggests that the mirror transformation of the quantum cohomology of a projective Calabi-Yau hypersurface has a close relation with the ring of symmetric functions, or with Schur polynomials. With this result in mind, we propose a generalized mirror transformation on the quantum cohomology of a hypersurface with negative first Chern class and construct an explicit prediction formula for three-point Gromov-Witten invariants up to cubic rational curves. We also construct a projective space resolution of the moduli space of polynomial maps, which is in good correspondence with the terms that appear in the generalized mirror transformation. quantum cohomology ring; hypersurface; mirror transformation; Calabi-Yau hypersurface; Schur polynomials; Gromov-Witten invariants Jinzenji, M., On the quantum cohomology rings of general type projective hypersurfaces and generalized mirror transformation, Int. J. Mod. Phys., A15, 1557-1596, (2000) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Hypersurfaces and algebraic geometry, Calabi-Yau manifolds (algebro-geometric aspects) On the quantum cohomology rings of general type projective hypersurfaces and generalized mirror transformation | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper is a survey of Čech and sheaf cohomologies, presenting expansions for Grothendieck toposes with examples of applications in other cohomology theories and other areas of mathematics. The last section (\S 5) discusses establishing a cohomology theory for elementary toposes, citing the works in [\textit{J. Wildeshaus}, J. Pure Appl. Algebra 150, No. 2, 207--213 (2000; Zbl 0966.18008); \textit{A. Joyal} and \textit{M. Tierney}, An extension of the Galois theory of Grothendieck. Providence, RI: American Mathematical Society (AMS) (1984; Zbl 0541.18002); \textit{S. Posur}, Constructive category theory and applications to equivariant sheaves. Siegen: Universität Siegen (PhD Thesis) (2017), \url{https://d-nb.info/113738087X/34}; \textit{M. Shulman}, ``Linear logic for constructive mathematics'', Preprint, \url{arXiv:1805.07518}]. sheaf and Čech cohomologies; Grothendieck topos; topos cohomology Research exposition (monographs, survey articles) pertaining to category theory, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Grothendieck topologies and Grothendieck topoi, Sheaves in algebraic geometry, Sheaf cohomology in algebraic topology, Topoi On sheaf cohomology and natural expansions | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Certifying function nonnegativity is a ubiquitous problem in computational mathematics, with especially notable applications in optimization. We study the question of certifying nonnegativity of signomials based on the recently proposed approach of Sums-of-AM/GM-Exponentials (SAGE) decomposition due to the second author and Shah. The existence of a SAGE decomposition is a sufficient condition for nonnegativity of a signomial, and it can be verified by solving a tractable convex relative entropy program. We present new structural properties of SAGE certificates such as a characterization of the extreme rays of the cones associated to these decompositions as well as an appealing form of sparsity preservation. These lead to a number of important consequences such as conditions under which signomial nonnegativity is equivalent to the existence of a SAGE decomposition; our results represent the broadest-known class of nonconvex signomial optimization problems that can be solved efficiently via convex relaxation. The analysis in this paper proceeds by leveraging the interaction between the convex duality underlying SAGE certificates and the face structure of Newton polytopes. After proving our main signomial results, we direct our machinery toward the topic of globally nonnegative polynomials. This leads to (among other things) efficient methods for certifying polynomial nonnegativity, with complexity independent of the degree of a polynomial. arithmetic-geometric-mean inequality; certifying nonnegativity; fewnomials; SAGE; signomials; sparse polynomials Nonconvex programming, global optimization, Nonlinear programming, Real-analytic and semi-analytic sets Newton polytopes and relative entropy optimization | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The double ramification locus in the moduli space \(\mathcal M_{g,n}\) parameterizes curves of genus \(g\) with \(n\) marked points such that the marked points form the zeros and poles with prescribed order for some rational function on the underlying curve. There have been great interests and many efforts recently, dating back to a question of Eliashberg, towards understanding the extension of the double ramification locus to (part of) the boundary of the Deligne-Mumford moduli space \(\overline{\mathcal M}_{g,n}\) as well as computing the related cycle class, see [\textit{R. Cavalieri} et al., J. Pure Appl. Algebra 216, No. 4, 950--981 (2012; Zbl 1273.14053); \textit{G. Farkas} and \textit{R. Pandharipande}, J. Inst. Math. Jussieu 17, No. 3, 615--672 (2018; Zbl 1455.14056); \textit{S. Grushevsky} and \textit{D. Zakharov}, Proc. Am. Math. Soc. 142, No. 12, 4053--4064 (2014; Zbl 1327.14132); \textit{S. Grushevsky} and \textit{D. Zakharov}, Duke Math. J. 163, No. 5, 953--982 (2014; Zbl 1302.14039); \textit{F. Janda} et al., Publ. Math., Inst. Hautes Étud. Sci. 125, 221--266 (2017; Zbl 1370.14029); \textit{F. Müller}, Math. Nachr. 286, No. 11--12, 1255--1266 (2013; Zbl 1285.14032)].
In this paper, the author studies the compactified universal Jacobian \(\overline{\mathcal J}_{g,n}\) over \(\overline{\mathcal M}_{g,n}\) and computes the class of its zero section in the rational Chow ring of \(\overline{\mathcal J}_{g,n}\) after extending some variants of the Abel-Jacobi map to the locus of tree-like curves. As a consequence, this calculation gives an expression of the double ramification cycle in a more general setting, studied first by \textit{R. Hain} [Advanced Lectures in Mathematics (ALM) 24, 527--578 (2015; Zbl 1322.14049)] and also see [\textit{J. L. Kass} and \textit{N. Pagani}, Adv. Math. 321, 221--268 (2017; Zbl 1387.14085)] for a closely related work. moduli space of curves; double ramification cycle; universal Jacobian Families, moduli of curves (algebraic), Jacobians, Prym varieties Compactified universal Jacobian and the double ramification cycle | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a complex reductive affine algebraic group and \(\Gamma\) a finitely generated group. The set of group homomorphisms \(\mathrm{Hom}(\Gamma, G)\) is an affine algebraic set on which \(G\) acts rationally by conjugation. The (affine) geometric invariant theoretic (GIT) quotient \[\mathfrak{X}(\Gamma,G):=\mathrm{Hom}(\Gamma,G)/\!\!/G\] is called the \textit{\(G\)-character variety of \(\Gamma\)}.
In [\textit{S. Lawton}, J. Algebra 313, No. 2, 782--801 (2007; Zbl 1119.13004)] the reviewer explicitly describes \(\mathfrak{X}(F_2,\mathrm{SL}(3,\mathbb{C}))\) where \(F_2\) is a free group of rank 2. In particular, the reviewer shows that \(\mathfrak{X}(F_2,\mathrm{SL}(3,\mathbb{C}))\) is a normal hypersurface in \(\mathbb{C}^9\) that branch double covers \(\mathbb{C}^8\), and the branching locus properly contains the singular locus (shown to be exactly the reducible homomorphisms). This theorem is proved using non-commutative algebra, in particular, the theory of polynomial identity algebras (PI Theory). Later, in [\textit{C. Florentino} and \textit{S. Lawton}, Math. Ann. 345, No. 2, 453--489 (2009; Zbl 1200.14093)] the reviewer geometrically describes the branching locus as exactly the symmetric homomorphisms (Proposition 6.11, ibid) and uses this as a step in proving that \(\mathfrak{X}(F_2,\mathrm{SL}(3,\mathbb{C}))\) strong deformation retracts onto a topological 8-sphere despite the fact that \(\mathfrak{X}(F_2,\mathrm{SL}(3,\mathbb{C}))\) is not a (topological) manifold (in the analytic topology).
There has been much work since then on \(\mathrm{SL}(3,\mathbb{C})\)-character varieties. In particular, the reviewer has described global coordinates [\textit{S. Lawton}, J. Algebra 320, No. 10, 3773--3810 (2008; Zbl 1157.14030)], local coordinates [\textit{S. Lawton}, J. Algebra 324, No. 6, 1383--1391 (2010; Zbl 1209.14036)], Poisson structures [\textit{S. Lawton}, Trans. Am. Math. Soc. 361, No. 5, 2397--2429 (2009; Zbl 1171.53052)], and \(E\)-polynomials [\textit{S. Lawton} and \textit{V. Muñoz}, Pac. J. Math. 282, No. 1, 173--202 (2016; Zbl 1335.14003)] on \(\mathfrak{X}(F_r,\mathrm{SL}(3,\mathbb{C}))\) for arbitrary rank free groups \(F_r\). There are many other examples by many other authors too.
For \(\mathrm{SL}(2,\mathbb{C})\)-character varieties of \(\Gamma\) a complete algebraic picture is given in [\textit{C. Ashley} et al., Geom. Dedicata 192, 1--19 (2018; Zbl 1390.14187)]. In other words, an explicit presentation of the (reduced) coordinate ring of \(\mathfrak{X}(\Gamma, \mathrm{SL}(2,\mathbb{C}))\) is given for any finitely presented group \(\Gamma\). This is only possible since the free group case is solved for all ranks for \(\mathrm{SL}(2,\mathbb{C})\). The only case where generators and relations (of the coordinate ring of the character variety) are known for free groups and \(\mathrm{SL}(3,\mathbb{C})\) is the reviewer's theorem for \(F_2\) referenced above. So one naturally wonders if the 2-generator group problem can likewise be solved for \(\mathfrak{X}(\Gamma, \mathrm{SL}(3,\mathbb{C}))\); that is, can one describe the coordinate ring of \(\mathfrak{X}(\Gamma, \mathrm{SL}(3,\mathbb{C}))\) for any \(\Gamma\) finitely presentable with 2-generators.
Using the results and methods of the reviewer mentioned above, in particular the explicit description of \(\mathfrak{X}(F_2, \mathrm{SL}(3,\mathbb{C}))\) and PI theory, the author of the paper under review presents a method for determining the irreducible locus of \(\mathfrak{X}(\Gamma, \mathrm{SL}(3,\mathbb{C}))\) for any 2-generated finitely presented group \(\Gamma\) and demonstrates this method with some examples (various link complements). character variety; irreducible representation; two-generator group; double twist link; symmetric slice; asymmetric slice Character varieties, Invariants of 3-manifolds (including skein modules, character varieties) \(\mathrm{SL}(3,\mathbb{C})\)-character varieties of two-generator groups | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Based on the basis theorem of Bruhat--Chevalley and the formula for multiplying Schubert classes obtained by \textit{H. Duan} [Invent. Math. 159, No. 2, 407--436 (2005; Zbl 1077.14067)] and programmed by \textit{H. Duan} and \textit{X. Zhao} [Int. J. Algebra Comput. 16, 1197--1210 (2006; Zbl 1107.14047)], the authors introduce a new method computing the Chow rings of flag varieties (resp. the integral cohomology of homogeneous spaces). In this paper they demonstrate this technique to descrtibe the Chow rings of the seven generalized Grassmannians (resp. the seven homogeneous spaces of rank one) for the exceptional Lie groups \(F_4\), \(E_6\), \(E_7\) and \(E_8\). flag manifolds; Schubert varieties; Chow ring H. Duan, X. Zhao, Appendix to ''The Chow rings of generalized Grassmannians'', arXiv:math.AG/0510085 . (Equivariant) Chow groups and rings; motives, Grassmannians, Schubert varieties, flag manifolds The Chow rings of generalized Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(X=G/P\) be a Schubert variety, defined as the quotient of the Lie group \(G\) by a parabolic subgroup \(P\). \(X\) is minuscule when \(P\) is associated with a weight \(w\) such that \(<a^*, w>\leq 1\), for any positive root \(a\) with respect to a Borel subgroup. A first example of minuscule Schubert variety is the set of lines meeting a fixed line in \(\mathbb P^3\).
