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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 The study of Schubert polynomials is an important and interesting subject in algebraic combinatorics. One of the possible methods for studying Schubert polynomials is through the modules introduced by \textit{W. Kraskiewicz} and \textit{P. Pragacz} [C. R. Acad. Sci., Paris, Sér. I 304, 209--211 (1987; Zbl 0642.13011)]. In this paper the authors show that any tensor product of Kraśkiewicz-Pragacz modules admits a filtration by Kraśkiewicz-Pragacz modules. This result can be seen as a module-theoretic counterpart of a classical result that the product of Schubert polynomials is a positive sum of Schubert polynomials, and gives a new proof to this classical fact. Schubert polynomials; Schubert functors; Kraśkiewicz-Pragacz modules; Schubert calculus Watanabe, M.: Tensor product of kraśkiewicz-pragacz modules. J. algebra 443, 422-429 (2015) Classical problems, Schubert calculus Tensor product of Kraśkiewicz-Pragacz modules | 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 purpose of this paper is to show that the multiplication of a Schubert polynomial of finite type \(A\) by a Schur function can be understood from the multiplication in the space of dual \(k\)-Schur functions. Using earlier work by the second author, we encode both problems by means of quasisymmetric functions. On the Schubert vs. Schur side, we study the \(r\)-Bruhat order given by Bergeron-Sottile, along with certain operators associated to this order. On the other side, we connect this poset with a graph on dual \(k\)-Schur functions given by studying the affine Grassmannian order of Lam-Lapointe-Morse-Shimozono. Also, we define operators associated to the graph on dual \(k\)-Schur functions which are analogous to the ones given for the Schubert vs. Schur problem. Schubert polynomials; \(k\)-Schur functions; affine Grassmannian; \(r\)-Bruhat order; strong order Symmetric functions and generalizations, Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds Schubert polynomials and \(k\)-Schur functions (extended abstract). | 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 derive an explicit formula, with no cancellations, for expanding in the basis of Grothendieck polynomials the product of two such polynomials, one of which is indexed by an arbitrary permutation, and the other by a simple transposition; hence, this is a Monk-type formula, expressing the hyperplane section of a Schubert variety in \(K\)-theory. Our formula is in terms of increasing chains in the \(k\)-Bruhat order on the symmetric group with certain labels on its covers. An intermediate result concerns the multiplication of a Grothendieck polynomial by a single variable. As applications, we rederive some known results, such as Lascoux's transition formula for Grothendieck polynomials. Our results are reformulated in the context of recently introduced Pieri operators on posets and combinatorial Hopf algebras. In this context, we derive an inverse formula to the Monk-type one, which immediately implies a new formula for the restriction of a dominant line bundle to a Schubert variety. Grothendieck polynomials; Lascoux's transition formula; posets Cristian Lenart, A \?-theory version of Monk's formula and some related multiplication formulas, J. Pure Appl. Algebra 179 (2003), no. 1-2, 137 -- 158. Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Geometric applications of topological \(K\)-theory A \(K\)-theory version of Monk's formula and some related multiplication formulas | 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 maximal flag of \({\mathbb R}^n\) is a sequence \(F=(F_1\subset\cdots\subset F_n)\), where \(F_i\) is a subvector space of \({\mathbb R}^n\) of dimension \(i\). Two such flags \(F\) and \(G\) are called transversal, if
\[
\dim F_i\cap G_j = \begin{cases} i+j-n,&\text{ if }i+j\geq n,\\ 0, &\text{ if }i+j<n.\end{cases}
\]
The set \(U_F\) of all flags transversal to \(F\) is an open Schubert cell. The main result of the paper identifies the number \(N_n\) of connected components of the intersection \(U_F\cap U_G\) for two transversal flags \(F\) and \(G\), with the number of orbits of a certain finite group acting on some \({\mathbb F}_2\)-vector space. For \(n=3\), \(4\) and \(5\) one gets \(6\), \(20\) and \(52\) for \(N_n\), respectively. Based on computer calculations, the authors conjecture that this number should be \(N_n=3\cdot 2^{n-1}\) if \(n>5\). maximal flag; open Schubert cell; Kazhdan-Lusztig polynomials; intersection of transversal flags B. Shapiro, M. Shapiro, and A. Vainshtein, Connected components in the intersection of two open opposite Schubert cells in \?\?_{\?}(\?)/\?, Internat. Math. Res. Notices 10 (1997), 469 -- 493. Grassmannians, Schubert varieties, flag manifolds Connected components in the intersection of two open opposite Schubert cells in \(SL_n(\mathbb{R})/B\) | 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 0517.00010.]
This work is a short version of a paper which will be published by the author and \textit{J. L. Verdier} in Ann. Sci. Éc. Norm. Supér., IV. Sér. 16, 409-449 (1983; Zbl 0538.14033). The results are given without proofs. Let G be a finite subgroup of SL(2,\({\mathbb{C}})\) and S the surface obtained as the a quotient \(S={\mathbb{C}}^ 2/G\), having the origin as double rational singularity. If \(q:\widetilde S\to S\) is a minimal resolution of the singularity, D the exceptional fiber of q and Irr(D) the union of the irreducible components of D, there are well-known results of M. Artin connecting the group G and the dual graph \(\Gamma\) of Irr(D). In this paper are considered the ring R(G) of the representations of G and the Grothendieck ring \(K(\widetilde S)\) and are announced the following results: (1) The rings R(G) and \(K({\mathbb{C}}^ 2,0)\) are isomorphic. - (2) There is a bijection \(\pi *R(G)\to K(\tilde S)\). - (3) If \(c\in R(G)\) is the canonical representation there is a ring homomorphism \(R(G)/(2-c)R(G)\to K(\tilde S)/K_ D(\tilde S)\). Grothendieck ring; representations of group of automorphisms; double rational singularity Singularities of surfaces or higher-dimensional varieties, Group actions on varieties or schemes (quotients), Representation theory for linear algebraic groups, Homogeneous spaces and generalizations, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) Representation of polyhedral groups and singularities of surfaces | 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 \textit{A. S. Buch} and \textit{R. Rimányi} [C. R., Math., Acad. Sci. Paris 339, No. 1, 1--4 (2004; Zbl 1051.14062)] proved a formula for a specialization of double Grothendieck polynomials based on the Yang-Baxter equation related to the degenerate Hecke algebra. A geometric proof was found by \textit{A. Woo} and \textit{A. Yong} [Am. J. Math. 134, No. 4, 1089--1137 (2012; Zbl 1262.13044)] by constructing a Gröbner basis for the Kazhdan-Lusztig ideals. In this note, we give an elementary proof for this formula by using only divided difference operators. Buch-Rimányi formula; double Grothendieck polynomial; specialization Combinatorial aspects of algebraic geometry, Classical problems, Schubert calculus, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) A note on specializations of Grothendieck 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 In this nice survey, the author presents recent development in the theory of real analytic singularities with particular emphasis on the use of analytic arcs and the structure of the spaces of such arcs.
Arc-symmetric sets and arc-analytic functions were introduced by \textit{K. Kurdyka} [Math. Ann. 282, No. 3, 445--462 (1988; Zbl 0686.14027)]. Recall that a subset \(A\) of a real analytic manifold \(M\) is said to be \textit{arc-symmetric} if for any analytic arc \(\gamma : (-1, 1) \rightarrow M\) with \(\gamma (-1, 0) \subset A\) it holds that \(\gamma (-1, 1) \subset A.\) A map \(f : M \rightarrow N\) between real analytic manifolds \(M\) and \(N\) is called \textit{arc-analytic} if \(f \circ \gamma\) is analytic for every analytic arc \(\gamma.\)
Section~2 contains the background on analytic arcs, arc-symmetric sets and arc-analytic functions.
Section~3 presents the virtual Betti numbers and virtual Poincaré polynomial for a large class of subsets of finite dimensional affine spaces such as algebraic sets and arc-symmetric semi-algebraic sets.
In Section~4, the author studies map germs which become real analytic after composing with a locally finite number of blowing-ups.
In Section 5 the author discusses relationships between the notions of arc analytic map and of blow analytic map. Lipschitz properties of arc- and blow- analytic maps are provided also in this section.
In Section~6 the author describes phenomena of analytic arcs in the theory of hyperbolic polynomials with analytic coefficients and analytic families of symmetric (and antisymmetric) matrices.
This survey is very well written and gives a nice introduction for a wide audience to modern real algebraic and analytic geometry. real algebraic sets; arc-symmetric sets; spaces of arcs; Grothendieck ring; virtual Bett numbers; zeta functions; classification of analytic germs; hyperbolic polynomials Nash functions and manifolds, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Germs of analytic sets, local parametrization, Singularities of surfaces or higher-dimensional varieties Analytic arcs and real analytic 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 Hall algebra \(\mathbf H_X\) of the category of coherent sheaves on an elliptic curve \(X\) defined over a finite field \(\mathbb F_l\) contains a natural `spherical' subalgebra \(\mathbf U_X^+\) which is a two-parameter deformation of the ring of diagonal invariants
\[
\mathbf R_n^+=\mathbb C[x_1,\dots,x_n,y_1^{\pm 1},\dots,y_n^{\pm 1}]^{\Sigma_n},
\]
where \(\Sigma_n\) acts simultaneously on the \(x\)-variables and the \(y\)-variables. For any \(n\geq 1\) the authors construct a surjective algebra homomorphism between the Drinfeld double \(\mathbf{DU}_X^+\) and the spherical subalgebra of Cherednik's double affine Hecke algebra of type \(\text{GL}_n\). This leads to a geometric construction of the Macdonald polynomials \(P_\lambda(q,t^{-1})\) in terms of certain Eisenstein series on the moduli space of semistable vector bundles on \(X\). elliptic curves; finite fields; Hall algebras; Cherednik Hecke algebras; Macdonald polynomials; double affine Hecke algebras Schiffmann (O.), and Vasserot (E.).â The elliptic Hall algebra, Cherednik Hecke algebras and Macdonald polynomials, Compos. Math. 147, p. 188-234 (2011). Hecke algebras and their representations, Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.), Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Quantum groups (quantized enveloping algebras) and related deformations The elliptic Hall algebra, Cherednik Hecke algebras and Macdonald 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 This paper relates a formula for quiver varieties found by \textit{A. S. Buch} and \textit{W. Fulton} [Invent. Math. 135, 665-687 (1999; Zbl 0942.14027)] to Stanley symmetric functions. A Stanley symmetric function is a symmetric homogeneous power series in infinitely many variables used to determine how many reduced words a given permutation has. The connection between the two is found using Schubert polynomials. Let \(w\in S_{m}.\) A Stanley symmetric function has been shown to be a limit of Schubert polynomials [\textit{A. Lascoux} and \textit{M.-P. Schützenberger}, C. R. Acad. Sci., Paris, Sér. I 294, 447-450 (1982; Zbl 0495.14031) and \textit{I. G. MacDonald}, Lond. Math. Soc. Lect. Note Ser. 166, 73-99 (1991; Zbl 0784.05061)]: Such a polynomial can be written using Schur functions as a basis---the coefficients are denoted \(\alpha _{w\lambda }\) where \(\lambda \) is a partition of the length of \(w\). The quiver variety formula given in the author's previous work specializes to the double Schubert polynomial for the permutation \(w.\) The coefficients of this double Schubert polynomial are denoted \(c_{w}(a,b,\lambda),\) where \(a\) and \(b\) are exponents. The \(c_{w}(a,b,\lambda)\) are special cases of Littlewood-Richardson coefficients. The author shows that the constant term \(c_{w}(0,0,\lambda)\) corresponds to Stanley's coefficient \(\alpha _{w^{-1}\lambda },\) thereby directly relating the double Schubert polynomial and the symmetric function. The first half of the paper is a review of the ideas found in the author's previous paper cited above.