The author studies the set \(\Hom_\alpha(\mathbb P^1,X)\) of maps whose image lies in a fixed class \(\alpha\in A_1(X)\). In other words, he studies the variety of rational curves in a given class \(\alpha\). If \(U\) is the dense orbit on \(X\) under the stabilizer of \(X\), then \(U\) is homogeneous, and the variety of maps \(\mathbb P^1\to U\) can be studied using general results on homogeneous varieties. \(X-U\) has codimension at least \(2\) and the author proves that, after a deformation, one may assume that the image of a map \(f\in \Hom_\alpha(\mathbb P^1,X)\) lies in \(U\). Using this fact, the author proves that the irreducible components of \(\Hom_\alpha(\mathbb P^1,X)\) are parametrized by the classes \(\beta\in \text{Pic}(U)^*\) such that \(\Hom_\beta(\mathbb P^1,U)\) is non empty and \(\beta\) maps to \(\alpha\) under the natural morphism \( \text{Pic}(U)^*\to A_1(X)\). Schubert varieties Perrin N., Rational curves on minuscule Schubert varieties, J. Algebra, 2005, 294(2), 431--462 Grassmannians, Schubert varieties, flag manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Rational curves on minuscule Schubert varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials A natural partition of flag superspaces into Schubert supercells indexed by Weyl supergroups is described. The closures and the singularities of these Schubert cells are closely studied. To resolve these singularities the authors generalize the classical Bott-Samelson construction. flag superspaces; Schubert supercells Manin, Y.I., Voronov, A.A.: Supercellular partitions of flag superspaces. Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Nov. Dostizh. VINITI Moscow \textbf{32}, 27-70 (1988) Supermanifolds and graded manifolds, Complex supergeometry, Supervarieties, Grassmannians, Schubert varieties, flag manifolds Supercell partitions of flag superspaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Refering to Hilbert's 12th problem the author presents some elementary aspects of what is today called arithmetical geometry, a theory which shows the close relationship of number theory, algebraic geometry, and the theory of functions. In a very readable exposé, he presents the following highlights for ``the educated layman'': analogies between \(\mathbb{Z}\) and \(\mathbb{C}[X]\); the theorems of Riemann-Roch and Minkowski (according to Weil); theory of height; Grothendieck schemes and Arakelov geometry. Arakelov geometry; arithmetical geometry; theorems of Riemann-Roch; Minkowski; Grothendieck schemes Hindry, M.: Géométrie arithmétique. (1991) Development of contemporary mathematics, History of mathematics in the 20th century, History of algebraic geometry Arithmetical geometry | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors introduce the canonical vector field as an infinitesimal device to study the bi-Lipschitz equisingularity of families of essentially isolated singularities. Let \(\tilde {F}:\mathbb C \times \mathbb C^q \to \mathbb C \times\mathrm{Hom}(\mathbb C^m,\mathbb C^n)\), \(\tilde{F}(y,x)=(y, \tilde {f}(y,x))\), \(\tilde {f}(y, x)=f(x)+y \theta(x)\), be a one-parameter family of map-germs defining a
family of essentially isolated determinantal singularities in \(\mathbb C^q\). The canonical vector field of \(\tilde {F}\) is defined as
\(\frac{\partial}{\partial y}+\Sigma^q_{i=1}\frac{\partial {\tilde f}}{\partial y} \frac{\partial }{\partial x_i}\). The authors give necessary and sufficient conditions for the canonical vector field of \(\tilde F\) to be Lipschitz in terms of the integral closure of the double of the ideal generated by the components of \(\tilde f\). Moreover, they prove that this always holds when \(\theta \) is a constant deformation. Several examples illustrate the results. bi-Lipschitz equisingularity; double of an ideal; canonical vector field Equisingularity (topological and analytic), Singularities of surfaces or higher-dimensional varieties, Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects) The bi-Lipschitz equisingularity of essentially isolated determinantal singularities | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In J. Differ. Geom. 20, 389--431 (1984; Zbl 0565.17007), \textit{S. Kumar} described the Schubert classes which are the dual to the closures of the Bruhat cells in the flag varieties of the Kac-Moody groups associated to the infinite dimensional Kac-Moody algebras. These classes are indexed by affine Weyl groups and can be chosen as elements of integral cohomologies of the homogeneous space \(\widehat{L}_{\text{pol}} G_{\mathbb C}/ \widehat{B}\) for any compact simply connected semi-simple Lie group \(G\). Later, \textit{S. Kumar} and \textit{B. Kostant} [Adv. Math. 62, 187--237 (1986; Zbl 0641.17008)] gave explicit cup product formulas of these classes in the cohomology algebras by using the relation between the invariant-theoretic relative Lie algebra cohomology theory (using the representation module of the nilpotent part) with the purely nil-Hecke rings. These explicit product formulas involve some BGG-type operators \(A^i\) and reflections. Using some homotopy equivalences, we determine cohomology ring structures of \(LG/T\) where \(LG\) is the smooth loop space on \(G\). Here, as an example we calculate the products and explicit ring structure of \(LSU_2/T\) using these ideas.
Note that these results grew out of a chapter of the author's thesis [On the complex cobordism of flag varieties associated to loop groups, PhD thesis, University of Glasgow (1998)]. divided-power algebras; Schubert classes; Kac-Moody groups; Kac-Moody algebras; BGG-type operators; homotopy equivalences; cohomology ring C. Özel, ``On the cohomology ring of the infinite flag manifold LG/T,'' Turkish Journal of Mathematics, vol. 22, no. 4, pp. 415-448, 1998. Cohomology of Lie (super)algebras, Homology and cohomology of homogeneous spaces of Lie groups, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Grassmannians, Schubert varieties, flag manifolds, Loop groups and related constructions, group-theoretic treatment On the cohomology ring of the infinite flag manifold \(LG/T\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials algebraic geometry; systems theory; McMillan degree; Kronecker indices; transfer functions; system invariants; Grassmann manifold; Grothendieck invariants R. Hermann and C. Martin, ''Application of algebraic geometry to systems theory: Part II: The McMillan degree and Kronecker indices as topological and holomorphic invariants,''SIAM J. Contr. Optimiz.,16, 743--755 (1978). Algebraic methods, Vector and tensor algebra, theory of invariants, Multivariable systems, multidimensional control systems, Grassmannians, Schubert varieties, flag manifolds Applications of algebraic geometry to systems theory: the McMillan degree and Kronecker indices of transfer functions as topological and holomorphic system invariants | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \({\mathcal T}'\) be the thick triangulated monoidal subcategory of compact objects in the homotopy category \({\mathcal T}=\text{Ho}({\mathcal C})\) of a simplicial model symmetric monoidal category \({\mathcal C} \). Assume that all \textit{Hom} groups in \({\mathcal T}\) are vector spaces over \({\mathbb Q}\). The author proves that the wedge and symmetric powers induce two special \(\lambda\)-structures in the Grothendieck ring \(K_0({\mathcal T}')\), which are opposite to each other. From this it follows that the motivic zeta function is multiplicative with respect to distinguished triangles in \({\mathcal T}\). As an application, the motivic zeta functions of all varieties whose motives lie in a thick triangulated tensor subcategory of Voevodsky's category \({\mathcal DM} \) with coefficients in \({\mathbb Q}\) generated by motives of quasi-projective curves over a field are rational. Together with a result due to P. O'Sullivan, this also gives an example of a variety whose motive is not finite-dimensional in the sense of Kimura and whose motivic zeta function is rational. motivic zeta function; finite-dimensional motives; monoidal category; triangulated category; homotopy category; Grothendieck group Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), (Equivariant) Chow groups and rings; motives, Monoidal categories (= multiplicative categories) [See also 19D23], Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects), Grothendieck groups (category-theoretic aspects), Derived categories, triangulated categories Zeta functions in triangulated categories | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We show that the Braden-MacPherson algorithm computes the stalks of parity sheaves. As a consequence we deduce that the Braden-MacPherson algorithm may be used to calculate the characters of tilting modules for algebraic groups and show that the \(p\)-smooth locus of a (Kac-Moody) Schubert variety coincides with the rationally smooth locus, if the underlying Bruhat graph satisfies a GKM-condition. modular representation theory; equivariant cohomology; moment graphs; constructible sheaves; tilting modules; Schubert varieties; \(p\)-smooth locus Fiebig, P. & Williamson, G., Parity sheaves, moment graphs and the p-smooth locus of Schubert varieties. \textit{Ann. Inst. Fourier (Grenoble)}, 64 (2014), 489-536. Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Classical groups (algebro-geometric aspects), Modular representations and characters, Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.), Intersection homology and cohomology in algebraic topology, Equivariant homology and cohomology in algebraic topology, Grassmannians, Schubert varieties, flag manifolds Parity sheaves, moment graphs and the \(p\)-smooth locus of Schubert varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials A loop group LG is the group of maps of the circle \(S^ 1\) into some topological group G (with group law coming from pointwise multiplication in G). The book under review is devoted to the study of structure and representations of LG in the case when G is either a compact or a complex Lie group.
Groups similar to LG enter mathematics in several points. Their Lie algebras form a class of Kac-Moody Lie algebras, the so-called affine algebras. In the last few years these algebras were extensively studied, and their deep relations with various branches of mathematics (combinatorics, finite groups) and physics (quantum field theory, especially string models) were displayed [see, for example, Vertex operators in mathematics and physics, \textit{J. Lepowsky}, \textit{S. Mandelstam} and \textit{I. M. Singer} (eds.) (Publ., Math. Sci. Res. Inst. 3) (1985; Zbl 0549.00013)].
The present book differs from the other sources in that it mostly adopts analytic and geometric, rather that algebraic and combinatoric, approaches.
The first part of the book studies the group LG itself. After the introduction (Chapter I) and a survey of the results about finite- dimensional representations of Lie groups (Chapter 2) the authors give in Chapter 3 general facts about infinite-dimensional Lie groups and consider LG from this viewpoint. In Chapter 4 they study one of the most important properites of loop groups, namely the existence of natural central extensions by the circle group T; these extensions are sometimes more important than loop groups themselves. In this chapter the extensions are constructed and studied by differential geometric methods. Chapter 5 contains a brief survey of Kac-Moody Lie algebras as Lie algebras of loop groups.
Chapter 6 is one of the main chapters in the first part of the book. In this chapter LG is represented as a group of operators in an appropriate Hilbert space, namely in the space \(H=L^ 2(S^ 1,V)\) of \(L^ 2\)- functions on the circle with values in some finite-dimensional representation of G; LG acts in H pointwise. The idea (coming from quantum field theory) is to consider the polarization of H, i.e. the decomposition \(H=H_+\oplus H_-\) where \(H_+\) (resp. \(H_-)\) is the space of functions with vanishing negative (resp. positive) Fourier coefficients. Properties of operators from LG with respect to this decomposition form a very interesting and important part of the theory.
Another very important concept in the first part of the book is the notion of the Grassmannian Gr(H) of a polarized Hilbert space H, introduced in Chapter 7. The authors study the canonical (determinant) line bundle on Gr(H), Schubert cell decomposition of Gr(H), etc.
Chapter 8 introduces the fundamental homogeneous space X of LG that is defined as LG/G where \(G\subset LG\) is considered as the subgroup of constant loops. Two main properties of X are as follows. First, X can be considered as a (infinite-dimensional) complex manifold via the identification \(X=LG_{{\mathbb{C}}}/L_+G_{{\mathbb{C}}}\) where \(G_{{\mathbb{C}}}\) is the complexification of G and \(L_+G_{{\mathbb{C}}}\) consists of boundary values of analytic mappings of the disk \(| z| <1\) into \(G_{{\mathbb{C}}}\). Second, X can be canonically imbedded into Gr(H), thus inheriting from Gr(H) the stratification by Schubert cells and other nice features.
The remaining Chapters 9-14 deal with the representation theory of loop groups. We will not describe the content of these chapters and restrict ourselves to giving their titles: Ch. 9: Representation theory. Ch. 10: The fundamental representation. Ch. 11: The Borel-Weil theory. Ch. 12: The spin representation. Ch. 13: ''Blips'' or ''vertex'' operators. Ch. 14: The Kac character formula and the Bernstein-Gel'fand-Gel'fand resolution. determinant line bundle; vertex operators; loop group; Kac-Moody Lie algebras; affine algebras; infinite-dimensional Lie groups; central extensions; circle group; Grassmannian; polarized Hilbert space; Schubert cell decomposition; homogeneous space; complex manifold; Borel-Weil theory; spin representation; Kac character formula; Bernstein-Gel'fand- Gel'fand resolution A. Pressley and G. Segal, \textit{Loop Groups}, Clarendon Press, Oxford (1986). Infinite-dimensional Lie groups and their Lie algebras: general properties, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Research exposition (monographs, survey articles) pertaining to topological groups, Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.), Grassmannians, Schubert varieties, flag manifolds, Harmonic analysis on homogeneous spaces, Homogeneous complex manifolds Loop groups | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The main results are two generalizations of Pieri's formula to cycles in \(SO_{2n+1}/U_ n\) and \(Sp_ n/U_ n\). Both formulae give the intersection class of a general Schubert cycle with a special Schubert cycle as a linear combination of classes of Schubert cycles. Unlike the classical formula (i.e., for cycles in a grassmannian), the present formulae involve multiplicities which are computed in the paper. Pieri formula; classes of Schubert cycles Howard Hiller and Brian Boe, Pieri formula for \?\?_{2\?+1}/\?_{\?} and \?\?_{\?}/\?_{\?}, Adv. in Math. 62 (1986), no. 1, 49 -- 67. Classical groups (algebro-geometric aspects), Enumerative problems (combinatorial problems) in algebraic geometry, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Pieri formula for \(SO_{2n+1}/U_ n\) and \(Sp_ n/U_ n\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(k\) be a field of characteristic 0, and let \(G_{m,n}\) be the \(k\)-algebra of \(m\) generic \(n\times n\) matrices \(X_1,\dots,X_m\). This algebra is one of the most important objects in PI theory; it is free of rank \(m\) in the variety of algebras generated by \(M_n(k)\). It is well known that \(G_{m,n}\) is a domain. The general linear group acts naturally on \(G_{m,n}\) by first defining its action on the span of the \(X_i\).
The main result of the paper under review is Theorem 1.2; it states that whenever \(2\leq m\leq n^2-2\) the algebra of \(\text{SL}_m\)-invariants \((G_{m,n})^{\text{SL}_m}\) is of PI degree \(n\).
There are several rings of importance related to \(G_{m,n}\). Some of these are the trace ring \(T_{m,n}\) and their centers \(Z(G_{m,n})\), \(Z(T_{m,n})\). All these rings possess natural \(\mathbb{Z}\)-gradings. As consequence of the above theorem the authors obtain that if \(R\) is any of the above four rings, \(\limsup_{d\to\infty}(\dim R^{\text{SL}_m}[d])/d^{(m-1)m^2-m^2+1})\) is a nonzero number (finite). Here \(R[d]\) stands for the homogeneous component of \(R\) of degree \(d\).
Furthermore, the authors show that if \(2\leq m\leq n^2-2\), and \(n\geq 3\) then for the universal division algebra \(\text{UD}(m,n)\) one has that \(\text{UD}(m,n)^{\text{GL}_m}\) is a division algebra of degree \(n\). generic matrices; universal division algebras; central polynomials; PI-degrees; group actions; geometric actions; invariants; concomitants; Gelfand-Kirillov dimension Reichstein Z., Adv. Appl. Math. 37 pp 481-- (2006) Trace rings and invariant theory (associative rings and algebras), Geometric invariant theory, Group actions on varieties or schemes (quotients), Finite-dimensional division rings, Actions of groups and semigroups; invariant theory (associative rings and algebras), Vector and tensor algebra, theory of invariants Group actions and invariants in algebras of generic matrices. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Double Poisson algebras were introduced by M. Van den Bergh as a generalization of classical Poisson geometry to the setting of noncommutative geometry. A double Poisson algebra \(A\) is an associative unital algebra equipped with a linear map \(\{\{-,-\}\}:A\otimes A\rightarrow A\otimes A\) that is a derivation in its second argument for the outer \(A\)-bimodule structure on \(A\otimes A\), together with an explicitly defined double Jacobi identity. This map is called a double Poisson bracket.
A double Poisson bracket yields, for each \(n\), a classical Poisson bracket on the coordinate ring \(\mathbb C[\text{rep}_n(A)]\) of the variety of \(n\)-dimensional representations of \(A\). This bracket restricts to a Poisson bracket on \(\mathbb C[\text{rep}_n(A)]^{\text{GL}}\).
This article study the the double Poisson brackets on a direct sum \(S=M_{d_1}(\mathbb{C})\oplus\cdots\oplus M_{d_k}(\mathbb{C})\) of matrix algebras over \(\mathbb C\). On such algebras, all double Poisson brackets are determined by double Poisson tensors, that is, elements of degree \(2\) in \(\mathbb{D}S=T_S\mathbb D\text{er}(S)\) where \(D\text{er}(S)=\text{Der}(S,S\otimes S)\) is the module of double derivations.
The first result of the article is the explicit description of \(\mathbb D\text{er}(S)\) and \(\mathbb D\text{er}_T(S)\), the last one the bimodule of \(T\)-linear double derivations with \(T\subset S\) a subalgebra:
\[
\mathbb D\text{er}(S)\cong\bigoplus_{i=1}^k M_{d_1}(\mathbb C)^{\oplus d_i^2-1}\oplus\bigoplus_{i\neq j}M_{d_i\times d_j}(\mathbb C)^{\oplus d_id_j}.
\]
Also, if \(T=M_{e_1}(\mathbb C)\oplus\cdots\oplus M_{e_l}\) a finite dimensional semi-simple subalgebra of \(S\) with Bratelli diagram with respect to \(S\) given by \((a_{ij})^{(k,l)}_{(i,j)=(1,1)},\) then
\[
\mathbb D\text{er}_T(S)\cong\bigoplus_{i=1}^k M_{d_1}(\mathbb C)^{\oplus r_i}\oplus\bigoplus_{i\neq j}M_{d_i\times d_j}(\mathbb C)^{\oplus r_{ij}}
\]
as bimodules, with \(r_i=\sum_{u=1}^l a_{iu}^2-1\) and \(r_{ij}=\sum_{u=1}^la_{iu}a_{ju}.\)
Using this results, the author is able to formulate an explicit description of the graded Lie algebra \(\mathbb D S/[\mathbb DS,\mathbb DS][1],\) where the bracket on \(\mathbb D S/[\mathbb DS,\mathbb DS][1]\) is the bracket associated to the double Schouten-Nijenhuis bracket on \(\mathbb DS\). This description it explicitly formulated in terms of a assigned duo-tone quiver.
This result can be used to determine all monomials of degree \(2\) in \(\mathbb DS\) that yield nontrivial double Poisson structures on \(S\), and is in turn used to compute the first double Poisson-Lichnerowicz cohomology groups for \(S\).
Double Poisson structures on these algebras yield interesting noncommutative geometry as they can be extended to double Poisson structures on the free product of such algebras. For such products \(S\ast T\), the quotient variety \(\text{iss}_n(S\ast T)\) is no longer trivial and double Poisson structures can yield nontrivial Poisson structures on this variety. In particular, an explicit description of the Poisson brackets on the quotient variety \(\text{iss}_n(\mathbb C^{\oplus p}\ast\mathbb C^{\oplus q})\) is given.
The article is reasonable detailed, and explains the theory good. It gives nice examples on the applications of quivers. double derivations; double Schouten bracket; double Poisson bracket; quivers; double Poisson algebras G. Van de Weyer, Double Poisson structures on finite dimensional semi-simple algebras, math.AG/0603533. Noncommutative algebraic geometry Double Poisson structures on finite dimensional semi-simple algebras | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This work is about degenerate flag varieties of type \(A_n\) and \(C_n\) over \(\mathbb{Z}\). Given an irreducible module \(V(\lambda)\) for a complex Lie algebra, in 2010 Evgeny Feigin defined the degenerate flag variety \(Fl(\lambda)^a\) as the closure of a certain highest weight orbit in the projectivization of a degenerate version \(V(\lambda)^a\) of \(V(\lambda)\).
In the case of type \(A_n\) and \(C_n\) the authors have previously shown (see [\textit{G. Cerulli Irelli} and \textit{M. Lanini}, Int. Math. Res. Not. 2015, No. 15, 6353--6374 (2015; Zbl 1349.14157)]) that \(Fl(\lambda)^a\) can be realized as Schubert varieties in a partial flag variety of the same type and bigger rank. In this work they show that \(V(\lambda)^a\) itself is isomorphic to a Demazure representation for a group of type respectively \(A_{2n}\) and \(C_{2n}\). As corollary they redemonstrate the previous result. Furthermore their results are characteric free.
In the case of \(\mathrm{SL}_n(\mathbb{C})\) the result is as follow. Let \(\mathcal{N}^-\), \(\mathcal{H}^-\), \(\mathcal{N}^+\), \(\mathcal{B}\) the Lie subalgebra of \(\mathcal{SL}_n(\mathbb{C})\) of strictly lower triangular, diagonal strictly upper triangular and upper triangular matrices. Let \(\tilde{\mathcal{N}}^-\), \(\tilde{\mathcal{H}}^-\), \(\tilde{\mathcal{N}}^+\) and \(\tilde{\mathcal{B}}\) be the correspond Lie subalgebra of \(\mathcal{SL}_{2n}\) of lower triangular, diagonal and upper triangular matrices. Let \(\mathcal{N}\) be the unipotent subgroup of lower triangular matrices with determinant 1.
Consider the following subalgebra of \(\mathrm{Lie}(\mathrm{SL}_{2n})\):
\[
\mathcal{N}^a=\Bigg\{ \begin{pmatrix}0&A\\
0&0\end{pmatrix}\mid A\in \mathcal{N}\Bigg\}
\]
and the following subgroup of \(\mathrm{SL}_{2n}\):
\[
\mathcal{N}^a=\Bigg\{ \begin{pmatrix}\mathbf{1}&A\\
0&\mathbf{1}\end{pmatrix} \mid A\in \mathcal{N}\Bigg\}.
\]
\(\mathcal{N}^a\) is an abelianization of \(\mathcal{N}\) and \(\mathcal{N}^a\) is an abelianization of \(\mathcal{N}\).
The embedding \(\mathcal{N}^{-,a}\rightarrow \tilde{b}\) induces an embedding of the enveloping algebra of \(\mathcal{N}^{-,a}\) into \(U(\tilde{b})\). Let \(\tilde{V}(\mu)\) be a irreducible representation for \(\mathrm{SL}_{2n}\) with highest weight \(\mu\) and let \(w\) be an element of the Weyl group. The Demazure submodule \(\tilde{V}(\mu)_{w\mu}\) is the cyclic \(U(\tilde{b})\)-module generated by a weight vector of weight \(w\mu\).
The PBW filtration on \(U(\mathcal{N}^-)\) induces a filtration on the cyclic \(U(\mathcal{N}^-)\)-module \(V(\lambda)=U(\mathcal{N}^-).v_\lambda\) and the associated graded space \(V^a(\lambda)=gr(V(\lambda))\) becomes a module for the associated graded algebra
\[
S^\bullet(\mathcal{N}^-) =gr U(\mathcal{N}^-)\cong S^\bullet(\mathcal{N}^{-,a} ).
\]
The action of \(n^{-,a}\) on \(V^a(\lambda)\) can be integrated to an action of \(N^{-,a}\). The closure of the orbit \(N^{-,a}.[V_\lambda]\subset P(V^a(\lambda))\) is called the degenerate flag variety \(Fl(\lambda)^a\).
Let \(\lambda^*\) be the dual dominant weight of \(\lambda\) and let \(\Psi:h^*\rightarrow \tilde{h}^*\) be the linear map who sends i-th fundamental weight to the 2i-fundamental weight. After defining explicitly an element \(\tau\in \tilde{W}\) of the Weyl group of \(\mathrm{SL}_{2n}\) the authors prove the following results:
i) The Demazure submodule \(\tilde{V}(\psi(\lambda^*))_\tau\) is isomorphic to the abelianized module \(V^a(\lambda)\) as \(n^{-,a}\) module.
ii) The Schubert Variety \(X(\tau)\subset \mathbb{P}(\tilde{V}(\psi(\lambda^*))_\tau)\) is isomorphic to the degenerate Flag variety \(F^a(\lambda)\). Schubert varieties; degenerate flag varieties; Demazure modules Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Grassmannians, Schubert varieties, flag manifolds, Simple, semisimple, reductive (super)algebras, Representation theory for linear algebraic groups, Linear algebraic groups over global fields and their integers Degenerate flag varieties and Schubert varieties: a characteristic free approach | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The purpose of this note is to give a survey of \textit{M. Borovoi}'s work on the Hasse principle and approximation theorems for homogeneous spaces. An ingenious tool to study such arithmetic is abelianization of Galois cohomology of affine algebraic groups [Mem. Am. Math. Soc. 626 (1998; Zbl 0918.20037)].
For principal homogeneous spaces, this theme is, in the algebraic aspect, the theory of Galois cohomology of affine algebraic groups, which has been developed by several authors. Sansuc discussed these object in light of the brilliant idea of using the Brauer-Grothendieck group, introduced by Manin for the Hasse principle and by Colliot-Thélène-Sansuc for weak approximation. The problem here is the uniqueness of the Brauer-Manin obstruction.
On the other hand, Kottwitz showed some analogies for reductive groups to duality theorems for Galois cohomology of Abelian algebraic groups. He described the results in terms of the Langlands dual group \(\widehat{G}\). Since the correspondence \(G\to \widehat{G}\) is not functorial in \(G\), his duality theorems were stated in the category where the morphisms are normal. Borovoi (loc. cit.) reconstructed Kottwitz's results based on the theory of non-Abelian hypercohomology with coefficients in crossed modules and described them in a functorial manner using the algebraic fundamental groups. So, the results are naturally expected to be applied to some arithmetic questions on homogeneous spaces. Moreover, since the Picard group of an affine algebraic group is expressed by the algebraic fundamental group, we can describe these cohomological results in terms of the Brauer-Grothendieck group. Hasse principle; approximation theorems for homogeneous spaces; abelianization of Galois cohomology; affine algebraic groups; non-Abelian hypercohomology; Brauer-Grothendieck group Morishita, M.: Hasse principle and approximation theorems for homogeneous spaces. Algebraic number theory and related topics, Kyoto 1996 998, 102-116 (1997) Galois cohomology of linear algebraic groups, Rational points, Cohomology theory for linear algebraic groups, Research exposition (monographs, survey articles) pertaining to number theory, Galois cohomology Hasse principle and approximation theorems for homogeneous spaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We show an explicit formula for the Chow class of the image of a complex smooth projective variety by a holomorphic section of a Grassmann bundle. Then we show how the general proof works, by analyzing in detail a significative example. Grassmann bundles; Schubert calculus A. Nigro, Sections of Grassmann Bundles, Ph.D. Thesis, Politecnico di Torino, 2010. Classical problems, Schubert calculus, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Chow classes of sections of Grassmann bundles | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Isospectral torus plays a fundamental role in the study of orthogonal polynomials on the real line with measures whose essential support is a finite union of disjoint closed intervals. The authors describe the Jost functions of elements of the isospectral torus, relate Jost functions to theta functions and discuss Jost solutions and the associated Bloch waves. Finally they apply these solutions to the study of Green's function. The key to their analysis is a machinery developed by \textit{M. Sodin} and \textit{P. Yuditskii} [J. Geom. Anal. 7, No. 3, 387--435 (1997; Zbl 1041.47502)] and exploited by \textit{F. Peherstorfer} and \textit{P. Yuditskii} [Proc. Am. Math. Soc. 129, No. 11, 3213--3220 (2001; Zbl 0976.42012)]. isospectral torus; covering map; orthogonal polynomials J.S. Christiansen, B. Simon and M. Zinchenko, \textit{Finite gap Jacobi matrices, I. The} \textit{isospectral torus. }Constr. Approx. 32 (2010), 1--65. Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis, Isospectrality, Coverings of curves, fundamental group Finite gap Jacobi matrices. I: The isospectral torus | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials There are many variants of Gromov-Witten theory and many remarkable formulas relating the various invariants. This was discovered back when \textit{E. Witten} [in: Surveys in differential geometry. Vol. I: Proceedings of the conference on geometry and topology. Providence, RI: American Mathematical Society; Bethlehem, PA: Lehigh University. 243--310 (1991; Zbl 0757.53049)] described 2D quantum gravity and his conjecture was proved by \textit{M. Kontsevich} [Commun. Math. Phys. 147, No. 1, 1--23 (1992; Zbl 0756.35081)] using the matrix Airy function. Various Gromov-Witten invariants may be combined together into generating functions that satisfy various systems of PDE such as the KdV hierarchy. One recent progress is the development of the double ramification hierarchy in the paper [Commun. Math. Phys. 336, No. 3, 1085--1107 (2015; Zbl 1329.14103)] by \textit{A. Buryak}. In fact that paper provides nice exposition on the background for the paper under review. The double ramification hierarchy is based on covers of \(P^1\) branched over zero and infinity with given partitions.