After proving that \(c_{w}(0,0,\lambda)=\alpha _{w^{-1}\lambda },\) the results are related to the Littlewood-Richardson conjecture, a conjecture which states that generalized Littlewood-Richardson coefficients are nonnegative. quiver varieties; Stanley symmetric functions; Schubert polynomials; Schur functions; Littlewood-Richardson conjecture; Littlewood-Richardson coefficients A.S. Buch, ''Stanley symmetric functions and quiver varieties,'' J. Algebra 235 (2001), 243--260. Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds Stanley symmetric functions and quiver 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 We introduce an algorithm to describe Pieri's Rule for multiplication of Schubert polynomials. The algorithm uses tower diagrams introduced by the authors and another new algorithm that describes Monk's rule. Our result is different from the well-known descriptions (and proofs) of the rule by Bergeron-Billey and Kogan-Kumar and uses Sottile's version of Pieri's Rule. Schubert polynomials; Pieri's rule; Monk's rule; tower diagram Combinatorial aspects of representation theory, Symmetric functions and generalizations, Classical problems, Schubert calculus Tower diagrams and Pieri's rule | 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 ring $K_0(\mathrm{Var}/{\mathbb C})$ is defined as the quotient of the group of formal linear combinations with integer coefficients modulo the relations $[Z]-[U]-[Z\backslash U] $ for all open subvarieties $U\subseteq Z.$ This is a ring with the product induced by the Cartesian product. The main result of the paper asserts that the class of the affine line is a zero divisor in the Grothendieck ring of varieties over ${\mathbb C}.$ The question is important since it has significant consequences. One of them is that a rational smooth cubic fourfold in ${\mathbb P}^5$ must have its Fano variety of lines birational to a symmetric square of a $K3$ surface (cf. [\textit{S. Galkin} and \textit{E. Shinder}, ``The Fano variety of lines and rationality problem for a cubic hypersurface'', Preprint, \url{arXiv:1405.5154}]). The other consequence is the fact that cut-and-paste conjecture of Larsen and Lunts fails (cf. Zbl [\textit{M. Larsen} and \textit{V. A. Lunts}, Mosc. Math. J. 3, No. 1, 85--95 (2003; Zbl 1056.14015)]). The proof of the main theorem is based on the Pfaffian-Grassmannian double mirror correspondence. Grothendieck ring; affine line; Pfaffian-Grassmanian double mirror correspondence Applications of methods of algebraic \(K\)-theory in algebraic geometry The class of the affine line is a zero divisor in the Grothendieck ring | 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 ``flip-and-reversal'' involution arising in the study of quasisymmetric Schur functions provides a passage between what we term ``Young'' and ``reverse'' variants of bases of polynomials or quasisymmetric functions. Building on this perspective, which has found recent application in the study of \(q\)-analogues of combinatorial Hopf algebras and generalizations of dual immaculate functions, we develop and explore Young analogues of well-known bases for polynomials. We prove several combinatorial formulas for the Young analogue of the key polynomials, show that they form the generating functions for left keys, and provide a representation-theoretic interpretation of Young key polynomials as traces on certain modules. We also give combinatorial formulas for the Young analogues of Schubert polynomials, including their crystal graph structure. We moreover determine the intersections of (reverse) bases and their Young counterparts, further clarifying their relationships to one another. quasisymmetric Schur functions; Schubert polynomials Symmetric functions and generalizations, Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds The ``Young'' and ``reverse'' dichotomy of 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 \(K\)-theoretic Littlewood-Richardson rule due to \textit{A. S. Buch} [Acta Math. 189, No. 1, 37--78 (2002; Zbl 1090.14015)] describes the product structure constants for the Grothendieck polynomials of Grassmannian type. We present a simple self-contained proof of the rule by generalizing Stembridge's cancelation argument which was applied for the classical Littlewood-Richardson rule. Grothendieck polynomials of Grassmannian type; Stembridge's cancelation argument Ikeda, T.; Shimazaki, T., A proof of \textit{K}-theoretic Littlewood-Richardson rules by Bender-Knuth-type involutions, Math. res. lett., 21, 2, 333-339, (2014) Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds, \(K\)-theory of schemes A proof of \(K\)-theoretic Littlewood-Richardson rules by Bender-Knuth-type involutions | 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 introduce weak dual equivalence and use it to introduce skew key polynomials which, when skewed by a partition, expand nonnegatively in the key basis. We also give a new nonnegative Littlewood-Richardson rule for the key positive of the product of a key polynomial and a Schur polynomial, recovering a result of \textit{J. Haglund} et al. [Trans. Am. Math. Soc. 363, No. 3, 1665--1686 (2011; Zbl 1229.05269)]. dual equivalence; key polynomials; slide polynomials; Schubert polynomials Symmetric functions and generalizations, Exact enumeration problems, generating functions, Combinatorial identities, bijective combinatorics, Combinatorial aspects of representation theory, Group actions on combinatorial structures, Classical problems, Schubert calculus Weak dual equivalence for 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 Let \(G\) be a classical complex Lie group, \(P\) any parabolic subgroup of \(G\), and \(X = G/P\) the corresponding homogeneous space, which parametrizes (isotropic) partial flags of subspaces of a fixed vector space. In the mid 1990s, Fulton, Pragacz, and Ratajski asked for global formulas which express the cohomology classes of the universal Schubert varieties in flag bundles -- when the space {\(X\)} varies in an algebraic family -- in terms of the Chern classes of the vector bundles involved in their definition. This has applications to the theory of degeneracy loci of vector bundles and is closely related to the Giambelli problem for the torus-equivariant cohomology ring of \(X\). In this article, we explain the answer to these questions which was obtained the author [J. Lond. Math. Soc., II. Ser. 82, No. 1, 89--109 (2010; Zbl 1204.14023); Math. Z. 268, No. 1--2, 355--370 (2011; Zbl 1223.14057)], in terms of combinatorial data coming from the Weyl group. Schubert calculus; Giambelli formulas; Schubert polynomials; degeneracy loci; equivariant cohomology Tamvakis, H., \textit{Giambelli and degeneracy locus formulas for classical G/P spaces}, Mosc. Math. J., 16, 125-177, (2016) Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Homogeneous spaces and generalizations, Classical problems, Schubert calculus, Symmetric functions and generalizations Giambelli and degeneracy locus formulas for classical \(G/P\) 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 The Schubert polynomials lift the Schur basis of symmetric polynomials into a basis for \(\mathbb{Z}[x_1,x_2,\dots]\). We suggest the prism tableau model for these polynomials. A novel aspect of this alternative to earlier results is that it directly invokes semistandard tableaux; it does so as part of a colored tableau amalgam. In the Grassmannian case, a prism tableau with colors ignored is a semistandard Young tableau. Our arguments are developed from the Gröbner geometry of matrix Schubert varieties. Schubert polynomials; Gröbner geometry; Young tableau Combinatorial aspects of representation theory, Classical problems, Schubert calculus, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) The prism tableau model 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 The symmetric Grothendieck polynomials generalize Schur polynomials and are Schur-positive by degree. Combinatorially this is manifested as the generalization of semistandard Young tableaux by set-valued tableaux. We define a (weak) symmetric \(P\)-Grothendieck polynomial which generalizes \(P\)-Schur polynomials in the same way. Combinatorially this is manifested as the generalization of shifted semistandard Young tableaux by a new type of tableau which we call shifted multiset tableaux. symmetric Grothendieck polynomials; semistandard Young tableaux; set-valued tableaux Combinatorial aspects of representation theory, Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds \(P\)-Schur positive \(P\)-Grothendieck 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 nonvanishing problem asks if a coefficient of a polynomial is nonzero. Many families of polynomials in algebraic combinatorics admit combinatorial counting rules and simultaneously enjoy having saturated Newton polytopes (SNP). Thereby, in amenable cases, nonvanishing is in the complexity class (NP intersection coNP) of problems with ``good characterizations''. This suggests a new algebraic combinatorics viewpoint on complexity theory. This paper focuses on the case of Schubert polynomials. These form a basis of all polynomials and appear in the study of cohomology rings of flag manifolds. We give a tableau criterion for nonvanishing, from which we deduce the first polynomial time algorithm. These results are obtained from new characterizations of the Schubitope, a generalization of the permutahedron defined for any subset of the \(n \times n\) grid, together with a theorem of \textit{A. Fink} et al. [Adv. Math. 332, 465--475 (2018; Zbl 1443.05179)], which proved a conjecture of \textit{C. Monical} et al. [Sel. Math., New Ser. 25, No. 5, Paper No. 66, 37 p. (2019; Zbl 1426.05175)]. Schubert polynomials; Newton polytopes; computational complexity Combinatorial aspects of representation theory, Symmetric functions and generalizations, Exact enumeration problems, generating functions, Grassmannians, Schubert varieties, flag manifolds, Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.) Computational complexity, Newton polytopes, and 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 Phase transitions in Potts models are studied using motivic invariants of the zero sets of multivariable Tutte polynomials arising from a nested family of finite graphs. The topological complexity is estimated via the computation of the compactly supported Euler characteristics. The main tool is a deletion-contraction formula for the corresponding Grothendieck classes of algebraic varieties. phase transitions; Grothendieck rings; Tutte polynomials Aluffi, P.; Marcolli, M., A motivic approach to phase transitions in Potts models, J. Geom. Phys., 63, 6-31, (2013) (Equivariant) Chow groups and rings; motives, Drinfel'd modules; higher-dimensional motives, etc., Applications of methods of algebraic \(K\)-theory in algebraic geometry A motivic approach to phase transitions in Potts models | 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 Schubert polynomials give explicit polynomial representatives for the Schubert classes in the cohomology ring of the complete flag variety, with the goal of facilitating computations of intersection numbers. \textit{A. Lascoux} and \textit{M.-P. Schützenberger} [C. R. Acad. Sci., Paris, Sér. I 294, 447--450 (1982; Zbl 0495.14031)] first defined Schubert polynomials indexed by permutations in terms of divided difference operators, and later \textit{S. C. Billey} et al. [J. Algebr. Comb. 2, No. 4, 345--374 (1993; Zbl 0790.05093)] and \textit{S. Fomin} and \textit{R. P. Stanley} [Adv. Math. 103, No. 2, 196--207 (1994; Zbl 0809.05091)] gave direct monomial expansions. \textit{N. Bergeron} and \textit{S. Billey} [Exp. Math. 2, No. 4, 257--269 (1993; Zbl 0803.05054)] reformulated this again to give a beautiful combinatorial definition of Schubert polynomials as generating functions for \(RC\)-graphs, often called pipe dreams.
In this paper, the authors introduce a new tool to aid in the study of Schubert polynomials.They define two new families of polynomials they call the monomial slide polynomials and fundamental slide polynomials. Both monomial and fundamental slide polynomials are combinatorially indexed by weak compositions, and both families form a basis of the polynomial ring. Moreover, the Schubert polynomials expand positively into the fundamental slide basis, which in turn expands positively into the monomial slide basis. While there are other bases that refine Schubert polynomials, most notably key polynomials, it has two main properties that make it a compelling addition to the theory of Schubert calculus. First, these polynomials exhibit a similar stability to that of Schubert polynomials, and so they facilitate a deeper understanding of the stable limit of Schubert polynomials, which, as originally shown by Macdonald, are Stanley symmetric functions. Second, and in sharp contrast to key polynomials, their bases themselves have positive structure constants, and so their Littlewood-Richardson rule for the fundamental slide expansion of a product of Schubert polynomials takes one step closer to giving a combinatorial formula for Schubert structure constants. Schubert polynomials; Stanley symmetric functions; pipe dreams; reduced decompositions; quasisymmetric functions S. Assaf and D. Searles. ''Schubert polynomials, slide polynomials, Stanley symmetric func tions and quasi-Yamanouchi pipe dreams''. Adv. Math. 306 (2017), pp. 89--122.DOI. Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus Schubert polynomials, slide polynomials, Stanley symmetric functions and quasi-Yamanouchi pipe dreams | 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 algebraic group and \(W\) be its Weyl group. We prove that if \(W\) is of type \(A_{n}\), \(F_{4}\), or \(G_{2}\) and \(w\), \(w'\) are disjoint involutions in \(W\), then the corresponding Kostant-Kumar polynomials do not coincide. As a consequence, we deduce that the tangent cones to the Schubert subvarieties \(X_{w}\), \(X_{w'}\) of the flag variety of \(G\) do not coincide as well. tangent cones; involutions in Weyl groups; Kostant-Kumar polynomials; Schubert varieties Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus Kostant-Kumar polynomials and tangent cones to Schubert varieties for involutions in \(A_{n}\), \(F_{4}\), and \(G_{2}\) | 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 Arakelov theory is a way of completing a variety defined over the ring of integers of a number field by adding some kind of fibers over the archimedean places. In this way one obtains an intersection theory which generalizes the concept of height. This was initiated by Arakelov in the case of relative dimension one, and generalized to higher dimensions by \textit{H. Gillet} and \textit{Soulé} [Publ. Math., Inst. Hautes Étud. Sci. 72, 93-174 (1990; Zbl 0741.14012)]. Unfortunately, in the higher dimensional case there exist few examples where explicit computations can be made. The arithmetic Chow ring of projective space was studied by \textit{H. Gillet} and \textit{C. Soulé} [Ann. Math., II. Ser. 131, No. 1, 163-203 (1990; Zbl 0715.14018) and No. 2, 205-238 (1990; Zbl 0715.14006)] and arithmetic intersection on Grassmannian by \textit{V. Maillot} [Duke Math. J. 80, No. 1, 195-221 (1995; Zbl 0867.14024)].
The paper under review deals with the case of general flag varieties, thus generalizing the results of V. Maillot. In section 2 the author reviews arithmetic intersection theory as well as Bott-Chern forms; then Schubert polynomials are introduced, following \textit{A. Lascoux} and \textit{H.-P. Schuetzenberger} [C. R. Acad. Sci., Paris Sér. I 294, 447-450 (1982; Zbl 0495.14031)]. The main results are contained in sections 7 and 8. First an effective procedure for computing arithmetic intersection numbers on a complete flag variety is provided. Then the structure of the arithmetic Chow ring is described via an ``arithmetic Schubert calculus''. As an application, in section 9 it is proved that the height of a complete flag variety with respect to its natural pluri-Plücker embedding is a rational number which can be computed explicitly. The last section explains how to generalize the previous results to partial flag varieties. Arakelov theory; flag varieties; arithmetic intersection theory; Bott-Chern forms; Schubert polynomials; arithmetic Schubert calculus; height Tamvakis H.: Arithmetic intersection theory on flag varieties. Math. Ann. 314, 641--665 (1999) Grassmannians, Schubert varieties, flag manifolds, Arithmetic varieties and schemes; Arakelov theory; heights Arithmetic intersection theory on flag 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 Given a vector bundle \(V\) of rank \(n\), the Grothendieck ring of classes of vector bundles of the relative flag manifold \({\mathcal F}(V)\) is generated by the classes \(a_1,\dots,a_n\) of the so-called tautological line bundles on \({\mathcal F}(V)\). The structure sheaf of a Schubert variety in \({\mathcal F}(V)\), having a finite resolution by vector bundles, can be expressed as a (Laurent) polynomial in the \(a_i\), \(1/a_i\). Explicit representatives \(G_\sigma\), \(\sigma\in{\mathfrak S}_n\) were defined by the author and \textit{M.-P. Schützenberger} [in: Invariant theory. Lect. Notes Math. 996, 118--144 (1983; Zbl 0542.14031) and C. R. Acad. Sci., Paris, 295, 629--633 (1982; Zbl 0542.14030)] under the name ``Grothendieck polynomials''. We describe how general Grothendieck of polynomials are related to those for Grassmann manifolds, which themselves are deformations of Schur functions. The geometry of Grassmann varieties is well understood thanks to Schubert, Giambelli and their successors. It is hoped that relating the Schubert subvarieties of a flag manifold to those of a Grassmannian will be of some help to understand those varieties, the singularities of which we still do not know how to describe. Schubert polynomials; 0-Hecke algebra; Grassmann manifolds A. Lascoux, Transition on Grothendieck polynomials, Physics and combinatorics, 2000 (Nagoya), World Sci. Publ., River Edge, NJ, 2001, pp. 164 -- 179. Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations, Combinatorial aspects of representation theory Transition of Grothendieck 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 For a flag variety \(\text{Fl}_n\), the classes of structure sheaves of Schubert varieties form an integral basis in the Grothendieck ring. A major open problem in the modern Schubert calculus is to determine the \(K\)-theory Schubert structure constants, which express the product of two Schubert classes in terms of this basis.
The authors derive explicit Pieri-type formulae in the Grothendieck ring of a flag variety, which generalize both the \(K\)-theory Monk formula [see \textit{C.~Lenart}, J. Pure Appl. Algebra 179, No. 1--2, 137--158 (2003; Zbl 1063.14060)] and the cohomology Pieri formula [see \textit{F.~Sottile}, Ann. Inst. Fourier 46, No. 1, 89--110 (1996; Zbl 0837.14041)]. These expand the product of an arbitrary Schubert class and a special Schubert class in the basis of Schubert classes. These special classes are indexed by cycles of the form \((k-p+1, k-p+2,\dots,k+1)\) or \((k+p, k+p-1, \dots ,k)\), and are pulled back from the projection of \(\text{Fl}_n\) to the Grassmannian of \(k\)-planes.
The formula is expressed in terms of certain labelled chains in the \(k\)-Bruhat order of the symmetric group, and the multiplicities in it are certain binomial coefficients. The proof exploits algebraic-combinatorial setting of Grothendieck polynomials and a Monk-like formula for multiplying a Grothendieck polynomial by a variable. Grothendieck polynomial; Schubert variety; Bruhat order; Pieri's formula DOI: 10.1090/S0002-9947-06-04043-8 Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics, Geometric applications of topological \(K\)-theory A Pieri-type formula for the \({K}\)-theory of a flag manifold | 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 quantum Bruhat graph, which is an extension of the graph formed by covering relations in the Bruhat order, is naturally related to the quantum cohomology ring of \(G/B\). We enhance a result of Fulton and Woodward by showing that the minimal monomial in the quantum parameters that occurs in the quantum product of two Schubert classes has a simple interpretation in terms of directed paths in this graph.
We define path Schubert polynomials, which are quantum cohomology analogs of skew Schubert polynomials recently introduced by \textit{C. Lenart} and \textit{F. Sottile} [Proc. Am. Math. Soc. 131, 3319--3328 (2003; Zbl 1033.05097)]. They are given by sums over paths in the quantum Bruhat graph of type \(A\). The 3-point Gromov-Witten invariants for the flag manifold are expressed in terms of these polynomials. This construction gives a combinatorial description for the set of all monomials in the quantum parameters that occur in the quantum product of two Schubert classes. quantum Bruhat graph; Bruhat order; quantum cohomology; Schubert classes; path Schubert polynomials Postnikov, A., Quantum Bruhat graph and Schubert polynomials. Proc. Amer. Math. Soc., 133 (2005), 699--709. Symmetric functions and generalizations, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds Quantum Bruhat graph and 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 Pipe dreams and bumpless pipe dreams for vexillary permutations are each known to be in bijection with certain semistandard tableaux via maps due to \textit{C. Lenart} [J. Algebr. Comb. 20, No. 3, 263--299 (2004; Zbl 1056.05146)] and \textit{A. Weigandt} [J. Comb. Theory, Ser. A 182, Article ID 105470, 52 p. (2021; Zbl 1475.05172)], respectively. Recently, \textit{Y. Gao} and \textit{D. Huang} [``The canonical bijection between pipe dreams and bumpless pipe dreams'', Preprint, \url{arXiv:2108.11438}] have defined a bijection between the former two sets. In this note we show for vexillary permutations that the Gao-Huang bijection preserves the associated tableaux, giving a new proof of Lenart's result [loc. cit.]. Our methods extend to give a recording tableau for any bumpless pipe dream. semistandard tableaux; vexillary permutations; Schubert polynomials Symmetric functions and generalizations, Permutations, words, matrices, Combinatorial aspects of representation theory, Classical problems, Schubert calculus Lenart's bijection via bumpless pipe dreams | 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 ``Consider the system of equations \(P_1=\dots= P_n=0\) in \((\mathbb{C}\setminus 0)^n\), where \(P_1,\dots, P_n\) are Laurent polynomials with the Newtonian polyhedrons \(\Delta_1,\dots, \Delta_n\). Let us associate each Laurent polynomial \(Q\) with the \(n\)-form
\[
\omega= Q\Biggl/ \Biggl( P\frac{dz_1}{z_1} \wedge\dots\wedge \frac{dz_n}{z_n} \Biggr),
\]
where \(z_1,\dots, z_n\) are independent variables and \(P= P_1\cdot \dots\cdot P_n\).