The paper under review develops several new recursion relations for the double ramification hierarchy. The new recursion relations recover the full hierarchy from just one of the Hamiltonians. The paper ends with a proof of the Miura equivalence between the double ramification hierarchy and the Dubrovin-Zhang hierarchy for the Gromov-Witten theory of the complex projective line. Gromov-Witten; double ramification; hierarchy Buryak, A.; Rossi, P., Recursion relations for double ramification hierarchies, Commun.Math. Phys., 342, 533-568, (2016) Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Hodge theory in global analysis Recursion relations for double ramification hierarchies | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this third edition of his well known book [for the first edition (1966) see Zbl 0144.448, for the second edition (1975) see Zbl 0307.55015], the author has introduced two new chapters: 7 and 19. In Chapter 7 he considers the gauge group of a principal bundle, which has numerous applications in differential geometry and mathematical physics. In Chapter 19, starting from A. Weil's viewpoint, the author calculates the characteristic classes of smooth manifolds and vector bundles, using the invariant polynomials associated to the curvature forms for a linear connection. This bibliography has been enlarged and updated. gauge group; principal bundle; characteristic classes; smooth manifolds; vector bundles; invariant polynomials; curvature forms; linear connection Husemöller, D., Fibre Bundles, (1993), Springer Research exposition (monographs, survey articles) pertaining to algebraic topology, Fiber bundles in algebraic topology, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], de Rham cohomology and algebraic geometry Fibre bundles. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We study the class of bihomogeneous polynomials \(r(z, \bar z)\) on \(\mathbb{C}^n \times \mathbb{C}^n\) for which there is a positive integer \(d\) such that \(r(z, \bar z) \|z\|^{2d}\) can be written as a Hermitian sum of squares. We reinterpret this problem in terms of commutative algebra, and we give necessary algebraic conditions on this class of polynomials.
For Part II, see [the authors et al., Proc. Am. Math. Soc. 150, No. 8, 3471--3476 (2022; Zbl 1496.32002)]. sums of squares; real polynomials; Betti numbers CR manifolds, Fields related with sums of squares (formally real fields, Pythagorean fields, etc.), Real algebraic sets Algebraic properties of Hermitian sums of squares | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials For a semialgebraic set \(S\subseteq {\mathbb{R}}^n\) the \textit{Zariski closure at infinity of }\(S\) is the smallest Zariski closed \(Z\) such that there is a compact \(K\subseteq {\mathbb{R}}^n\) with \(S\subseteq Z\cup K\). The author gives a new proof that if \(S\subseteq {\mathbb{R}}^n\) is semialgebraic and such that its Zariski closure at infinity is nonempty and a proper subset of \({\mathbb{R}}^n\) then the algebra of polynomials bounded on \(S\) does not have a finite set of generators. This is originally due to \textit{D. Plaumann} and \textit{C. Scheiderer} [Trans. Am. Math. Soc. 364, No. 9, 4663--4682 (2012; Zbl 1279.14072)]. The proof in the paper under review is of a more geometric nature than the original proof and uses, amongst other things, a result of Jelonek's on the set of points at which a polynomial map is not proper. bounded polynomials; semialgebraic set; finitely generated algebra; proper mapping Michalska, M., Algebra of bounded polynomials on a set Zariski closed at infinity cannot be finitely generated, Bull. Sci. Math., 137, 705-715, (2013) Real algebraic and real-analytic geometry, Semialgebraic sets and related spaces Algebra of bounded polynomials on a set Zariski closed at infinity cannot be finitely generated | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We study the enumerative significance of the \(s\)-pointed genus zero Gromov-Witten invariant on a homogeneous space \(X\). For that, we give an interpretation in terms of rational curves on \(X\). Gromov-Witten invariants; Grassmannians; Schubert varieties; classical problems Martín, López, A.: Gromov-Witten invariants and rational curves on Grassmannians, J. Geom. Phys., 61, 213-216, (2011) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus Gromov-Witten invariants and rational curves on Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The representation theory for infinite dimensional classical groups is relatively well understood and the same is true for infinite symmetric groups, but this is not the case for groups of infinite matrices over finite fields \(\mathrm{GL}(\infty,\mathbb{F}_q)\). In the paper under review the author gives some steps towards the development of this theory. There are several attempts to extend the well known techniques for studying \(\mathrm{GL}(n,\mathbb{F}_q)\) to the groups \(\mathrm{GL}(\infty,\mathbb{F}_q)\), here the author chooses the way through inverse limits of homogeneous spaces (c.f. \textit{D. Pickrell} [J. Funct. Anal. 70, 323--356 (1987; Zbl 0621.28008)]).
Let \(\ell\) be the direct sum of countable copies of \(\mathbb{F}_q\) and let \(\ell^\diamond\) be the direct product of countable copies of \(\mathbb{F}_q\). The author consider the linear space \(\ell\oplus\ell^\diamond\) equipped with a natural topology coming from the extension of the discrete topology of \(\mathbb{F}_q\). Set \(\mathrm{GL}(\ell\oplus\ell^\diamond)\) for the group of continuous invertible linear operators in \(\ell\oplus\ell^\diamond\), this group has a natural homomorphism \(\theta\) into \(\mathbb{Z}\) whose kernel is denoted by \(\mathrm{GL}^0(\ell\oplus\ell^\diamond)\). Take \(\mathcal{M}\) as the set of subspaces of \(\ell\oplus\ell^\diamond\) of the form \(v\oplus vT\), for some infinite matrix \(T\) over \(\mathbb{F}_q\). A subspace \(M\) of \(\ell\oplus\ell^\diamond\) is said semi-infinite if there exists \(L\in\mathcal{M}\) such that \(\alpha(L)=\dim\left(L/(L\cap M)\right)\) and \(\beta(L)=\dim\left(M/(L\cap M)\right)\), are both finite. The relative dimension of \(L\) is then defined as \(\dim(L)=\alpha(L)-\beta(L)\). The primary target of the paper is to understand the set \(\mathrm{Gr}^\alpha\) of subspaces \(L\in\mathcal{M}\) of relative dimension \(\alpha\), these subspaces are called the semi-infinite Grassmannians and we have that \(\mathrm{GL}^0(\ell\oplus\ell^\diamond)\) acts transitively on each \(\mathrm{Gr}^\alpha\).
In the paper it is shown that there exists a unique, up to a scalar factor, and finite \(\mathrm{GL}^0(\ell\oplus\ell^\diamond)-\)invariant Borel measure \(\mu\) on \(\mathrm{Gr}^0\). When \(\mu\) is normalized by the condition \(\mu(\mathcal{M})=1\), then
\[
\mu(\mathrm{Gr}^0)=\prod_{j=1}^{\infty}(1-q^{-j})^{-1}.
\]
Given \(L\in \mathrm{Gr}^0\), it is defined \(\Sigma_L=\{K\in \mathrm{Gr}^0\mid \dim\left(L/(L\cap K)\right)=\dim\left(K/(L\cap K)\right)=1\}\), and it is shown that there exists a unique probability measure \(\nu_L\) on \(\Sigma_L\) invariant with respect to the stabilizer of \(L\) in \(\mathrm{GL}^0(\ell\oplus\ell^\diamond)\). Then the operator
\[
\Delta f(L)=\int\limits_{\Sigma_L}f(K)d\nu_L(K),
\]
is a bounded self adjoint \(\mathrm{GL}^0(\ell\oplus\ell^\diamond)-\)intertwining operator in \(L^2(\mathrm{Gr}^0,\mu)\) with spectrum \(\{1,q^{-1},q^{-2},\ldots\}\). Finally the author gives the invariant functions in terms of the Al-Salam--Carlitz orthogonal polynomials, and describes a measure on the flags of \(\ell\oplus\ell^\diamond\). infinite-dimensional groups; infinite matrices over finite fields; Grassmanians in representation theory; invariant measures; Al-Salam--Carlitz polynomials; \(q-\)Hahn polynomials; flags Neretin, YA, The space \(L\)\^{}\{2\} on semi-infinite Grassmannian over finite field, Adv. Math., 250, 320-350, (2014) Grassmannians, Schubert varieties, flag manifolds, Representations of infinite symmetric groups, General groups of measure-preserving transformations, Group structures and generalizations on infinite-dimensional manifolds, Linear algebraic groups over finite fields, Finite ground fields in algebraic geometry The space \(L^2\) on semi-infinite Grassmannian over finite field | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper, we introduce the notions of \(S_\phi\)-polynomial and \(S\)-minimal value set polynomial where \(\phi\) is a polynomial over a finite field \(\mathbb{F}_q\) and \(S\) is a finite subset of an algebraic closure of \(\mathbb{F}_q\). We study some properties of these polynomials and we prove that the polynomials used by Garcia, Stichtenoth and Thomas in their work on good recursive tame towers are \(S_\phi\)-minimal value set polynomials for \(\phi= x^m\), whose \(S\)-value sets can be explicitly computed in terms of the monomial \(x^m\). value sets; finite fields; polynomials; towers of function fields Polynomials over finite fields, Algebraic functions and function fields in algebraic geometry A link between minimal value set polynomials and tamely ramified towers of function fields over finite fields | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors study knots as real smooth space curves in the 3-sphere with parametrization given by the classical Chebyshev polynomials \(T_n(t) = \cos(n\arccos(t))\). In view of their own theorem [J. Knot Theory Ramifications 20, No. 4, 575--593 (2011; Zbl 1218.57009)] any knot is, in fact, isotopic to a Chebyshev knot \(C(a,b,c,\varphi)\) with parametrization \(x = T_{a}(t),\, y = T_{b}(t), \, z = T_{c}(t+\varphi),\) where \(t\in \mathbb R,\) \(a\) and \(b\) are coprime integers, \(c\) is an integer and \(\varphi\) is a real constant. When \(a,b,c\) are coprime then \(C(a,b,c,0)\) is called a harmonic knot. In the paper under review a complete classification of harmonic knots with \(a \leq 3\) is given. In addition, the authors describe an explicit parametrization for all two-bridge knots [\textit{J. H. Conway}, in: Comput. Probl. abstract Algebra, Proc. Conf. Oxford 1967, 329--358 (1970; Zbl 0202.54703)]. It should be underlined that the paper contains many interesting pictures and examples such as the torus knots, twist knots, Stevedore knots, etc. Moreover, almost all results are derived from elementary properties of continued fractions and their matrix representations. real smooth curves; Chebyshev polynomials; rational knots; Lissajous knots; harmonic knots; two-bridge knots; continued fractions Koseleff, P.-V.; Pecker, D., Chebyshev diagrams for two-bridge knots, Geom. Dedic., 150, 405-425, (2011) Knots and links in the 3-sphere, Plane and space curves, Real algebraic and real-analytic geometry, Continued fractions Chebyshev diagrams for two-bridge knots | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The present work is an extension of the author's results obtained in [Transform. Groups 10, No.1, 63--132 (2005; Zbl 1080.33018)], especially concerning the Koorwinder polynomials, which are generalized to a family of biorthogonal abelian functions. A number of properties of these functions are proven, most notably analogues of Macdonald's conjectures. More precisely, the author begins this work with the definition and basic properties of balanced interpolation polynomials, in particular, considering four of the main perfect cases satisfying extra vanishing properties; the fifth (elliptic) perfect case is the subject of the remainder of the article. The author defines a family of \(BC_{n}\)-symmetric theta functions, parametrized by partitions contained in a rectangle and defined via vanishing conditions. The key property of these functions is a difference equation, from which the extra vanishing property follows. He also defines a corresponding family of abelian functions indexed by all partitions of length at most \(n\). Evaluating an interpolation abelian function at a partition gives a generalized binomial coefficient. The author studies these coefficients and obtains a pair of identities generalizing Jackson's summation and Bailey transformation respectively; each identity involves a sum over partitions contained in a skew Young diagram of a product of binomial coefficients. Using these, some unexpected symmetries of binomial coefficients are derived, relating a coefficient to coefficients with conjugated or complemented partitions. By this, a number of properties of interpolation functions, including a connection coefficient formula, a branching rule, a Pieri identity, and a Cauchy identity are proven. Then the author introduces the biorthogonal functions, defined via an expansion in interpolation functions. Using the interpolation function identities he proves a number of identities for the biorthogonal functions; in addition to the analogue of Macdonald's conjectures for Koornwinder polynomials, he obtains connection coefficients, a quasi-branching rule, a quasi-Pieri identity and a Cauchy identity. Next the author shows that the interpolation and biorthogonal functions can be defined purely algebraically. He sketches a purely algebraic derivation of the identities obtained earlier in this paper. Using his results on interpolation functions the author defines a family of perfect bigrids based on elliptic curves. Degeneration classes of these bigrids are also obtained. Finally, the author considers some open problems relating to interpolation and biorthogonal functions. symmetric theta functions; biorthogonal functions; interpolation functions; symmetric abelian functions; generalized binomial coefficients; Jackson's summation formula; Bailey's transformation formula; Macdonald's conjectures for Koornwinder's polynomials; elliptic curves Rains, E., \(B C_n\)-symmetric abelian functions, Duke Math. J., 135, 99-180, (2006) Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.), Elliptic curves, Theta functions and abelian varieties \(BC_n\)-symmetric abelian functions | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We revisit what we call the fibred topology on a fibred category over a site and we prove a few basic results concerning this topology. We give a general result concerning the invariance of a 2-category of stacks under change of base.
For part I, cf. [\textit{A. E. Stanculescu}, Theory Appl. Categ. 29, 654--695 (2014; Zbl 1302.18017)]. Grothendieck topology; fibred category; stack; model category Nonabelian homotopical algebra, Fibered categories, Generalizations (algebraic spaces, stacks) Stacks and sheaves of categories as fibrant objects. II. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The Grothendieck-Teichmüller group \(\widehat{GT}\) acts naturally on the profinite completion \(\widehat K(0,n)\) of the mapping class group of a sphere with \(n\) marked points. The elements of \(\widehat{GT}\) are, by definition, parameterized by the pairs \((\lambda,f)\in\widehat\mathbb{Z}^\times\times\widehat F_2\) subject to certain conditions. In this paper, it is shown that the principal parameter \(f\) evaluated in \(\widehat K(0,4)\) (resp. \(\widehat K(0,5)\)) admits certain decompositions like non-commutative 1-coboundaries under certain standard actions of finite cyclic groups of order 2, 3 (resp. 5). As a by-product of their method, it is also proven that the ``complex conjugation'' in \(\widehat{GT}\) is self-centralizing. profinite groups; non-commutative cohomology; Grothendieck-Teichmüller group; profinite completions; mapping class groups; actions of finite cyclic groups P. Lochak and L. Schneps, ''A cohomological interpretation of the Grothendieck-Teichmüller group,'' Invent. Math., vol. 127, iss. 3, pp. 571-600, 1997. Limits, profinite groups, Cohomology of groups, Coverings in algebraic geometry, Other groups related to topology or analysis A cohomological interpretation of the Grothendieck-Teichmüller group. Appendix by C. Scheiderer | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper deals with the problem of characterizing complete intersections among smooth codimension two subvarieties in projective space. The first characterization is given by means of the splitting of the normal bundle. The authors prove this result for surfaces (so that this characterization holds for dimension at least two, being clearly false for curves). The key point is to prove that the summands of the normal bundle depend numerically on the hyperplane section, and the result is then obtained by restricting to a general hyperplane section.