For general sets of the polyhedrons \(\Delta_1,\dots, \Delta_n\), the sum of the Grothendieck residues of the form \(\omega\) over all roots of the system of equations is evaluated''. Newton polyhedra; Grothendieck residues; Laurent polynomials Gel'fond, O. A.; Khovanskii, A. G.: Newtonian polytopes and Grothendieck residues. Dokl. math. 54, 700-702 (1996) Residues for several complex variables, Toric varieties, Newton polyhedra, Okounkov bodies Newtonian polyhedra and Grothendieck residues | 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 article contains some remarkable necessary and sufficient conditions for a Schubert variety X in the flag manifold SL(n)/B to be non-singular. A striking special case is that X is non-singular if and only if local Poincaré duality holds for X.
The proofs contains in putting together results of Kazdan-Lusztig and Seshadri-Lakshmibai with methods of the author concerning the Bruhat ordering. Kazdan-Lusztig polynomials; non-singular Schubert variety; local Poincaré duality; Bruhat ordering Deodhar, Vinay V., Local Poincaré duality and nonsingularity of Schubert varieties, Comm. Algebra, 13, 6, 1379-1388, (1985), MR 788771 (86i:14015) Grassmannians, Schubert varieties, flag manifolds, Homogeneous spaces and generalizations, Singularities in algebraic geometry Local Poincaré duality and non-singularity 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 Given an integer \(n\) and \(n\) functions of a single variable \(x\), the Wrońskian is the determinant \(|f_j^{(i)}(x)|_{i=0,\ldots,n-1\atop {j=1,\ldots,n}}\) which is of fundamental importance in the theory of linear differential equations. Instead of derivatives, one can also take their divided differences which furnish discrete Wrońskians which is a symmetric function. Varying the functions \(f_1,\ldots,f_n\) one can obtain Schur functions, factorial Schur functions, and more generally Grassmannian Scubert polynomials as the Wrońskians.
In this paper, the author extends the Wrońskian to symmetric functions. As in the classical case, the Wrońskian on a set of symmetric functions can be written in terms of the images of the initial functions under permutations of the variables, and it possesses the factorization property with respect to multiplication by a single symmetric function. Then the author applies the Wrońskian to specific symmetric functions, and he also evaluates the determinant of some complicated resultants by recognizing in them a Wrońskian of Schubert polynomials. Symmetric functions; Schur functions; Schubert polynomials; resultant Lascoux, A.: Wroñskian of symmetric functions Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds, Convolution, factorization for one variable harmonic analysis Wrońskian of symmetric 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 In this small paper, the author studies the positivity of polynomials in the Chern classes of ample or nef vector bundles \(E\) with a filtration. Let \(n= \text{rank } E\), \(0= E_k \subset E_{k-1} \subset \cdots \subset E_0 =E\) a filtration such that \(\text{rank} E/E_j= r_j\), \(0= r_0< r_1< \cdots< r_k=n\). Fix \(n\) and \(\underline {r}= (r_0, \dots, r_k)\). Choose Chern roots \(y_1, \dots,y_n\) of \(E\) so that \(y_{r_j+1}, \dots, y_{r_{j+1}}\) correspond to the Chern roots of \(E_j/ E_{j+1}\). Let \(P(r,d)\) denote the vector space of homogeneous polynomials of degree \(d\) in \(n\) variables \(x_1, \dots, x_n\) with rational coefficients and symmetric in the variables \(x_i\) and \(x_{i+1}\) for all \(i\) such that \(r_j< i< r_{j+1}\) for some \(j= 0, \dots, k-1\). For \(P\in P(r, d)\) let \(P(E)= P(y_1, \dots, y_n)\). Then \(P(E)\in H^{2d} (X; \mathbb{Q})\) if \(X\) is a complex variety, \(P(E)\in A^d (X; \mathbb{Q})\) the Chow cohomology group for an algebraic scheme \(S\) in general. Hence if \(X\) is complete of dimension \(d\), then \(P(E)\cap [X]\) has a degree. A polynomial \(P\in P(r, d)\) is called numerically positive for \(\underline {r}\)-filtered ample (resp. nef) vector bundles if degree\((P (E)\cap [X])\geq 0\) for all ample (resp. nef) vector bundles with filtration of the above type. The Schubert polynomials \(S_\omega\), as \(\omega\) varies over all permutations of length \(d\) with \(\omega (i)< \omega (i+1)\) if \(r_j< i< r_{j+1}\) for some \(j\), form a basis of \(P(r, d)\) over \(\mathbb{R}\). The main results are:
Theorem 1: A nonzero polynomial \(P= \sum a_\omega S_\omega\in P(r,d)\) is numerically positive for \(\underline {r}\)-filtered ample vector bundles if and only if every coefficient \(a_\omega\geq 0\).
Theorem 2: Let \(X\) be a \(d\)-dimensional complex compact Kähler manifold. Then any \(P\in P(r, d)\) with \(P= \sum a_\omega S_\omega\), \(a_\omega\geq 0\) is numerically positive for \(\underline {r}\)-filtered nef vector bundles.
In case \(k=1\), the Schubert polynomials are Schur polynomials. In this case, theorem 1 was proved by the author and \textit{R. Lazarsfield} [Ann. Math., II. Ser. 118, 35-60 (1983; Zbl 0537.14009)] while theorem 2 was proved by \textit{J.-P. Demailly}, \textit{T. Peternell} and \textit{M. Schneider} [J. Algebr. Geom. 3, No. 2, 295-345 (1994; Zbl 0827.14027)]. Chern classes; vector bundles; filtration; vector space of homogeneous polynomials; Schubert polynomials; complex compact Kähler manifold; Schur polynomials Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Grassmannians, Schubert varieties, flag manifolds Positive polynomials for filtered ample vector 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 The Schubert polynomials lift the Schur basis of symmetric polynomials into a basis for \(\mathbb{Z}[x_1,x_2,\ldots]\). We suggest the prism tableau model for these polynomials. A novel aspect of this alternative to earlier results is that it directly invokes semistandard tableaux; it does so as part of a colored tableau amalgam. In the Grassmannian case, a prism tableau with colors ignored is a semistandard Young tableau. Our arguments are developed from the Gröbner geometry of matrix Schubert varieties. Schubert polynomials; bi-Grassmannian permutations; semistandard tableaux A. Weigandt and A. Yong, The Prism tableau model for Schubert polynomials, J. Combin. Theory Ser. A 154 (2018), 551--582. Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds The Prism tableau model 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 It is well-established that Schur functions are related to the cohomology of Grassmannians. They can be seen as particular cases of Schubert polynomials, which describe the cohomology of flag varieties. Moreover, Schubert polynomials are generalized by Grothendieck polynomials which are defined via \(K\)-theory rather than cohomology. While semi-standard tableaux give a combinatorial description of Schur functions, \textit{A. S. Buch} [Acta Math. 189, No. 1, 37--78 (2002; Zbl 1090.14015)] has shown that set-valued tableaux give a combinatorial description of (stable) Grothendieck polynomials.
This idea of considering set-valued (rather than integer valued) objects was further extended to the theory of P-partition by \textit{T. Lam} and \textit{P. Pylyavskyy} [Int. Math. Res. Not. 2007, No. 24, Article ID rnm125, 48 p. (2007; Zbl 1134.16017)]. The main goal of the present work is to give an ``enriched'' analog of their results, which means that the underlying combinatorics is similar to that appearing in the theory of Schur \(P\)- and \(Q\)-functions (where two copies of \(\mathbb{N}\) are used as labels in the combinatorial objects). This is motivated by \textit{J. R. Stembridge}'s theory of enriched \(P\)-partitions [Trans. Am. Math. Soc. 349, No. 2, 763--788 (1997; Zbl 0863.06005)].
The authors consider the generating functions of their enriched set-valued \(P\)-partitions (in the same way as Schur functions can be seen as generating functions of semi-standard tableaux). What they obtain are symmetric functions that in some sense generalize \textit{T. Ikeda} and \textit{H. Naruse}'s shifted stable Grothendieck polynomials [Adv. Math. 243, 22--66 (2013; Zbl 1278.05240)]. Along the way, they consider various related Hopf algebras, such as an algebra of labeled posets and some subalgebras of quasisymmetric functions. symmetric functions; quasisymmetric functions; Hopf algebras; posets; Grothendieck polynomials Partitions of sets, Symmetric functions and generalizations, Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds Enriched set-valued \(P\)-partitions and shifted stable Grothendieck 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 We define the equivariant Kazhdan-Lusztig polynomial of a matroid equipped with a group of symmetries, generalizing the nonequivariant case. We compute this invariant for arbitrary uniform matroids and for braid matroids of small rank. Kazhdan-Lusztig polynomials; matroids; equivariant matroids; representation theory; symmetric functions; log concavity; linear species; Schubert calculus; representation stability K. Gedeon, N. Proudfoot, and B. Young. ''The equivariant Kazhdan--Lusztig polynomial of a matroid''. 2016. arXiv:1605.01777. Combinatorial aspects of representation theory, Combinatorial aspects of matroids and geometric lattices, Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.), Symmetric functions and generalizations, Classical problems, Schubert calculus The equivariant Kazhdan-Lusztig polynomial of a matroid | 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 Gysin map introduced by W. Gysin, originally associated to a map $f:M\rightarrow N$ of closed oriented manifolds, a push-forward covariant homomorphism $f_*: H^*(M)\rightarrow H^*(N)$ between cohomology groups and was later generalized to many different settings. The Gysin homomorphism for fiber bundles and de Rham cohomology of differential forms has a natural interpretation as integration along fibers. The Gysin map was also defined as a push-forward in Chow groups of varieties along proper morphisms of non-singular algebraic varieties.
Gysin maps proved useful in singularity theory and have provided a tool to study the degeneracy loci of morphisms of flag bundles which are related to Schubert manifolds by the Thom-Porteous formula. Most of the results on push-forwards for flag bundles relied on inductive procedures reducing the problem to studying projective bundles. The study of the degeneracy loci of morphisms of flag bundles leads to the development of combinatorial techniques concentrating on the study of Schur polynomials and their generalizations and modifications.
Another direction of study of Gysin homomorphisms was motivared by Quillen's description of the push-forward maps in complex cobordism using a certain type of a residue, which provided a background for the results of Damon and Akyildiz and Carrell expressing Gysin maps for fiber bundles as Grothendieck residues.
The development of equivariant cohomology had enriched the theory with new tools and a different perspective. Many classical theorems have been rephrased in terms of equivariant characteristic classes. The question of computing Gysin maps for projective bundles can be reduced to studying push-forward maps in equivariant cohomology. A powerful tool to study Gysin maps in equivariant cohomology are localization theorems.
In this paper the author gives a review and an example of computational application of an adaptation in the context of equivariant cohomology of the Pragacz-Ratajski formula for push-forwards of Schur classes for Lagrangian Grassmaniann manifolds. Gysin maps; push-forward maps; characteristic classes; equivariant cohomology; Schubert manifolds; flag manifolds; localization; Grothendieck residue; Lagrangian Grassmanianns Equivariant algebraic topology of manifolds, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Pushing-forward Schur classes using iterated residues at infinity | 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 is a surprising number of fascinating connections between combinatorics, algebra and geometry. Among the most beautiful illustrations of such ties are the relations between symmetric functions, and in particular \textit{Schur polynomials}, on the one hand, the \textit{representation theory} of the \textit{symmetric}- and \textit{general linear} groups, on the other-, and the theory of \textit{Schubert polynomials}, and the geometry and cohomology theory of \textit{flag varieties} and \textit{Schubert varieties}, on the third hand. These connections have intrigued mathematicians for more than hundred years, and have proved extremely fruitful for the development of the fields. The area is still active and expanding. New surprising ties appear amazingly often, and there remain many interesting problems and conjectures. This book contains an introduction to the basic material of the fields and their interrelations. It offers a nice complement to the books ``Symmetric functions and Hall polynomials'' (1998; Zbl 0899.05068), and the notes on ``Schubert polynomials'', Lond. Math. Soc. Lect. Note Ser. 166, 73-99 (1991; Zbl 0784.05061) by \textit{I. G. Macdonald}, and to \textit{W. Fulton}'s book ``Young tableaux. With applications to representation theory and geometry'' [Lond. Math. Soc. Student Texts. 35 (1997; Zbl 0878.14034)]. At many places the book gives a different point of view and chooses different techniques. Is is written in a clear and quick style. Sometimes so quick that more examples would have been useful. The book consists of three separate parts:
Symmetric functions and Schur polynomials.
Schubert polynomials.
The geometry of Schubert varieties and the cohomology of flags varieties.
The contents is roughly as follows:
In the first part the Schur functions are introduced in the original way of C. Jacobi and interpreted in terms of Young tableaux. The Pieri formula, the Jacobi-Trudi formulas, and Giambelli's formula are proved in the usual way. Combinatorial correspondences like those of Knuth, Schensted and Robinson are explained, and the Plactic monoid is defined. Using these tools the Littlewood-Richardson rule for multiplication of Schur polynomials is given. The presentation is an alternative to that used by Macdonald in the book on symmetric functions and Hall polynomials mentioned above.
Several applications of the theory are mentioned and the important Kostka-Foulkes polynomials are defined and their main properties are proved. In a separate section the classical connection between Schur polynomials and the irreducible characters of the symmetric group is given.
The second part of the book is devoted to Schubert polynomials. There are several possible approaches to Schubert polynomials. Here the author chooses the one proposed by S. Fomin and A. N. Kirillov via the Yang-Baxter equation and the Hecke algebras of the symmetric groups. It is shown how Schubert polynomials can be computed, and the main properties of the Schubert polynomials, like symmetries, the Cauchy formula, bases, interpolation, and specialization are proved. The lattice path method of I. Gessel and G. Viennot is explained and used. An interesting problem is to determine how the Schubert polynomials are multiplied. The partial formulas of Monk and the Pieri Formula for Schubert polynomials are proved.
In the third part of the book Grassmann varieties and their Schubert varieties are introduced and studied. Their coordinate rings are described, and the fundamental properties of the singularities of the Schubert varieties are given. Also the cohomology ring of the Grassmann variety is determined. The well known correspondence between Chern classes and the classes of the special Schubert varieties is described, and the well known Thom-Porteous formula is proved. In order to study degeneracy loci the more general theory of flag varieties and their Schubert varieties is studied, and the cohomology rings of the flag varieties are described. One of the highlights of the book is the study of degeneracy loci of maps between vector bundles, and a proof of the beautiful result of Fulton that gives the relation between the classes of certain degeneracy loci and the Schubert polynomials. Young tableaux; symmetric functions; Schur polynomials; Schubert polynomials; Bruhat order; Hecke algebras; Grassmannians; flag varieties; Schubert varieties; plactic ring; Kostka-Foulkes polynomials; symmetric group; general linear group; singularity; Pieri's formula; Jacobi-Trudi formulas; Giambelli's formula; Yang-Baxter equation; Thom-Porteous formula; degeneracy loci; partitions; Littlewood-Richardson rule; Monk's formula Manivel, Laurent, Symmetric functions, Schubert polynomials and degeneracy loci, SMF/AMS Texts and Monographs 6, viii+167 pp., (2001), American Mathematical Society, Providence, RI; Société Mathématique de France, Paris Grassmannians, Schubert varieties, flag manifolds, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Research exposition (monographs, survey articles) pertaining to combinatorics, Combinatorial aspects of representation theory, Enumerative problems (combinatorial problems) in algebraic geometry, Symmetric functions and generalizations, Representations of finite symmetric groups Symmetric functions, Schubert polynomials and degeneracy loci. Transl. from the French by John R. Swallow | 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 [Part I, cf. Invent. Math. 79, 499-511 (1985; Zbl 0563.14023).]
We want to develop a theory of polynomials for the parabolic setup for any Coxeter group (W,S) and the subgroup \(W_ J\) generated by any subset \(J\subseteq S\). This is the starting point of our investigation. It turns out that one does get a set \(\{P^ J_{\tau,\sigma}\}\) of polynomials in \({\mathbb{Z}}[q]\) which is indexed by a pair \(\tau\), \(\sigma\) of elements in \(W^ J\), the set of minimal coset representatives of \(W/W_ J\). These polynomials give the dimensions of the intersection cohomology modules of Schubert varieties in G/P (see Theorem 4.1) for any P. They are related to \(P_{x,y}'s\) when the subgroup corresponding to P is finite (see Propositions 3.4 and 3.5). Incidentally, Proposition 3.5 provides a method for computing \(P_{x,y}'s\) which is very efficient since the number of intermediate steps is considerably smaller than that in the original setup (for any Coxeter group (W,S)). Kazhdan Lusztig polynomials; Verma modules; semisimple algebraic group; Kac-Moody groups; Coxeter group; coset representatives; intersection cohomology modules; Schubert varieties V.V. Deodhar, \textit{On some geometric aspects of Bruhat orderings. II. The parabolic analogue of Kazhdan-Lusztig polynomials}, \textit{J. Algebra}\textbf{111} (1987) 483. Infinite-dimensional Lie groups and their Lie algebras: general properties, Cohomology theory for linear algebraic groups, Grassmannians, Schubert varieties, flag manifolds, Semisimple Lie groups and their representations, Representation theory for linear algebraic groups, Other algebraic groups (geometric aspects) On some geometric aspects of Bruhat orderings. II: The parabolic analogue of Kazhdan-Lusztig 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 Kazhdan and Lusztig have introduced the so-called \(P\)-polynomials \(P_{x,y}(q)\) for each pair \((x,y)\) of elements of a Coxeter group \(W\) with \(x\prec y\), where \(\prec\) denotes the Bruhat order of \(W\). The polynomial \(P_{x,y}(q)\) is a measure for the singularity of the Schubert variety \(V_y\) at the generic point of \(V_x\) in the sense that \(V_y\) is smooth along the generic point of \(V_x\) if and only if \(P_{x,y}(q)=1\).
In general it is very hard to calculate the polynomials \(P_{x,y}(q)\). The authors give some explicit formulas for \(P_{x,y}(q)\) in the case \(W\) is the symmetric group \(S_n\) and \(y\) is a particular permutation associated to any flag variety, while \(x\) is arbitrary. Kazhdan-Lusztig polynomials; \(R\)-polynomials; Schubert cells; Coxeter group; \(P\)-polynomials; symmetric group; singularity of a Schubert variety Shapiro, B.; Shapiro, M.; Vainshtein, A., Kazhdan-Lusztig polynomials for certain varieties of incomplete flags, \textit{Discrete Math.}, 180, 345-355, (1998) Grassmannians, Schubert varieties, flag manifolds, Reflection and Coxeter groups (group-theoretic aspects) Kazhdan-Lusztig polynomials for certain varieties of incomplete 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 Kazhdan-Lusztig polynomials \(P_{x,w}(q)\) play an important role in the study of Schubert varieties as well as the representation theory of semisimple Lie algebras. We give a lower bound for the values \(P_{x,w}(1)\) in terms of ``patterns''. A pattern for an element of a Weyl group is its image under a combinatorially defined map to a subgroup generated by reflections. This generalizes the classical
definition of patterns in symmetric groups. This map corresponds geometrically to restriction to the fixed point set of an action of a one-dimensional torus on the flag variety of a semisimple group \(G\). Our lower bound comes from applying a decomposition theorem for ``hyperbolic localization'' [\textit{T. Braden}, Transform. Groups 8, No. 3, 209-216 (2003; Zbl 1026.14005)] to this torus action. This gives a geometric explanation for the appearance of pattern avoidance in the study of singularities of Schubert varieties. Kazhdan-Lusztig polynomials; Schubert varieties; representations of semisimple Lie algebras; Weyl groups; symmetric groups; flag varieties; semisimple groups; torus actions \beginbarticle \bauthor\binitsS. C. \bsnmBilley and \bauthor\binitsT. \bsnmBraden, \batitleLower bounds for Kazhdan-Lusztig polynomials from patterns, \bjtitleTransform. Groups \bvolume8 (\byear2003), no. \bissue4, page 321-\blpage332. \endbarticle \OrigBibText Sara C. Billey and Tom Braden, Lower bounds for Kazhdan-Lusztig polynomials from patterns , Transform. Groups 8 (2003), no. 4, 321-332. \endOrigBibText \bptokstructpyb \endbibitem Reflection and Coxeter groups (group-theoretic aspects), Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Representation theory for linear algebraic groups Lower bounds for Kazhdan-Lusztig polynomials from patterns. | 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 An old problem in Algebraic Geometry asks about the maximum number \(\mu(n)\) of isolated singular points on a projective surface \(F\) of degree \(n\), contained in \(\mathbb{P}^3=\mathbb{P}^3(\mathbb{C})\). The paper under review is a survey on this subject, written by one of its a foremost experts.
Although its statement is elementary, this problem is very hard. The value of \(\mu(n)\) has been found for \(n<6\). For \(n>6\) only upper bounds for the number of isolated singularities were obtained, but probably they (specially for \(n\) large) are not sharp. Many ingenious examples of surfaces of degree \(n>6\) with many nodes were found, but the number is probably \(<\mu(n)\).
The article contains interesting historical information, for instance on a failed attempt by Severi to get a simple bound, based on the study of certain moduli, which turned out to be incorrect. There is a discussion of the proof of \textit{A. Beauville} [Invent. Math. 55, 121--140 (1979; Zbl 0403.14006)] of the formula \(\mu(5)=31\), using a method that (with some homological arguments) reduces the problem to one on ``binary codes'', i.e., finite dimensional vector spaces over \(\mathbb{Z}_2\). This method, with more labour, yields the formula \(\mu(6)=65\), and might work for higher values of \(n\). The author also explains the work of Miyaoka, who found interesting general bounds and studied the asymptotic behaviour of \(\mu(n)/n^3\).
In addition the author discusses numerous examples of surfaces with many nodes, and several other related topics.