One could also try to characterize complete intersections by giving the normal bundle as an extension of two line bundles. This holds for dimension at least four, but it is not true for surfaces (although the authors prove it when the line bundles depend numerically on the hyperplane section). In the case of a threefold \(X\), the authors consider a natural double structure on \(X\), which appears from the given extension and is the zero section of a rank two vector bundle on \(\mathbb{P}^ 5\). Then they prove that \(X\) is a complete intersection if and only if this bundle splits. characterizing complete intersections; codimension two subvarieties; splitting of the normal bundle; threefold; double structure Basili B. and Peskine C., Décomposition du fibré normal des surfaces lisses de \({\mathbb{P}^{4}}\) et structures doubles sur les solides de \({\mathbb{P}^{5}}\), Duke Math. J. 69 (1993), 87-95. Complete intersections, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Low codimension problems in algebraic geometry, \(3\)-folds, Projective techniques in algebraic geometry Decomposition of normal bundle of smooth surfaces of \(\mathbb{P}_ 4\) and double structures on solids in \(\mathbb{P}_ 5\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Studying integration along Bott-Samelson cycles the author proves the following beautiful explicit formula for the degree of the Schubert variety \(X_w\) in the flag manifold \(G/T\) associated to an element \(w\) in the Weyl group:
Let \(b_w\) be the Kähler sequence and \(A_w\) the Cartan matrix then,
\[
\sum \frac{k!b_1^{r_1}\cdots b_k^{r_k}}{r_1!\cdots r_k!} T_{A_w}(x_1^{r_1}\cdots x_k^{r_k}), \quad k=I(w),
\]
where the sum is over all \(r_1+\cdots +r_k=k\) with \(r_i\geq 0\), and \(r_1+\cdots +r_i\leq i\), and where, for each \(k\times k\)-matrix \(A=(a_{ij})\), the homomorphism \(T_A:\mathbb Z[x_1,\dots, x_k]^{(k)}\to \mathbb Z\) on the homogeneous polynomials of degree \(k\) is defined recursively by:
(1) For \(h\in \mathbb Z[x_1,\dots, x_{k-1}]^{(k)}\) we have \(T_A(h)=0\).
(2) If \(k=1\) then \(T_A(x_1)=1\).
(3) For \(h\in \mathbb Z[x_1,\dots, x_{k-1}]^{(k-r)}\) with \(r\geq 1\) we have
\[
T_A(hx_k^r) =T{A'}(h(a_{1,k}x_1+\cdots +a_{k-1,k}x_{k-1})^{r-1}),
\]
where \(A'\) is the matrix obtained from \(A\) deleting the \(k\)'th column and the \(k\)'th row.
The author also provides two generalizations of this result. Bott-Samelson cycles; Cartan Numbers; Schubert varieties; degrees; compex structures; Weyl group; Kaehler sequence; Cartan matrix; flag manifolds Duan, H.: The degree of a Schubert variety. Advances in Mathematics 180(1), 112--133 (2003) Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus The degree of a Schubert variety. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(k\) be a field, char \(k=p>0.\) Let \(n,r>0,\) and let
\[
R\left( n,r\right) =k\left[ x_{1},\dots,x_{n}\right] /\left( x_{1}^{p^{r}},\dots,x_{n}^{p^{r} }\right).
\]
Then \(R\left( n,r\right) \) represents the \(r^{\text{th}}\) Frobenius kernel on \(n\)-dimensional affine space. Let \(G\left( n,r\right) \) be the automorphism scheme of \(R\left( n,r\right) .\) Associated to each \(G\left( n,r\right) \)-representation is a canonical vector bundle obtained by twisting with the \(r^{\text{th}}\) Frobenius morphism. In \(K\)-theory, these bundles are studied using the Grothendieck ring of \(G\left( n,r\right) .\)
In the work under review, the author provides a description of the irreducible \(G\left( n,r\right) \)-representations which form a \(\mathbb{Z}\)-basis of \(K_{0}\left( G\left( n,r\right) -\text{rep}\right) \) as an abelian group. This is accomplished by taking a triangular decomposition \(G\left( n,r\right) =G^{-}\text{GL}_{n}G^{+}.\) A surjective map \(K_{0}\left( \text{GL}_{n}-\text{rep}\right) ^{\oplus r+1}\rightarrow K_{0}\left( G\left( n,r\right) -\text{rep}\right) \) is constructed recursively, and its kernel is computed. Much of the work uses the more general language of triangulated group schemes, which facilitates the recursive definition. Grothendieck rings; automorphism schemes; triangulated group schemes Group schemes, Representation theory for linear algebraic groups, \(K\)-theory of schemes, Modular Lie (super)algebras, Lie algebras of linear algebraic groups The Grothendieck ring of the structure group of the geometric Frobenius morphism | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials A complex reductive (algebraic) monoid \(M\) is called rationally smooth if it has sufficiently mild singularities as a topological space. The author characterizes this class of monoids in combinatorial terms, and then uses his results to calculate the Betti numbers of certain projective, rationally smooth group embeddings using the ``monoid BB-decomposition''. Betti numbers; group embeddings; idempotents; monoid BB-cells; Poincaré polynomials; rationally smooth reductive monoids; semisimple algebraic monoids; toric varieties L. Renner, \textit{Rationally smooth algebraic monoids}, Semigroup Forum \textbf{78} (2009), 384-395. Algebraic monoids, Group actions on varieties or schemes (quotients), Toric varieties, Newton polyhedra, Okounkov bodies, Linear algebraic groups over arbitrary fields Rationally smooth algebraic monoids. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Schubert polynomials were introduced in the context of the geometry of flag varieties. This paper investigates some of the connections not yet understood between several combinatorial structures for the construction of Schubert polynomials; we also present simplifications in some of the existing approaches to this area. We designate certain line diagrams for permutations known as rc-graphs as the main structure. The other structures in the literature we study include: semistandard Young tableaux, Kohnert diagrams, and balanced labelings of the diagram of a permutation. The main tools in our investigation are certain operations on rc-graphs, which correspond to the coplactic operations on tableaux, and thus define a crystal graph structure on rc-graphs; a new definition of these operations is presented. One application of these operations is a straightforward, purely combinatorial proof of a recent formula due to \textit{A. S. Buch, A. Kresch, H. Tamvakis}, and \textit{A. Yong} [Duke Math. J. 122, 125--143 (2004; Zbl 1072.14067)], which expresses Schubert polynomials in terms of products of Schur polynomials. In spite of the fact that it refers to many objects and results related to them, the paper is mostly self-contained. Schubert polynomial; Young tableau; rc-graph; crystal graph; Kohnert diagram Lenart, C.: A unified approach to combinatorial formulas for Schubert polynomials. J. Algebr. Comb. 20, 263--299 (2004) Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of representation theory A unified approach to combinatorial formulas for Schubert polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Siehe JFM 32.0418.04 Double integrals; algebraic functions Surfaces and higher-dimensional varieties On the periods of double integrals in the theory of algebraic functions of two variables. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(\text{Fl}({\mathbb C}^n)\) be the flag variety i.e. the variety of complete flags \(0=V_0\subset V_1\subset \cdots \subset V_n=\mathbb C^n.\) The classes of Schubert structure sheaves \([{\mathcal O}_{X_{\pi}}]\) constitute an additive \(\mathbb Z\)-basis of the \(K\)-theory ring \(K(\text{Fl} ({\mathbb C}^n)).\) \(X_\pi\) denotes here the Schubert variety corresponding to a permutation \(\pi \in {\mathcal S}_n.\) The Schubert constants are defined by the following formula:
\[
[{\mathcal O}_{X_{\pi}}]\cdot [{\mathcal O}_{X_{\rho}}]= {\sum}_{\pi \in {\mathcal S}_n} {\mathcal C}^{\pi}_{{\sigma},{\rho}}[{\mathcal O}_{X_{\pi}}].
\]
This formula stabilizes with respect to inclusions \({\mathcal S}_n\hookrightarrow {\mathcal S}_{n+1}\) therefore the Schubert constants may be defined for \(({\sigma}, {\rho},{\pi})\in {\mathcal S}_{\infty }^3\). The analogous problem for the Grassmannians was settled in the cohomology case by the Littlewood-Richardson rule and in the case of the \(K\)-theory by \textit{A. S. Buch} [Acta Math. 189, No. 1, 37--78 (2002; Zbl 1090.14015)]. The main result of the authors is a substraction-free formula for the family they call truncation Schubert problems. This formula has some nice specializations. It allows one to compute the \(K\)-theory generalizations of numbers considered by \textit{M. Kogan} [Int. Math. Res. Not. 15, 765--782 (2001; Zbl 0994.05150)] and the \(K\)-theory Littlewood-Richardson coefficients considered by Buch [loc. cit.]. The proof is combinatorial. Schubert problem; \(K\)-theory Knutson, A.; Yong, A., A formula for K-theory truncation Schubert calculus, Int. Math. Res. Not., 70, 3741-3756, (2004) Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Applications of methods of algebraic \(K\)-theory in algebraic geometry, \(K\)-theory of schemes A formula for \(K\)-theory truncation Schubert calculus | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The intersection ring of a complex Grassmann manifold is generated by Schubert varieties, and its structure is governed by the Littlewood-Richardson rule. Given three Schubert varieties \(S_{1}, S_{2}, S_{3}\) with intersection number equal to one, we show how to construct an explicit element in their intersection. This element is obtained generically as the result of a sequence of lattice operations on the spaces of the corresponding flags, and is therefore well defined over an arbitrary field of scalars. Moreover, this result also applies to appropriately defined analogues of Schubert varieties in the Grassmann manifolds associated with a finite von Neumann algebra.
The arguments require the combinatorial structure of honeycombs, particularly the structure of the rigid extremal honeycombs. It is known that the eigenvalue distributions of selfadjoint elements \(a,b,c\) with \(a+b+c=0\) in the factor \(\mathcal R^{\omega}\) are characterized by a system of inequalities analogous to the classical Horn inequalities of linear algebra. We prove that these inequalities are in fact true for elements of an arbitrary finite factor. In particular, if \(x,y,z\) are selfadjoint elements of such a factor and \(x+y+z=0\), then there exist selfadjoint \(a,b,c \in \mathcal R^{\omega}\) such that \(a+b+c=0\) and \(a\) (respectively, \(b,c\)) has the same eigenvalue distribution as \(x\) (respectively, \(y,z\)). A (`complete') matricial form of this result is known to imply an affirmative answer to an embedding question formulated by Connes.
The critical point in the proof of this result is the production of elements in the intersection of three Schubert varieties. When the factor under consideration is the algebra of \(n\times n\) complex matrices, our arguments provide new and elementary proofs of the Horn inequalities, which do not require knowledge of the structure of the cohomology of the Grassmann manifolds. Schubert variety; hive; honeycomb; factor Bercovici, H., Collins, B., Dykema, K., Li, W.S., Timotin, D.: Intersections of Schubert varieties and eigenvalue inequalities in an arbitrary finite factor. J. Funct. Anal. 258, 1579--1627 (2010) Classification of factors, Grassmannians, Schubert varieties, flag manifolds, Eigenvalue problems for linear operators, Symmetric functions and generalizations Intersections of Schubert varieties and eigenvalue inequalities in an arbitrary finite factor | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We develop numerical homotopy algorithms for solving systems of polynomial equations arising from the classical Schubert calculus. These homotopies are optimal in that generically no paths diverge. For problems defined by hypersurface Schubert conditions we give two algorithms based on extrinsic deformations of the Grassmannian: One is derived from a Gröbner basis for the Plücker ideal of the Grassmannian and the other from a SAGBI basis for its projective coordinate ring. The more general case of special Schubert conditions is solved by delicate intrinsic deformations, called Pieri homotopies, which first arose in the study of enumerative geometry over the real numbers. Computational results are presented and applications to control theory are discussed. numerical homotopy algorithms; systems of polynomial equations; Schubert calculus; SAGBI basis Huber, B; Sottile, F; Sturmfels, B, Numerical Schubert calculus, J. Symb. Comp., 26, 767-788, (1998) Enumerative problems (combinatorial problems) in algebraic geometry, Configurations and arrangements of linear subspaces, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Global methods, including homotopy approaches to the numerical solution of nonlinear equations, Computational aspects of higher-dimensional varieties, Symbolic computation and algebraic computation Numerical Schubert calculus | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this extended abstract, we give a complete description and enumeration of smooth and rationally smooth Schubert varieties in finite type. In particular, we show that rationally smooth Schubert varieties are in bijection with a new combinatorial data structure called staircase diagrams. Schubert varieties; rational smoothness; Billey-Postnikov decompositions Combinatorial aspects of algebraic geometry, Classical problems, Schubert calculus, Exact enumeration problems, generating functions Staircase diagrams and the enumeration of smooth Schubert varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The problem of semialgebraic Lipschitz classification of quasihomogeneous polynomials on a Hölder triangle is studied. For this problem, the ``moduli'' are described completely in certain combinatorial terms. Lipschitz classification; quasihomogeneous polynomials; Hölder triangle; moduli L. Birbrair, A. Fernandes, and D. Panazzolo, Lipschitz classification of functions on a Hölder triangle, Algebra i Analiz 20 (2008), no. 5, 1 -- 8; English transl., St. Petersburg Math. J. 20 (2009), no. 5, 681 -- 686. Equisingularity (topological and analytic), Semialgebraic sets and related spaces, Singularities in algebraic geometry, Local complex singularities Lipschitz classification of functions on a Hölder triangle | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a simply-connected, semi-simple algebraic group over an algebraic closed field, let \(T\) be a maximal torus of \(G\) and let \(B\) be a Borel subgroup of \(G\) containing \(T\). The authors are interested to determine what Schubert varieties of a projective homogeneous variety \(G/P\) contain a semistable point with respect to the action of \(T\) and to a fixed ample line bundle \(\mathcal{L}\). In this work there are two main results. In the first part of this article the authors restrict themselves to the case where \(G\) is a simple group of type \(B\), \(C\) or \(D\) and \(P\) is a maximal parabolic subgroup. With these hypotheses they classify the minimal elements \(w\in W/W_{P}\) such that \(X(w)_{T}^{ss}(\mathcal{L})\neq\emptyset\). Here \(X(w)=\overline{BwP/P}\) is the Schubert variety associated to \(w\). The authors affirm that, when \(G\) exceptional or \(P\) non-maximal, the same problem is more complicated.
In the second part of this work, the authors classify the Coxeter elements \(\tau\) of \(W\) such that there is a non-trivial line bundle \(\mathcal{L}\) on \(G/B\) with \(X(\tau)_{T}^{ss}(\mathcal{L})\neq\emptyset\). An element of \(W\) is a Coxeter element if it can be written as a product of distinct simple reflections. The authors are interested to such elements for the following reason: a Schubert variety \(X(w)\) contains a (rank \(G\))-dimensional \(T\)-orbit if and only if \(w\geq \tau\) for some Coxeter element \(\tau\). In this part, they do not make special assumption on \(G\).
Let \(\chi\) be the \(B\)-character associated to a fixed globally generated line bundle \(\mathcal{L}\). Supposing that \(\chi\) belongs to the root lattice, the authors prove that \(X(w)_{T}^{ss}(\mathcal{L})\neq\emptyset\) if and only if \(w\chi\leq0\). This fact allows them to prove their main theorems in a combinatorial way.
Remark that in the Proposition 3 (and only in that Proposition) the authors use a definition of \(\omega\geq0\) different from the usual one. They say that a weight \(\omega\) is greater or equal to 0 if it can written as a positive linear combination of simple roots. In particular, they do not assume that the coefficients are integral. semistable points; line bundle; Coxeter element; Schubert varieties Kannan, S. S.; Pattanayak, S. K., Torus quotients of homogeneous spaces-minimal dimensional Schubert varieties admitting semi-stable points, \textit{Proc. Indian Acad. Sci. (Math. Sci.)}, 119, 4, 469-485, (2009) Geometric invariant theory, Grassmannians, Schubert varieties, flag manifolds Torus quotients of homogeneous spaces --- minimal dimensional Schubert varieties admitting semi-stable points | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials It is well known that both the classical Galois theory and the topological theory of coverings can be formulated as an equivalence between a category of ``subobjects'' and a category of ``actions''. In this paper, there are discussed variants of Grothendieck's axioms characterizing Galois theories in several contexts corresponding to categories of transitive/not-necessarily-transitive (continuous) actions of a discrete/profinite group (or of even a monoid). Especially, the authors give a presentation of \textit{A. Grothendieck}'s fundamental theorem in SGA1 [Séminaire de géométrie algébrique, 1960/61, Lect. Notes Math. 224 (1971; Zbl 0234.14002); exposé V] as a ``passing into the limit'' of Galois's Galois theory.
At the end, several possible extensions of the authors' results in view of Giraud's theorem on topoi, Joyal's spatial groups etc. are illustrated. Galois category; Grothendieck topos; strict epimorphism; Galois theory; coverings Dubuc, E. J.; De La Vega, C. S.: On the Galois theory of Grothendieck. Bol. acad. Nac. cienc. Cordoba 65, 111-136 (2000) Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Coverings of curves, fundamental group, Categories of topological spaces and continuous mappings [See also 54-XX], Separable extensions, Galois theory On the Galois theory of Grothendieck | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials It was shown by \textit{S. P. Smith} and \textit{J. T. Stafford} [Proc. Lond. Math. Soc., III. Ser. 56, No.2, 229-259 (1988; Zbl 0672.14017)] that \((1)\quad the\) ring of differential operators \({\mathcal D}(X)\) of an irreducible affine curve X over \({\mathbb{C}}\) is a noetherian affine domain of Krull dimension 1 which has only one minimal nonzero ideal J(X) and \((2)\quad if\) we denote by H(X) the quotient \({\mathcal D}(X)/J(X)\) and define by analogy \({\mathcal D},J,H\) for any local ring \({\mathfrak O}_{X,x}\), then H(X) is the direct sum of all H(X,x), x singular points of X.
In this paper one studies (1) the relationship between H(X,x) and the nature of the singularity x and (2) the relation between the structure of H(X) and that of \({\mathcal D}(X)\), in particular one gives a description of the Grothendieck groups of \({\mathcal D}(X)\) with the help of those of H(X) and of the ring of regular functions on the normalization of X. Artin algebras; differential operator; Krull dimension; Grothendieck groups; differential operators; Krull dimension 1; normalization. Singularities of curves, local rings, Commutative rings of differential operators and their modules, Commutative Artinian rings and modules, finite-dimensional algebras The Artin algebras associated with differential operators on singular affine curves | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(F=(F_1,\dots,F_n)\) be a maximal flag of subspaces of \({\mathbb R}^n\), i.e. \(F_i\) is a subspace of \({\mathbb R}^n\) of dimension \(i\) and \(F_i\subset F_{i+1}\). The set of all such flags is isomorphic to the flag variety \(SL_n({\mathbb R})/B\) and the subset \(U_F\) of all flags transversal to the given flag \(F\), is an open Schubert cell. In a previous paper, the authors proved that the number of connected components of the intersection \(U_F\cap U_G\) for two transversal flags \(F\) and \(G\) is equal to the number of orbits of some finite group \(G\) acting on a certain \({\mathbb F}_2\)-vector space \(V\). For \(n=3\), \(4\) and \(5\) this number is equal to \(6\), \(20\) and \(52\) respectively. For \(n>5\) it was conjectured, that the number of \(G\)-orbits in \(V\) should be equal to \(3\cdot 2^{n-1}\).
In this new paper, the authors give a detailed description of the action of \(G\) on \(V\), and in particular they prove their conjecture on the number of connected components of the intersection of two opposite open Schubert cells. Schubert cells; flags; number of orbits; action of finite group B. Shapiro, M. Shapiro, and A. Vainshtein, ''Skew-symmetric vanishing lattices and intersections of Schubert cells,'' Internat. Math. Res. Notices, 11(1998), 563--588 Grassmannians, Schubert varieties, flag manifolds, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Group actions on varieties or schemes (quotients) Skew-symmetric vanishing lattices and intersections of Schubert cells | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials A smooth complex projective variety is called \(\mathcal{B}\)-regular if it admits an algebraic action of the upper triangular Borel subgroup \(\mathcal{B}\subset \text{SL}(2)\) such that the unipotent radical in \(\mathcal{B}\) has a unique fixed point. An important example of a regular variety is the variety of full flags \(G/B\), where \(G\) is a semisimple group and \(B\) is a Borel subgroup in \(G\); here \(\mathcal{B}\) is associated with a regular nilpotent element \(e\in\text{Lie}(G)\).
It was proved by \textit{M.~Brion} and \textit{J. B.~Carrell} [Mich. Math. J. 52, No. 1, 189--203 (2004; Zbl 1084.14044)] that the equivariant cohomology algebra of a \(\mathcal{B}\)-regular variety \(X\) is the coordinate ring of a remarkable affine curve in \(X\times\mathbb{P}^1\). The main result of this paper uses this fact to classify the \(\mathcal{B}\)-invariant subvarieties \(Y\) of a \(\mathcal{B}\)-regular variety \(X\) for which the restriction map \(\imath_Y : H^*(X) \to H^*(Y)\) is surjective. This result may be applied to Schubert subvarieties, certain nulpotent Hessenberg varieties (including the Peterson variety) and certain Springer fibres in \(G/B\). regular varieties; subvarieties; flag varieties; Schubert subvarieties; equivariant cohomology J. B. Carrell and K. Kaveh, On the equivariant cohomology of subvarieties of a \({\mathfrak B}\) -regular variety, Transform. Groups 13 (2008), 495-505. Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Grassmannians, Schubert varieties, flag manifolds On the equivariant cohomology of subvarieties of a \(\mathfrak{B}\)-regular variety | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We prove that the Adams-Riemann-Roch theorem in degree one (i.e., at the level of the Picard group) can be lifted to an isomorphism of line bundles, compatibly with base change. Grothendieck-Riemann-Roch; vector bundles; equivariant geometry; fibration; fixed point formula Riemann-Roch theorems, Riemann-Roch theorems, Chern characters A local refinement of the Adams-Riemann-Roch theorem in degree one | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We prove that the étale fundamental group of a hyperbolic curve over an arithmetic field [e.g., a finite extension field of A\(\mathbb{Q}\) or \(A\mathbb{Q}_p\)] or an algebraically closed field is indecomposable [i.e., cannot be decomposed into the direct product of nontrivial profinite groups]. Moreover, in the case of characteristic zero, we also prove that the étale fundamental group of the configuration space of a curve of the above type is indecomposable.
Finally, we consider the topic of indecomposability in the context of the comparison of the absolute Galois group of \(\mathbb{Q}\) with the Grothendieck-Teichmüller group GT and pose the question: Is GT indecomposable? We give an affirmative answer to a pro-\(l\) version of this question. indecomposability; étale fundamental group; hyperbolic curve; configuration space; Grothendieck-Teichmüller group Coverings of curves, fundamental group Indecomposability of various profinite groups arising from hyperbolic curves | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Subword complexes were defined by \textit{A. Knutson} and \textit{E. Miller} [Adv. Math. 184, No. 1, 161--176 (2004; Zbl 1069.20026)] for describing Gröbner degenerations of matrix Schubert varieties. The facets of such a complex are indexed by pipe dreams, or, equivalently, by the monomials in the corresponding Schubert polynomial. \textit{S. Assaf} and \textit{D. Searles} [ibid. 306, 89--122 (2017; Zbl 1356.14039)] defined a basis of slide polynomials, generalizing Stanley symmetric functions, and described a combinatorial rule for expanding Schubert polynomials in this basis. We describe a decomposition of subword complexes into strata called slide complexes, that correspond to slide polynomials. The slide complexes are shown to be homeomorphic to balls or spheres. Gröbner degenerations of matrix Schubert varieties; Stanley symmetric functions Combinatorial aspects of simplicial complexes, Symmetric functions and generalizations, Reflection and Coxeter groups (group-theoretic aspects), Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus Slide complexes and subword complexes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper computes the characteristic cycles of intersection cohomology sheaves on transversal slices in a double affine Grassmannian. vanishing cycles; Poisson varieties; Uhlenbeck spaces; double affine Grassmannian Finkelberg, M. V.; Kubrak, D. V., Vanishing cycles on Poisson varieties, Funct. Anal. Appl.. Funct. Anal. Appl., Funktsional. Anal. i Prilozhen., 49, 2, 70-78, (2015), Translation of Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Grassmannians, Schubert varieties, flag manifolds, Loop groups and related constructions, group-theoretic treatment, Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects), Poisson manifolds; Poisson groupoids and algebroids, Vanishing theorems in algebraic geometry Vanishing cycles on Poisson varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This article proves a sharp version of Durfee's conjecture in the case of weighted homogeneous polynomials in 3 variables:
If \((V,0)\) is an isolated singularity defined in \(\mathbb{C}^ 3\) by a weighted homogeneous polynomial, then the inequality
\[
\mu-\nu+1 \geq 6Pg
\]
is valid, with equality iff we are in the homogeneous case. Here \(\mu\) is the Milnor number, \(\nu\) the multiplicity and \(Pg\) the geometric genus of \((V,0)\).
The signature \(\sigma\) of the Milnor fiber satisfies
\[
\sigma \leq-{\mu \over 3}-{2 \over 3} (\nu-1)
\]
(which implies \(\sigma \leq 0)\).