The paper is beautifully written, although some parts are expressed in the language of classical Italian Geometry, which might be a little challenge for some modern readers. The article contains a very extensive bibliography, that ranges from a 1750 article by Cramer to very recent preprints not published yet. surfaces in projetive 3-space; isolated singularities; rational double points Gallarati, D, Superficie algebriche con molti punti singolari isolati, Bull. Math. Soc. Sci. Math. Roumanie, 55, 249-274, (2012) Special surfaces, Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry, Computational aspects of algebraic surfaces, Hypersurfaces and algebraic geometry Algebraic surfaces with many isolated singular points (Superficie algebriche con molti punti singolari isolati) | 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 studies in detail the \(F\)-blowups of certain normal surface singularities. The \(F\)-blowup of a variety was introduced by \textit{T. Yasuda} [Am. J. Math. 134, No. 2, 349--378 (2012; Zbl 1251.14002)]. Its interaction to \(F\)-pure and \(F\)-regular singularities has not been fully expored. This article deals with understanding the properties of \(F\)-blowups of non \(F\)-regular rational normal double points and simple elliptic singularities, in relation to normality, smoothness, and stabilization of the \(F\)-blowup sequence. The techniques used combine classical results on normal surface singularities with computations performed with the help of Macaulay2 using two computational tools implemented here. Given a module, the first computes an ideal such that the blowups at the ideal and module coincide, based upon \textit{O. Villamayor U.}'s work [J. Algebra 295, No. 1, 119--140 (2006; Zbl 1087.14011)]. The second tool computes the Frobenius pushforward \(F_*M\) of a given module \(M\). Among other things, in the case of the rational normal double points, the authors exhibit two non-\(F\)-regular such surfaces for which the \(e\)th \(F\)-blowup is the minimal resolution, for \(e \geq 2\). For simple elliptic singularities, the authors determine the structure of \(F\)-blowups up to normalization. Some of the results build upon previous work of \textit{N. Hara} and \textit{T. Sawada} [RIMS Kôkyûroku Bessatsu B24, 121--141 (2011; Zbl 1228.13009)]. \(F\)-blowups; \(F\)-pure surface; \(F\)-regular surface; rational double points; simple elliptic singularities \beginbarticle \bauthor\binitsN. \bsnmHara, \bauthor\binitsT. \bsnmSawada and \bauthor\binitsT. \bsnmYasuda, \batitle\(F\)-blowups of normal surface singularities, \bjtitleAlgebra Number Theory \bvolume7 (\byear2013), page 733-\blpage763. \endbarticle \OrigBibText N. Hara, T. Sawada and T. Yasuda, \(F\)-blowups of normal surface singularities, Algebra Number Theory 7 (2013), 733-763. \endOrigBibText \bptokstructpyb \endbibitem Singularities in algebraic geometry, Positive characteristic ground fields in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) \(F\)-blowups of normal surface 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 This rather important paper indicates a precise concrete way to perform computations in the quantum equivariant ``deformation'' of the cohomology ring of \(G(k,n)\), the complex Grassmannian variety parametrizing \(k\)-dimensional vector subspaces of \({\mathbb C}^n\). It relies on the results of another important paper, regarding the same subject, by the same author [Adv. Math. 203, 1--33 (2006; Zbl 1100.14045)]. The usual singular cohomology ring of \(G(k,n)\) is a very well known object, studied since Schubert's time, at the end of the XIX Century. First of all, it is a finite free \({\mathbb Z}\)-module generated by the so-called Schubert cycles. Furthermore, the special Schubert cycles, the Chern classes of the universal quotient bundle over \(G(k,n)\), generates it as a \({\mathbb Z}\)-algebra. Multiplying two Schubert cycles then amounts to know how to multiply a special Schubert cycle with a general one (Pieri's formula) and a way to express any Schubert cycle as an explicit polynomial expression in the special Schubert cycles (Giambelli's formula).
The obvious way to deform the cohomology of a Grassmannian is to consider the cohomology of the total space of a Grassmann bundle, parametrizing \(k\)-planes in the fibers of a rank \(n\) vector bundle, which is a deformation of the cohomology of any fiber of it. In the last few decades, however, other ways to deform the cohomology ring of \(G(k,n)\) have been studied. \textit{E. Witten} [in: Geometry, topology and physics for Raoul Bott. Lectures of a conference in honor of Raoul Bott's 70th birthday, Harvard University, Cambridge, MA, USA 1993. Cambridge, MA: International Press. Conf. Proc. Lect. Notes Geom. Topol. 4, 357--422 (1995; Zbl 0863.53054)], introduced the small quantum deformation of the cohomology ring of the Grassmannian, whose structure constants were first determined by \textit{A. Bertram} [Adv. Math. 128, No. 2, 289--305 (1997; Zbl 0945.14031)]. Finally, \textit{A. Knutson} and \textit{T. Tao} [Duke Math. J. 119, No. 2, 221--260 (2003; Zbl 1064.14063)], studied the equivariant deformation of the cohomology of the Grassmannians via the combinatorics of puzzles.
In the beautiful paper under review the author recovers the quantum and equivariant Schubert calculus within a unified framework. Basing on the algebraic properties of the Schur factorial functions, the author realizes the equivariant quantum cohomology ring in terms of generators and relations and gives an explicit basis of polynomial representatives for the equivariant quantum Schubert classes. An alternative approach is offered by \textit{D. Laksov} [Adv. Math. 217, 1869--1888 (2008; Zbl 1136.14042)], where the author proves that the basic results of equivariant Schubert calculus, the basis theorem, Pieri's formula and Giambelli's formula can be obtained from the corresponding results of a more general and elementary framework, as in [\textit{D. Laksov}, Indiana Univ. Math. J., 56, No. 2, 825--845 (2007; Zbl 1136.14042)], by a change of basis.
The paper is organized as follows. Section 1 is the introduction, where the main results are clearly stated and motivated; Section 2 is a useful and very pleasant review of the algebra of factorial Schur functions. The quantum equivariant cohomology of Grassmannians is treated in Section 3, while the proof of the theorem about the presentation of the quantum equivariant cohomology ring is given in Section 4. Section 5 ends the paper with the discussion and the proof of Giambelli's formula in equivariant quantum cohomology. Giambelli's formulas; quantum equivariant Schubert calculus; factorial Schur functions L.C. Mihalcea, \textit{Giambelli formulae for the equivariant quantum cohomology of the Grassmannian}, \textit{Trans. AMS}\textbf{360} (2008) 2285 [math/0506335]. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Symmetric functions and generalizations, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Grassmannians, Schubert varieties, flag manifolds Giambelli formulae for the equivariant quantum cohomology of the Grassmannian | 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, the author develops a Bernstein-Sato theory for arbitrary ideals in an \(F\)-finite regular ring of positive characteristic, generalizing work of \textit{M. Mustaţă} [J. Algebra 321, No. 1, 128--151 (2009; Zbl 1157.32012); ``Bernstein-Sato polynomials for general ideals vs. principal ideals'', Preprint, \url{arXiv:1906.03086}] and \textit{T. Bitoun} [Sel. Math., New Ser. 24, No. 4, 3501--3528 (2018; Zbl 1423.13048)] in the case of principal ideals.
Let \(R\) be an \(F\)-finite regular ring of positive characteristic and let \(\mathfrak{a}=(f_1,\dots,f_r)\) be an ideal of \(R\). Consider the local cohomology module \(H_{(f_1-t_1,\dots,f_r-t_r)}^r(R[t_1,\dots,t_r])\). It is well-known that this module has a natural module structure over \(D_{R[t_1,\dots,t_r]}\), the ring of differential operators of \(R[t_1,\dots,t_r]\). The author defines a (decreasing) \(V\)-filtration on \(D_{R[t_1,\dots,t_r]}\), and more precisely on each \(D^e_{R[t_1,\dots,t_r]}=End_{R[t_1,\dots,t_r]^{p^e}}(R[t_1,\dots,t_r])\) (it is well-known that the union of these \(D^e\) is \(D\)). Then the author studies the module \(N_{\mathfrak{a}}^e\) which is defined as the quotient of \(V^0D^e_{R[t_1,\dots,t_r]} \cdot \delta\) by \(V^1D^e_{R[t_1,\dots,t_r]} \cdot \delta\), where \(\delta\) is the element \((f_1-t_1)^{-1}\cdots (f_r-t_r)^{-1}\in H_{(f_1-t_1,\dots,f_r-t_r)}^r(R[t_1,\dots,t_r])\), as well as their limit \(N_{\mathfrak{a}}:=\varinjlim_eN_{\mathfrak{a}}^e\). The higher Euler-type operator \(s_{p^m}:=\sum_{|\underline{a}|=p^m}\partial_{\underline{t}}^{[\underline{a}]}\underline{t}^{\underline{a}}\) acts on \(N_{\mathfrak{a}}^e\) for each \(0\leq m\leq e-1\). The key result which relates \(N_{\mathfrak{a}}^e\) with \(F\)-invariants is Theorem 3.11, where it was shown that the multi-eigenspace of \(N_{\mathfrak{a}}^e\) with eigenvalue \(\alpha=(\alpha_0,\dots,\alpha_{e-1})\) under the action \((s_{p^0},\dots, s_{p^{e-1}})\) is a direct sum of the modules in the set \(\{D_R^e\cdot \mathfrak{a}^{|\alpha|+s{p^e}} / D_R^e\cdot \mathfrak{a}^{|\alpha|+s{p^e}+1}\}\) where \(s=0, 1, \dots, r-1\) and \(|\alpha|=\alpha_0+p\alpha_1+\cdots+p^{e-1}\alpha_{e-1}\), and that each such module occurs in the direct sum. Since it is known (re-proved in this paper) that \(D_R^e\cdot \mathfrak{a}= (C_R^e\cdot \mathfrak{a})^{[p^e]}(=I_e(\mathfrak{a})^{[p^e]})\), it follows that the generalized eigenspace with eigenvalue \(\alpha\) is nonzero precisely when \(I_e(\mathfrak{a}^{|\alpha|+s{p^e}})\neq I_e(\mathfrak{a}^{|\alpha|+s{p^e}+1})\) for some \(0\leq s\leq r-1\). The latter is closely related to \(F\)-invariants such as \(F\)-jumping numbers, thus the author is able to obtain a series of results (Theorem 4.7, Theorem 4.12). The author further introduced Bernstein-Sato roots of \(N_{\mathfrak{a}}\) and proved that they are negative rational numbers (Theorem 6.7), and that there is a connection of these Bernstein-Sato roots with the \(F\)-jumping numbers of \(\mathfrak{a}\) (Theorem 6.11). The results obtained in this article can be viewed as analogs of Bernstein-Sato theory, multiplier ideals, and jumping numbers in characteristic zero, it would be interesting to explore the connections among them. \(F\)-jumping numbers; Bernstein-Sato polynomials; test ideals Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Singularities in algebraic geometry, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials Bernstein-Sato theory for arbitrary ideals in positive characteristic | 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 convex form of degree larger than one is always nonnegative since it vanishes together with its gradient at the origin. In 2007, Parrilo asked if convex forms are always sums of squares. A few years later, \textit{G. Blekherman} [J. Am. Math. Soc. 25, No. 3, 617--635 (2012; Zbl 1258.14067)] answered the question in the negative by showing through volume arguments that for high enough number of variables, there must be convex forms of degree as low as 4 that are not sums of squares. Remarkably, no examples are known to date. In this paper, we show that all convex forms in 4 variables and of degree 4 are sums of squares. We also show that if a conjecture of Blekherman related to the so-called Cayley-Bacharach relations is true, then the same statement holds for convex forms in 3 variables and of degree 6. These are the two minimal cases where one would have any hope of seeing convex forms that are not sums of squares (due to known obstructions). A main ingredient of the proof is the derivation of certain ``generalized Cauchy-Schwarz inequalities'' which could be of independent interest. convex polynomials; sum of squares of polynomials; Cauchy-Schwarz inequality Semialgebraic sets and related spaces, Projective techniques in algebraic geometry, Convex sets in \(n\) dimensions (including convex hypersurfaces), Inequalities and extremum problems involving convexity in convex geometry On sum of squares representation of convex forms and generalized Cauchy-Schwarz inequalities | 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 Reductive groups over finite fields are classified by root systems in a lattice with an action of the Frobenius; a way to get these data from the connected reductive algebraic group \(G\) is to take the projective variety \({\mathcal B}\) of Borel subgroups of \(G\), the set of isomorphism classes of G- equivariant line bundles over \({\mathcal B}\) forms a lattice \(X\); the subgroups containing a given \(B\in {\mathcal B}\) are not conjugate under \(G\), and the orbits of the minimal ones by conjugation under \(G\) form a finite set, each one giving \({\mathcal B}\) as the total space of a \({\mathbb{P}}_ 1\)-fibration over it, hence by taking the tangent bundle along the projections we get elements of \(X\): this is the basis of the root system. Now, each orbit of \(G\) in \({\mathcal B}\times {\mathcal B}\) defines an automorhism of X using the two projections on \({\mathcal B}\), and these orbits form the Weyl group \(W\) of \(G\). The Galois group of the algebraic closure of \(F_ q\) is naturally \({\bar {\mathbb{Z}}}\), generated topologically by the Frobenius F, and the set \(H^ 1({\bar {\mathbb{Z}}},W)\) classifies the maximal tori defined over \(F_ q\) in \(G\).
For each \(w\in W\), let \({\mathcal B}_ w\) be the set of Borel subgroups \(B\in {\mathcal B}\) for which \({}^ FB\) is in position w with respect to B; with a maximal tori \(T\subset B\) defined over \(F\), \textit{P. Deligne} and \textit{G. Lusztig} constructed a variety \({\mathcal B}^ T_ w\) projecting over \({\mathcal B}_ w\) with fibers \(T(F_ q)\) and compatible action of \(G(F_ q)\) so the alternate sum of the \(\ell\)-adic cohomology groups give a virtual representation of \(G(F_ q)\) commuting with the action of \(T(F_ q)\): this leads to the representations \(R^ T_{\theta}\) for the characters \(\theta\) of \(T(F_ q)\) in \({\bar {\mathbb{Q}}}^ x_{\ell}\) [Ann. Math., II. Ser. 103, 103-171 (1976; Zbl 0336.20029)]. Up to equivalence, all the irreducible representations of \(G(F_ q)\) occur in these \(R^ T_{\theta}\), when T and \(\theta\) vary. What was not given in this fundamental article, is an explicit formula for the multiplicities of the irreducible components in the \(R^ T_{\theta}\)'s. The book answers this question, completely in the case \(G\) has a connected center (since, Lusztig obtained the general case).
One of the main tools is the étale intersection cohomology of \textit{P. Deligne, A. A. Beilinson} and \textit{J. Bernstein} [Astérisque 100 (1982; Zbl 0536.14011)], applied to the closures of the varieties \({\mathcal B}_ w\), the Schubert cells. Another one is a deep understanding of the Weyl groups and their Hecke algebras; some properties on them are obtained through the theory of primitive ideals of enveloping algebras of complex reductive Lie algebras; the book uses systematically the results obtained by its author and by D. Kazhdan and its author in the theory of Weyl and Coxeter groups. He shows how the classification of irreducible representations reduces to the classification of unipotent representations of the ``endoscopic'' groups, where the solution comes from the Hecke algebra of the corresponding Weyl group. Also, the author, using the Springer correspondence [\textit{T. A. Springer}, Invent. Math. 36, 173-207 (1976; Zbl 0374.20054)], gives a parametrisation of the irreducible representations in terms of the special conjugacy classes of the dual group of \(G\). root systems; connected reductive algebraic group; projective variety; Borel subgroups; line bundles; orbits; tangent bundle; Weyl group; maximal tori; \(\ell\)-adic cohomology groups; virtual representation; characters; irreducible representations; multiplicities; irreducible components; intersection cohomology; Schubert cells; Weyl groups; Hecke algebras; enveloping algebras; complex reductive Lie algebras; unipotent representations G. Lusztig. Characters of reductive groups over a finite field, Ann. Math. Studies 107, Princeton University Press, 1984. ''BN13N22'' -- 2018/1/30 -- 14:57 -- page 225 -- #27 2018] QUANTIZATIONS OF REGULAR FUNCTIONS ON NILPOTENT ORBITS 225 Representation theory for linear algebraic groups, Linear algebraic groups over finite fields, Research exposition (monographs, survey articles) pertaining to group theory, Cohomology theory for linear algebraic groups, Universal enveloping (super)algebras, Étale and other Grothendieck topologies and (co)homologies, Group actions on varieties or schemes (quotients) Characters of reductive groups over a 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 The first edition of this textbook appeared three years ago (1993; Zbl 0781.00001). Since then, this introductory text on basic algebra has become quite popular with the German-speaking student of mathematics, mainly on account of its protrusive instructional features. The user-friendly style, in particular the skillfully arranged combination of historical background material, problem-oriented motivations, classical topics, modern conceptual theories, and outlooks to further applications of algebraic methods in geometry and number theory, has made this textbook into a valuable addition to the plenty of already existing standard books on modern algebra. The huge amount of carefully selected exercises, which very often reflect the informal, challenging discussions as they appear in classroom or exam situations, together with the comprehensive hints to their solutions, enhance the utility of this text for students in a rewarding manner.
The present second edition of the book shows one essential, methodical modification over the original version. Namely, the section on the Sylow theorems for groups has been completely rewritten. In other respects, the author has (-- apart from minor details and the elimination of misprints --) left the proved text all through intact. groups; rings; polynomials; fields; field extensions; Galois theory; Sylow theorems Mathematics in general, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to field theory, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to group theory, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry Algebra | 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 contains three theorems about algebraic surfaces in E 3 defined by homogeneous polynomials. The first one concerns a polynomial being a product of homogeneous positive definite quadratic forms; there are some sufficient conditions for the surface to be complete convex with non- negative Gauss curvature K. In the second one the spherical Gauss map of a homogeneous polynomial surface is examined. The third theorem contains the sharp estimate \(\iint | K| dS\geq 2\pi | m-1|,\) where m is the number of all linear factors in the polynomial determining of the surface. algebraic surfaces; homogeneous polynomials; Gauss curvature; Gauss map Surfaces in Euclidean and related spaces, Special surfaces Algebraic surfaces defined by a homogeneous polynomial | 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 First we describe the Weierstrass semigroups on a plane curve of degree \(\leqslant 6\). Using this description we determine the Weierstrass semigroups at a ramification point and a branch point on a double covering from a plane curve of degree \(\leqslant 6\). In the case of a double covering from a plane curve of degree 7 we determine all the Weierstrass semigroups at branch points. Weierstrass point; Weierstrass semigroup; smooth plane curve; double covering of a curve Komeda, J.; Kim, S.J., The Weierstrass semigroups on the quotient curve of a plane curve of degree \(###\)7 by an involution, J. Algebra, 322, 137-152, (2009) Special divisors on curves (gonality, Brill-Noether theory) The Weierstrass semigroups on the quotient curve of a plane curve of degree \(\leqslant 7\) by an involution | 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 Der Verfasser beweist, dass die gerade Polare eines Punktes \(P\) in Bezug auf eine Curve \(n^{\text{ter}}\) Klasse \(C\), deren Ordnung \(n\;(n-1)\) ist, auch gefunden wird, indem man von \(P\) die Tangenten an \(C\) zieht, deren Berührungspunkte paarweise verbindet, und die gerade Polare von \(P\) in Bezug auf diese zerfallene Curve sucht.
Ist die Curve nicht von der \(n\;(n-1)^{\text{ten}}\) Ordnung, sondern hat sie \(t\) Doppeltangenten und \(i\) Wendetangenten, so seien \(D, J, T, \varDelta\) die geraden Polaren von \(P\) in Bezug auf resp. die Curve \(C\), die Gesammtheit der Wendetangenten, die Gesammtheit der Doppeltangenten, und die Verbindungslinien der Berührungspunkte der von \(P\) ausgehenden Tangenten. Dann ist \(\varDelta\) die gerade Polare von \(P\) in Bezug auf das Dreieck, dessen Seiten \(D, J, T\) sind, wenn man diesen die Gewichte \(m, 3i, 2t\) beilegt.
Daraus werden mehrere specielle Sätze abgeleitet. Algebraic curves; polars; tangents and double tangents Questions of classical algebraic geometry, Plane and space curves On some properties of algebraic 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 We show that the decidability of an amplification of Hilbert's Tenth Problem in three variables implies the existence of uncomputably large integral points on certain algebraic curves. We obtain this as a corollary of a new positive complexity result: the Diophantine prefix \(\exists \forall \exists\) is generically decidable. This means that we give a precise geometric classification of those polynomials \(f \in\mathbb{Z}[v,x,y]\) for which the question
\[
\exists v \in \mathbb{N} \text{ such that } \forall x \in \mathbb{N} \exists y \in \mathbb{N} \text{ with } f(v,x,y)=0?
\]
may be undecidable, and we show that this set of polynomials is quite small in a rigorous sense. (The decidability of \(\exists \forall \exists\) was previously an open question.) We also show that if integral points on curves can be bounded effectively, then \(\exists \exists \forall \exists\) is generically decidable as well. We thus obtain a connection between the decidability of certain Diophantine problems, height bounds for points on curves, and the geometry of certain complex surfaces and 3-folds. decidability; Hilbert's Tenth Problem; uncomputably large integral points on algebraic curves; diophantine prefix; polynomials; height bounds; geometry of complex surfaces and 3-folds J.M. Rojas, Uncomputably large integral points on algebraic plane curves?, Theoret. Comput. Sci., 235 (this Vol.) (2000) 145--162. Decidability of theories and sets of sentences, Diophantine equations in many variables, Arithmetic problems in algebraic geometry; Diophantine geometry, Decidability (number-theoretic aspects), Curves of arbitrary genus or genus \(\ne 1\) over global fields, Curves in algebraic geometry, Rational and ruled surfaces, Undecidability and degrees of sets of sentences Uncomputably large integral points on algebraic plane 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 [For the entire collection see Zbl 0632.00005.]