As a corollary, this gives, using K. Saito's classical characterization of weighted homogeneous isolated hypersurfaces singularities, a characterization of (biholomorphic) homogeneous isolated hypersurfaces singularities in \(\mathbb{C}^ 3\). surface singularity; Durfee's conjecture; weighted homogeneous polynomials; Milnor number; multiplicity; genus; signature Némethi, A.: Dedekind sums and the signature of \(f(x,y)+z^N\). II. Selecta. Math. (N.S.) \textbf{5}, 161-179 (1999) Complex surface and hypersurface singularities, Singularities of surfaces or higher-dimensional varieties, Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants Durfee conjecture and coordinate free characterization of homogeneous singularities | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author discusses several constructions related to the Mori theory of quartic double solids, i.e., Fano varieties of index 2 arising as double covers of \(\mathbb{P}^3\) branched over a smooth quartic. Several modifications associated to special points are studied in detail, and decompositions into birational transformations are given. Sarkisov link; Francia antiflip; quartic double solid Grinenko, M M, New Mori structures on a double space of index 2, Russ. Math. Surveys, 59, 573-574, (2004) Fano varieties, \(3\)-folds New Mori structures on a double space of index 2. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The notion of t-structure in a triangulated category was introduced by \textit{A. A. Beilinson} et al. [Astérisque 100, 172 p. (1982; Zbl 0536.14011)] as a a pair of full subcategories satisfying some axioms which guarantee that their intersection is an abelian category, called the heart of the t-structure. In the literature about t-structures a relevant question is the following: given a t-structure, what are the conditions that permit to assert that its heart is a Grothendieck category with good finiteness conditions? In the paper under review, the author tackle the question for the locally coherent condition, assuming that the t-structure lives in the (unbounded) derived category \({\mathcal D}({\mathcal G})\) of a Grothendieck category \({\mathcal G}\) which is itself locally coherent. More concretely, if the t-structure restricts to \({\mathcal D}^{b}(fp({\mathcal G}))\), the bounded derived category of the category of finitely presented (i.e., coherent) objects, in Proposition 4.5, the author gives a precise list of sufficient conditions on a t-structure in \({\mathcal D}({\mathcal G})\) so that its heart \({\mathcal H}\) is a locally coherent Grothendieck category on which \({\mathcal H}\cap {\mathcal D}^{b}(fp({\mathcal G}))\) is the class of its finitely presented objects. As a consequence the following results are proved: \textbf{1)} (see Theorem 5.2) Let \({\mathcal G}\) be a locally coherent Grothendieck category and \(\mathbf t =({\mathcal T}, {\mathcal F})\) be a torsion pair in \({\mathcal G}\). The associated Happel-Reiten-Smalø-t-structure in \({\mathcal D}({\mathcal G})\) restricts to \({\mathcal D}^{b}(fp({\mathcal G}))\) and has a heart which is a locally coherent Grothendieck category if, and only if, \({\mathcal F}\) is closed under taking direct limits in \({\mathcal G}\) and \(\mathbf t\) restricts to \(fp({\mathcal G})\); \textbf{2)} (see Theorem 6.2) If \(R\) is a commutative noetherian ring, then any compactly generated t-structure in \({\mathcal D}(R)\) which restricts to \({\mathcal D}^{b}_{fg}(R)\backsimeq {\mathcal D}^{b}(R-mod)\) has a heart \({\mathcal H}\) which is a locally coherent Grothendieck category on which \({\mathcal H}\cap {\mathcal D}^{b}_{fg}(R)\) is the class of its finitely presented objects. Finally, as a corollary (see Corollary 6.4) of this last theorem, the author proved that, if \(R\) is a commutative noetherian ring, then the heart of each t-structure in\({\mathcal D}^{b}_{fg}(R)\) is equivalent to the category of finitely presented objects of some locally coherent Grothendieck category. locally coherent Grothendieck category; triangulated category; derived category; t-structure; heart of a t-structure Saorín, M., On locally coherent hearts, Pac. J. Math., 287, 1, 199-221, (2017) Grothendieck categories, Derived categories, triangulated categories, Homological methods in commutative ring theory, Foundations of algebraic geometry, Homological methods in associative algebras On locally coherent hearts | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This lecture of the proceedings presents the first half of Tamagawa's proof of the Grothendieck's anabelian conjecture for hyperbolic affine curves defined over finite fields [for the statement of Tamagawa's theorem see also his original paper: \textit{A. Tamagawa}, Compos. Math. 109, 135-194 (1997; Zbl 0899.14007)]. The author first presents the whole circle of ideas concerning the anabelian geometry (a terminology introduced by Grothendieck in the early eightieth), which, roughly speaking, consists in recovering informations about a given algebraic variety from the structure of its algebraic fundamental group. Before explaining Tamagawa's theorem the author also discusses some important previous work done in anabelian geometry by several people such as F. K. Schmidt, J. Neukirch, K. Uchida, and later on, F. Pop, S. Mochizuki, V. Voevodsky and others.
[See also part II of this paper by \textit{D. Harari}, same proceedings, Prog. Math. 187, 203-216 (2000; see the following review Zbl 0978.14015)]. algebraic fundamental group; hyperbolic affine curves; anabelian geometry; Grothendieck's anabelian conjecture T. Szamuely, Le théorème de Tamagawa I. In Courbes semi-stables et groupe fondamental en géométrie algébrique (Luminy, 1998), 185-201, Progr. Math. 187, Birkhäuser, Basel, 2000. Zbl0978.14014 MR1768101 Homotopy theory and fundamental groups in algebraic geometry, Coverings of curves, fundamental group, Global ground fields in algebraic geometry The Tamagawa theorem. I | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We enumerate smooth and rationally smooth Schubert varieties in the classical finite types \(A\), \(B\), \(C\), and \(D\), extending Haiman's enumeration for type \(A\). To do this enumeration, we introduce a notion of staircase diagrams on a graph. These combinatorial structures are collections of steps of irregular size, forming interconnected staircases over the given graph. Over a Dynkin-Coxeter graph, the set of ``nearly-maximally labelled'' staircase diagrams is in bijection with the set of Schubert varieties with a complete Billey-Postnikov (BP) decomposition. We can then use an earlier result of the authors showing that all finite-type rationally smooth Schubert varieties have a complete BP decomposition to finish the enumeration. flag manifold; Schubert varieties; smoothness; rational smoothness; Billey-Postnikov decomposition; Coxeter group; enumeration; generating function Edward Richmond and William Slofstra, Staircase diagrams and enumeration of smooth Schubert varieties. J. Combin. Theory Ser. A 150 (2017), 328--376. Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds, Reflection and Coxeter groups (group-theoretic aspects) Staircase diagrams and enumeration of smooth Schubert varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The differential equation on \(Z\) is considered in the paper as a hypersurface \(H\subseteq{\mathcal D}_k^rZ\) in a kind of iterated Grassmannian of the tangent bundle: \(D^0_kZ=Z\), \(D^1_kZ= \text{Gr}_k TZ,\dots,D^r_k Z\subset \text{Gr}_k TD_n^{r-1}Z\), defined in a way that one can call \(D_k^rZ\) the infinitesimal data of dimension \(k\) and order \(r\). Natural analogues of Schubert cycles \(C^r_i\) are then defined in \(D_k^r\mathbb{P}^n\), and it is shown that \(C^r_0\), \(C^r_1, \dots,C_r^r\) is a basis for \(A^1D_k^r \mathbb{P}^n\), the group of divisors of \(D_k^r \mathbb{P}^n\). For a subvariety \(S\subset \mathbb{P}^n\) the divisor of solutions of \(H\) is the push-forward to \(\mathbb{P}^n\) of the intersection \(D^r_kS\cap H\). It is shown how to find its degree if the decomposition of \(H\) with respect to \(C_0^r, C_1^r, \dots, C^r_r\) and some numerical characteristics of \(D^r_kS\) are known. divisor of solutions of a differential equation; Grassmannians of a tangent bundle; intersection numbers; Schubert cycles; divisor of solutions Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Grassmannians, Schubert varieties, flag manifolds, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Divisors, linear systems, invertible sheaves Degree of the divisor of solutions of a differential equation on a projective variety | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(V\) be a vector space of dimension \(n\) on an algebraically closed field \(K\). Let \(G = Gr(\ell,V) = Gr(\ell, n)\) be the Grassmann variety of subspaces of \(V\) of dimension \(\ell\). If \(V'\) is a linear subspace of \(V\), we denote, as usual, by \(\sigma (V') = \{\Lambda \in G | \Lambda \cap V' \neq 0\}\) the special Schubert variety associated to \(V'\). We prove the following:
Theorem. The intersection \(\bigcap \sigma (V_ j)\) of the special Schubert varieties associated to linear subspaces \(V_ j\), \(j = 1, \dots, m\), of dimension \(n - \ell - a_ j + 1\) such that \(\ell (n - \ell) - \sum^{j=m}_{j=1} a_ j > 0\), is connected. Moreover, the intersection is irreducible of dimension \(\ell (n - \ell) - \sum a_ j\) for a general choice of the subspaces \(V_ j\).
The second part of the theorem follows from its first part and the local irreducibility. \textit{J. Hansen} [Am. J. Math. 105, 633-639 (1983; Zbl 0544.14034)] proved a result about the connectedness of intersections of subvarieties of Grassmannians and flag varieties, but under the stronger hypothesis that the expected codimension of the intersection is smaller than \(n - 1\). As initial motivation for this work, let us remark that a particular case \((\dim V_ j = 2\) for all \(j)\) of this statement appears, without a proof, in a book by \textit{F. Enriques} and \textit{O. Chisini} [``Lezioni sulla teoria geometrica delle equazioni: e delle funzioni algebriche'' Vol. III (1924; reprint 1985; Zbl 0571.51001), page 524]. We conjecture that the irreducibility holds for intersections of Schubert varieties (not necessarily special), when they are in general position with nonempty intersection. (We have checked the conjecture for the case of two Schubert varieties.)
As an example of application we provide a new proof (over the field of complex numbers) of the following theorem. Recall that a curve of genus \(g\) is said \((d,r)\)-Brill-Noether if the variety of linear series of degree \(d\) and dimension \(r\) on \(C\) has dimension equal to the Brill- Noether number \(((r + 1)d - rg - r(r + 1) + r)\) and is empty if the number is negative. The general curve of a given genus is \((d,r)\)-Brill- Noether for all \((d,r)\).
Theorem. If the Brill-Noether number is strictly positive, then the varieties of special divisors of degree \(d\) and dimension \(r\) on a smooth, connected, \((d,r)\)-Brill-Noether curve \(C\) of genus \(g\) are connected. connectedness of intersections of special Schubert varieties; \((d,r)\)- Brill-Noether curve Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Topological properties in algebraic geometry Connectedness of intersections of special Schubert varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In \textit{T. Tao} et al. [J. Am. Math Soc. 17, No. 1, 19--48 (2001; Zbl 1043.05111)] the authors introduced a new rule called a puzzle rule for computing Schubert calculus. In this paper the authors give an independent and nearly self-contained proof of the puzzle rule, they also give a formula for equivariant Schubert calculus on Grassmannians that is manifestly positive in the sense of \textit{W. Graham} [Duke Math. J. 109, 599--614 (2001; Zbl 1069.14055)]. In section 1 the authors state their main result as theorem 2. In section 2, the authors give a combinatorial definition of the equivariant cohomology ring of Gr\(_k(\mathbb C^n)\) stated in section 2.1. In section 2.2 they introduce well known facts about T-equivariant cohomology to apply it in section 2.3 to construct equivariant Schubert classes. In section 2.4 they show that the equivariant Schubert classes form a basis for the equivariant cohomology ring of Gr\(_k(\mathbb C^n)\), concluding in lemma 2 with some properties of the structure constants. In section 3 they state in proposition 2 an equivariant version of the Pieri rule and in theorem 3 they provide a recurrence relation on the structure constants. In corollary 1 and lemma 4 of the same section they introduce four identities numbered (1)-(4). In section 4 they prove indentities (1) and (4). In section 5 they prove identity (3). Finally in section 6 they compare their results with those of \textit{A. I. Molev} and \textit{B. Z. Sagar} [Trans. Am. Math. Soc. 351, 4429--4443 (1999; Zbl 0972.05053)] for multiplying factorial Schur functions (which are equivariant Schubert polynomials for Grassmannian permutations). They introduce cohomological formulations of their problem and a reformulation of their rule in terms of ``Molev-Sagan puzzles''. In the appendix to this paper they extend the standard combinatorial proof of the existence of Schubert classes to equivariant Schubert classes. classical problems, Schubert calculus, groups acting on specific manifolds, equivariant algebraic topology of manifolds Allen Knutson & Terence Tao, ``Puzzles and (equivariant) cohomology of Grassmannians'', Duke Math. J.119 (2003) no. 2, p. 221-260 Classical problems, Schubert calculus, Groups acting on specific manifolds, Equivariant algebraic topology of manifolds Puzzles and (equivariant) cohomology of Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The book aims to describe the beautiful connection between Schubert varieties and their Standard Monomial Theory (SMT) on the one hand and Classical Invariant Theory (CIT) on the other. The roots of SMT are to be found in the work of Hodge, who described nice bases for the homogeneous coordinate ring of Schubert varieties of the Grassmannian in the Plücker embedding (over a field of characteristic zero). Grassmannians being precisely the homogeneous spaces that arise as quotients of special linear groups by maximal parabolic subgroups, it is natural to try to generalize Hodge's approach to projective embeddings of other spaces \(G/Q\), where \(G\) is a semisimple algebraic group and \(Q\) a parabolic subgroup. In the early '70s Seshadri initiated this generalization and called it SMT.
CIT concerns diagonal actions of classical groups on direct sums of the tautological representations and their duals. A description of the algebra of invariants for these actions comprises of two theorems. The First Fundamental Theorem specifies a finite set of generators for the algebra of invariants, and the Second Fundamental Theorem provides a finite set of generators of the ideal of relations among the algebra generators. A central role here plays the article [\textit{C.~De Concini} and \textit{C.~Procesi}, ``A characteristic free approach to Invariant Theory'', Adv. Math. 21, 330--354 (1976; Zbl 0347.20025)], where the First Fundamental Theorem for all classical groups and the Second Fundamental Theorem for general, orthogonal and symplectic linear groups were obtained (the case when the characteristic of the ground field is zero goes back to H.~Weyl). The idea to use SMT in the proof of the First and the Second Fundamental Theorems appeared in [\textit{V.~Lakshmibai} and \textit{C. S.~Seshadri}, Proc. Indian Acad. Sci., Sect. A 87, No. 2, 1--54 (1978; Zbl 0447.14011)] and turned out to be very fruitful. More precisely, one should realize the subalgebra of the algebra of invariants generated by ``basic'' invariants (which will in fact coincide with the algebra of invariants) as the algebra of regular functions on an affine variety related to a Schubert variety. Then there is a morphism from the spectrum of the algebra of invariants to this affine variety. Using Zariski's Main Theorem, one shows that this is an isomorphism. A difficult part of this program is to prove that our affine variety is normal. Normality follows from normality of Schubert varieties, and that is a consequence of SMT. Nowadays this approach is realized in complete generality, and the book under review provides an excellent account of there results.
The authors tried to make the presentation self-contained keeping in mind the needs of prospective graduate students and young researchers. After a detailed introduction, generalities on algebraic varieties and algebraic groups are given. Next chapters are devoted to classical, symplectic and orthogonal Grassmannians, determinantal varieties, Geometric Invariant Theory (GIT), basic results of SMT and their interrelations with CIT. The proof of the main theorem of SMT is given in an appendix. The authors also included some important applications of SMT: to the determination of singular loci of Schubert varieties, to the study of some affine varieties related to Schubert varieties --- ladder determinantal varieties, quiver varieties, varieties of complexes, etc. --- and to toric degenerations of Schubert varieties.
The book may be recommended as a nice introduction to SMT and related active research areas. It may be used for a year long course on Invariant Theory and Schubert varieties. Classical Invariant Theory; Grassmannians; Schubert varieties; homogeneous coordinates Lakshmibai, V.\!; Raghavan, K.\,N.\!, Standard monomial theory, Encyclopaedia of Mathematical Sciences (Invariant Theory and Alg. Transform. Groups VIII) 137, (2008), Springer-Verlag, Berlin Grassmannians, Schubert varieties, flag manifolds, Actions of groups on commutative rings; invariant theory, Rings with straightening laws, Hodge algebras, Determinantal varieties, Classical groups (algebro-geometric aspects) Standard monomial theory. Invariant theoretic approach | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The motivation of introducing the pro-étale topology is as follows: For a scheme \(X\), there are the étale cohomology groups \(\mathrm{H}^i(X_{\mathrm{et}},\mathbb{Q}_\ell)\) defined indirectly as \(\varprojlim_n\mathrm{H}^i(X_{\mathrm{et}},\mathbb{Z}/\ell^n) \otimes_{\mathbb{Z}_\ell} \mathbb{Q}_\ell\). The naive definition as \(\mathrm{H}^i(X_{\mathrm{et}},\mathbb{Q}_\ell)\) with \(\mathbb{Q}_\ell\) considered as a constant sheaf does not give the correct result, since e.\,g.\ for \(X\) a smooth projective connected curve over an algebraically closed field, one would have \(\mathrm{H}^i(X_{\mathrm{et}},\mathbb{Q}_\ell) = \mathbb{Q}_\ell\) for \(i = 0\) and \(= 0\) for \(i > 0\), which is not a good Weil cohomology theory.
The authors introduce a new Grothendieck topology \(X_{\mathrm{proet}}\) for a scheme \(X\) whose underlying topological space is Noetherian. The proétale site \(X_{\mathrm{proet}}\) is the site of weakly étale \(X\)-schemes, with covers given by fpqc covers. A morphism \(f: Y \to X\) is called weakly étale if \(f\) is flat and \(\Delta_f: Y \to Y \times_X Y\) is flat. (Definition 1.2) A morphism of rings is étale iff it is weakly étale and finitely presented, and an ind-étale morphism of rings is weakly étale. (Theorem 1.3) For a ring \(A\), the site of weakly étale \(A\)-algebras is equivalent to the site of ind-étale \(A\)-algebras.
A sheaf \(L\) of \(\bar{\mathbb{Q}}_\ell\)-modules on \(X_{\mathrm{proet}}\) is lisse if it is locally free of finite rank. A sheaf \(C\) of \(\bar{\mathbb{Q}}_\ell\)-modules on \(X_{\mathrm{proet}}\) is constructible if there is a finite stratification \(\{X_i \to X\}\) into locally closed subsets \(X_i \subseteq X\) such that \(C|_{X_i}\) is lisse. (Definition 1.1) An object \(K \in \mathrm{D}(X_{\mathrm{proet}}, \bar{\mathbb{Q}}_\ell)\) is constructible if it is bounded and all cohomology sheaves are constructible. Let \(\mathrm{D}_{\mathrm{cons}}(X_{\mathrm{proet}}, \mathbb{Q}_\ell)\) be the corresponding full triangulated subcategory.
Let \(x\) be a geometric point of a locally topologically Noetherian connected scheme \(X\) with \(\mathrm{ev}_x: \mathrm{Loc}_X \to \mathrm{Set}\) the associated functor \(F \mapsto F_x\), with \(\mathrm{Loc}_X\) the full subcategory of locally constant sheaves \(F \in \mathrm{Shv}(X_{\mathrm{proet}})\), i.\,e.\ such that there exists a cover \(\{X_i \to X\}\) in \(X_{\mathrm{proet}}\) with \(F|_{X_i}\) constant. The pro-étale fundamental group \(\pi_1^{\mathrm{proet}}(X,x)\) is defined as \(\mathrm{Aut}(\mathrm{ev}_x)\) using the compact-open topology on \(\mathrm{Aut}(S)\) for any set \(S\). (Definition 7.4.2) The category \(\mathrm{Loc}_X\) is equivalent to the category \(\mathrm{Cov}_X\) of étale \(X\)-schemes which satisfy the valuative criterion of properness. (Theorem 1.10)
Then: \(\mathrm{H}^i(X_{\mathrm{et}},\mathbb{Q}_\ell) = \mathrm{H}^i(X_{\mathrm{proet}},\mathbb{Q}_\ell)\). The full triangulated subcategory \(\mathrm{D}_{\mathrm{cons}}(X_{\mathrm{proet}}, \mathbb{Q}_\ell) \subseteq \mathrm{D}(X_{\mathrm{proet}}, \mathbb{Q}_\ell)\) of constructible objects is stable under Grothendieck's six operations, and there is a natural equivalence between \(\mathrm{D}_{\mathrm{cons}}(X_{\mathrm{proet}}, \mathbb{Q}_\ell)\) and \(\mathrm{D}^{\mathrm b}_{\mathrm c}(X,\mathbb{Q}_\ell)\). There is an equivalence between the category of finite dimensional continuous representations of \(\pi_1^{\mathrm{proet}}(X,x)\) and that of lisse \(\mathbb{Q}_\ell\)-sheaves on a locally topologically Noetherian connected scheme \(X\). If \(X\) is geometrically unibranch, \(\pi_1^{\mathrm{proet}}(X,x) \cong \pi_1^{\mathrm{et}}(X,x)\). (Lemma 7.4.10)
Most of the things above work for any local field \(E/\mathbb{Q}_\ell\) instead of \(\mathbb{Q}_\ell\). étale cohomology; higher regulators; zeta and L-functions; Grothendieck topologies; coverings; fundamental group Bhatt, Bhargav; Scholze, Peter, The pro-étale topology for schemes, Astérisque, 369, 99-201, (2015) Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects), Grothendieck topologies and Grothendieck topoi, Coverings of curves, fundamental group The pro-étale topology for schemes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors' goal is to compute the Chern-Schwartz-MacPherson and Segre-Schwartz-MacPherson classes of the orbits of a certain representation called the matrix Schubert cells. More precisely, let \(k\le n\) be nonnegative integers. Let us consider the group \(GL_k(\mathbb C)\times B_n^-\) acting on \(\text{Hom}(\mathbb C^k; \mathbb C^n)\) by \((A,B)\cdot M = BMA^{-1}\), where \(B_n^-\) denotes the Borel subgroup of \(n\times n\) lower triangular matrices. The finitely many orbits of this action are parametrized by \(d\)-element subsets \(J = \{j_1 < \dots < j_d\} \subset \{1, \dots, n\}\) where \(d \le k\). The corresponding orbits, denoted by \(\Omega_J\), are called matrix Schubert cells and their closures are usually called matrix Schubert varieties (see [\textit{L. M. Fehér} and \textit{R. Rimányi}, Cent. Eur. J. Math. 1, No. 4, 418--434 (2003; Zbl 1038.57008)]). Let us fix \(I\subset \{1,\dots,k\}\). The authors prove that the equivariant Chern-Schwartz-MacPherson class of \(\Omega_I\) is equal to the value of the weight function \(W_I(\alpha, \beta)\), considered in [\textit{V. Tarasov} and \textit{A. Varchenko}, Invent. Math. 128, No. 3, 501--588 (1997; Zbl 0877.33013)] or [\textit{R. Rimányi} and \textit{A. Varchenko}, Impanga 15. EMS Series of Congress Reports 225--235 (2018; Zbl 1391.14108)], where \(\alpha = (\alpha_1, \dots, \alpha_k)\) and \(\beta = (\beta_1, \dots, \beta_n)\) are suitable partitions. In turn, the weight function \(W_I(\alpha, \beta)\) can be expressed in terms of appropriated symmetric functions, which can be computed as values of the ``iterated residues'' of some generating functions parameterized by partitions, and so on. In a similar same way it is possible to compute the Segre-Schwartz-MacPherson classes. As an application, the authors perform the corresponding calculations and write out the exact formulas for these classes in the case of \(A_2\) quiver representation. characteristic classes; equivariant cohomology; Borel subgroup; symmetric functions; fundamental class; degeneracy loci; weight functions; Schubert cells; Schur expansion; iterated residues Grassmannians, Schubert varieties, flag manifolds, Global theory of complex singularities; cohomological properties, Global theory and resolution of singularities (algebro-geometric aspects), Classical problems, Schubert calculus, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Characteristic classes and numbers in differential topology Chern-Schwartz-MacPherson classes of degeneracy loci | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(p\) be an odd prime and \(m\) be a positive integer, \(q=p^m\), \(\mathcal{R} = \mathbb{F}_q + u\mathbb{F}_q\) with \(u^2 = u\), and \(\mathcal{S} = \mathbb{F}_q + u\mathbb{F}_q + v\mathbb{F}_q\) with \(u^2 = u\), \(v^2=v\), \(uv=vu=0\) and \(\Lambda = (\lambda_1, \lambda_2, \lambda_3)\in\mathbb{F}_q\mathcal{R}\mathcal{S}\). In this paper, we study the algebraic structure of constacyclic codes over \(\mathcal{R}\) and \(\mathcal{S}\). Further, we discuss the structure of \(\mathbb{F}_q\mathcal{R}\mathcal{S}\)-\(\Lambda\)-constacyclic codes of block length \((\alpha, \beta, \gamma)\). This family of codes can be viewed as \(\mathcal{S}[x]\)-submodules of \(\frac{\mathbb{F}_q[x]}{\langle x^\alpha - \lambda_1\rangle}\times\frac{\mathcal{R}[x]}{\langle x^\beta - \lambda_2\rangle}\times\frac{\mathcal{S}[x]}{\langle x^\gamma - \lambda_3\rangle}\). The generator polynomials of this family of codes are discussed. As application, we discuss the construction of quantum error-correcting codes (QECCs) from constacyclic codes over \(\mathbb{F}_q\mathcal{R}\mathcal{S}\) and obtain several new QECCs from this study. constacyclic codes; generator polynomials; separable codes; QECCs Quantum computation, Linear codes (general theory), Other types of codes, Algebraic coding theory; cryptography (number-theoretic aspects), Applications to coding theory and cryptography of arithmetic geometry Constacyclic codes over mixed alphabets and their applications in constructing new quantum codes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials If \(W\) is a vector space of dimension \(m\) and \(V\) is the direct sum of \(n\) copies of \(W\) then \(GL_ n\times GL_ m\) acts naturally on \(V\) and on a corresponding polynomial ring \(A\). The last ring decomposes as \(A=\oplus_ \lambda V_ \lambda(n)\otimes V^*_ \lambda(m)\), where, if \(m\leq n\), \(\lambda\) runs over all the Young diagrams with at most \(m\) columns. Therefore if \(G\leq GL_ m\) is a subgroup, the ring \(A^ G\) of the \(G\)-invariant polynomials decomposes, with respect to \(GL_ n\) as \(A^ G=\oplus_ \lambda V_ \lambda(n)\otimes V^*_ \lambda(m)^ G\). The author computes the multiplicities \(\mu_ \lambda\) of \(V_ \lambda(n)\) in \(A^ G\) in the case where \(V=M^ 0_ 3\) is the space of \(3\times 3\) matrices over \(\mathbb{C}\) with trace 0. The computation is complete in the sense that the author has found a rational function, which expresses the corresponding Poincaré series. general linear group; Young diagrams; invariant polynomials; Poincaré series Actions of groups on commutative rings; invariant theory, Geometric invariant theory, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series Poincaré series of n-tuples of \(3\times3\) matrices | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The classical Chevalley-Warning theorem in number theory states that a certain set of polynomial equations has a solution over a finite field as long as the total degree of the polynomials is smaller than the number of variables. The paper under review proposes a conjecture, which can be viewed as a geometric version of the Chevalley-Warning theorem. Given an algebraically closed base field \(k\), let \(K_0(var)\) be the Grothendieck ring of varieties and let \(\mathbb{L}\) be the class of \(\mathbb{A}_k^1\). The conjecture predicts that in \(K_0(var)\) the class of a variety, which is defined by a set of homogeneous polynomials of total degree less than the number of variables, is congruent to 1 modulo \(\mathbb{L}\). By a result of \textit{M. Larsen} and \textit{V. A. Lunts} [Mosc. Math. J. 3, No. 1, 85--95 (2003; Zbl 1056.14015)], the property of being ``congruent to 1 modulo \(\mathbb{L}\)'' is essentially the same as being stably rational. Hence the geometric Chevalley-Warning becomes interesting in the study of rationality problems. The author gives some examples of low degree hypersurfaces being conguent to 1 modulo \(\mathbb{L}\). These includes quadratic hypersurfaces and singular cubic hypersurfaces. stable birational equivalence; hypersurface of low degree; Grothendieck ring of varieties Rationality questions in algebraic geometry, Varieties of low degree, Computational aspects of algebraic surfaces Stable birational equivalence and geometric Chevalley-Warning | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The problem of constructing pairing-friendly elliptic curves is the key ingredients for implementing pairing-based cryptographic systems. In this paper, we aim at constructing such curves with \(\rho =1\). By offering a more generalized concept ``parameterized families'', we propose a method for constructing parameterized families of pairing-friendly elliptic curves which can naturally include many existent (and even more new) families of curves without exhaustive survey. We demonstrate the utility of the method by constructing concrete parameterized family in the cases of embedding degree 3, 4 and 6. An interesting result is proved that all the possible quadratic families of pairing-friendly elliptic curves of desired embedding degrees satisfying \(\rho =1\) have been covered in our parameterized families. As a by-product, we also revisit the supersingular elliptic curves from a new perspective. elliptic curves; pairing based cryptography; cyclotomic polynomials; parameterized families Cryptography, Applications to coding theory and cryptography of arithmetic geometry, Elliptic curves On constructing parameterized families of pairing-friendly elliptic curves with \(\rho =1\) | 0 |
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