The contribution of this paper is to show how to parametrize a cubic surface given its implicit equation. This capability makes cubic surfaces unique.
One parametrization is presented which works for any cubic surface, but which involves the use of a square root. Other parametrizations are presented which involve only rational polynomials but which do not apply to every cubic surface. - \(Section\quad 2\) in this paper reviews the development of piecewise algebraic surfaces. - \(Section\quad 3\) discusses the existence of \(27\quad straight\quad lines\) which occur on every nonsingular cubic surface, and which are used in the parametrization. - \(Section\quad 4\) discusses several methods of parametrizing a cubic surface. rational polynomials; parametrize a cubic surface Sederberg, T W; Snively, J, Parametrization of cubic algebraic surfaces, 299-2, (1987), New York Software, source code, etc. for problems pertaining to algebraic geometry, Parametrization (Chow and Hilbert schemes), Surfaces and higher-dimensional varieties Parametrization of cubic algebraic surfaces | 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 During the last century invariant theory was a subject of a bright interest, culminating in Hilbert's contributions like Hilbert's basis theorem and others. Here the point of view is a linear representation of a linear algebraic group, and looking at the ring consisting of those polynomials invariant under the group action. In fact this research pushed forwards the development of commutative algebra. In the work of Macaulay, E. Noether, Krull, Gröbner and others this development leads to new structural insights as syzygies, Cohen-Macaulay rings, factorial rings, and the birth of homological methods in commutative algebra. As a certain feedback of commutative algebra to invariant theory one might think of \textit{R. P. Stanley}'s article [see Bull. Am. Math. Soc., New Ser. 1, 475-511 (1979; Zbl 0497.20002)]. The present book grows out of the author's lectures at the Oxford University starting on the base of Stanley's article. While Stanley was primarily interested in characteristic zero the author of the book tried to indicate how much is true in arbitrary characteristics. Moreover, in addition there is a discussion of divisor class groups and unique factorization of rings of invariants, which is not covered by Stanley's article. Here the author extends \textit{P. Samuel}'s crucial results [see ``Lectures on unique factorization domains'' (1965; Zbl 0184.06601)].
The book is divided into eight chapters (`Finite generation of invariants', `Poincaré series', `Divisor classes, ramification and hyperplanes', `Homological properties of invariants', `Polynomial tensor exterior algebras', `Polynomial rings and regular local rings', `Groups generated by pseudoreflections', `Modular invariants') and two appendices (`Examples over the complex numbers', `Examples over finite fields'). An interesting feature is covered in section 3 (`Divisor classes, ramification and hyperplanes') by discussing the question of when unique factorization holds in rings of invariants. It culminates in Nakajima's result, that the ring of invariants is a unique factorization domain if and only if there are no non-trivial homomorphisms from the (finite) group to the multiplicative group of the field, taking every pseudoreflection to the identity element. A second feature is the question when the ring of invariants of representation of a finite group is a polynomial ring, answered by the classical Shephard-Todd theorem. Here the author follows \textit{N. Bourbaki}'s considerations [see ``Groupes et algèbres de Lie'', Chapitres 4, 5, 6 (1981; Zbl 0483.22001)], and a homological argument suggested by \textit{L. Smith} [Arch. Math. 44, 225-228 (1985; Zbl 0576.20032)].
Moreover there is a strong overlap to the book with the same title written by \textit{L. Smith} [see ``Polynomial invariants of finite groups'' (1995; see the following review)]. In his introduction the present author writes that his book is -- in a certain sense -- his reaction to the original project to write a book jointly with L. Smith. In fact the author gives credit to Smith for the discussion about the Shephard-Todd theorem as well as the description of the ring of invariants of the group \(SL_n(\mathbb{F}_q)\) in section 8.2. For an interesting result about a certain invariant of a graded module -- defined in the book -- see also \textit{D. J. Benson} and \textit{W. W. Crawley-Boevey} [Bull. Lond. Math. Soc. 27, No. 5, 435-440 (1995; Zbl 0860.13010)] to stimulating further research.
The reviewer recommends the booklet for a seminar course or a lecture for graduate students who have some basic knowledge in commutative algebra as one might find in the book by \textit{M. F. Atiyah} and \textit{I. G. Macdonald}, ``Introduction to commutative algebra'' (1969; Zbl 0175.03601) or \textit{H. Matsumura}, ``Commutative ring theory'' (1986; Zbl 0603.13001)]. -- A possible reader has the chance to compare the author's text at certain places with those of L. Smith's book (loc. cit.) at the corresponding arguments, a method that could be helpful for beginners and students as well as to learn something about different points of views on the same subject. The book under review gives a condensed presentation of the most important features of the theory. prime characteristic; invariant theory; polynomials invariant under the group action; factorization of rings of invariants; Shephard-Todd theorem D.J. Benson, \textit{Polynomial Invariants of Finite Groups, London Mathematical Society Lecture Notes Series}, vol. 190 (Cambridge University Press, Cambridge, 1993) Actions of groups on commutative rings; invariant theory, Geometric invariant theory, Research exposition (monographs, survey articles) pertaining to commutative algebra, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry Polynomial invariants of finite 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 We prove that the Lam-Shimozono ``down operator'' on the affine Weyl group induces a derivation of the affine Fomin-Stanley subalgebra of the affine nilCoxeter algebra. We use this to verify a conjecture of \textit{C. Berg} et al. [Electron. J. Comb. 19, No. 2, Research Paper P55, 20 p. (2012; Zbl 1253.05138)] describing the expansion of \(k\)-Schur functions of ``near rectangles'' in the affine nilCoxeter algebra. Consequently, we obtain a combinatorial interpretation of the corresponding \(k\)-Littlewood-Richardson coefficients. symmetric functions; \(k\)-Schur functions; affine Schubert calculus; dual graded graphs Symmetric functions and generalizations, Classical problems, Schubert calculus, Reflection and Coxeter groups (group-theoretic aspects) The down operator and expansions of near rectangular \(k\)-Schur 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 The authors consider \textit{proper permutations} \(w \in S_n\), i.e., ones which satisfy \(\ell(w) - \binom{d(w) + 1}{2} \leq n\), where \(\ell(w)\) is the number of inversions and \(d(w)\) is the number of left descents of~\(w\). One of the main results of the article is that the probability that a random permutation \(w \in S_n\) is proper goes to zero in the limit when \(n \rightarrow \infty\).
A very important aspect of this result is its relation to geometry: properness of \(w\) is related to the Schubert variety \(X_w\) being spherical. We say that \(X_w\) is spherical if it has a dense orbit of a Borel subgroup of some \(L_I\), a group of invertible block diagonal matrices, where blocks are determined by a set \(I\) of left descents of~\(w\). The authors conclude that the probability that for a random permutation \(w \in S_n\) the Schubert variety \(X_w\) is spherical goes to zero in the limit when \(n \rightarrow \infty\).
Finally, the authors consider the notion of \(w \in S_n\) being \(I\)-spherical, introduced by \textit{R. Hodges} and \textit{A. Yong} in [J. Lie Theory, 32(2), 447--474 (2022; Zbl 1486.14070)], and show that the probability of \(w\) being \(I\)-spherical goes to zero in the limit when \(n \rightarrow \infty\). This result settles a conjecture from the article cited above. Schubert varieties; spherical varieties; proper permutations Grassmannians, Schubert varieties, flag manifolds, Probabilistic methods in group theory Proper permutations, Schubert geometry, and randomness | 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 obtain an algorithm computing the Chern-Schwartz-MacPherson (CSM) classes of Schubert cells in a generalized flag manifold \(G/B\). In analogy to how the ordinary divided difference operators act on Schubert classes, each CSM class of a Schubert class is obtained by applying certain Demazure-Lusztig-type operators to the CSM class of a cell of dimension one less. These operators define a representation of the Weyl group on the homology of \(G/B\). By functoriality, we deduce algorithmic expressions for CSM classes of Schubert cells in any flag manifold \(G/P\). We conjecture that the CSM classes of Schubert cells are an effective combination of (homology) Schubert classes, and prove that this is the case in several classes of examples. We also extend our results and conjecture to the torus equivariant setting. Chern-Schwartz-MacPherson class; homogeneous space; Schubert variety; Demazure-Lusztig operator 10.1112/S0010437X16007685 Grassmannians, Schubert varieties, flag manifolds, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Classical problems, Schubert calculus, Classical groups (algebro-geometric aspects) Chern-Schwartz-MacPherson classes for Schubert cells in flag 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 We study the action of a differential operator on Schubert polynomials. Using this action, we first give a short new proof of an identity of \textit{I. Macdonald} [Notes on Schubert polynomials. Montréal: Publications du LACIM, Université du Québec (1991)]. We then prove a determinant conjecture of \textit{R. Stanley} [``Some Schubert shenanigans'', Preprint, \url{arXiv:1704.00851}]. This conjecture implies the (strong) Sperner property for the weak order on the symmetric group, a property recently established by \textit{C. Gaetz} and \textit{Y. Gao} [Proc. Am. Math. Soc. 148, No. 1, 1--7 (2020; Zbl 07144479)]. Sperner property; weak order; Schubert polynomial; Macdonald identity Combinatorial aspects of representation theory, Symmetric functions and generalizations, Group actions on combinatorial structures, Combinatorics of partially ordered sets, Determinants, permanents, traces, other special matrix functions, Grassmannians, Schubert varieties, flag manifolds Derivatives of Schubert polynomials and proof of a determinant conjecture of Stanley | 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 Grothendieck point residue is considered in the context of computational complex analysis. A new effective method is proposed for computing Grothendieck point residue mappings and residues. Basic ideas of our approach are the use of Grothendieck local duality and a transformation law for local cohomology classes. A new tool is devised for efficiency to solve the extended ideal membership problems in local rings. The resulting algorithms are described with an example to illustrate them. An extension of the proposed method to parametric cases is also discussed as an application. Grothendieck point residue mappings; local cohomology Residues for several complex variables, Local cohomology of analytic spaces, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Local cohomology and algebraic geometry An effective method for computing Grothendieck point residue 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 The authors turn their attention to the Gaussian map of a double cover. They make a cohomological study of the Gaussian map for double covers of smooth projective toric variety, and, in particular, for the case of Hirzebruch surfaces. In general, they focus on cohomological analyses for divisors, not on geometrical aspects. toric varieties; double cover; Gaussian map Duflot, J., Peters, P.: Gaussian maps for double covers of toric surfaces. Rocky Mt. J. Math. 42(5), 1471--1520 (2012) Toric varieties, Newton polyhedra, Okounkov bodies, Coverings in algebraic geometry Gaussian maps for double covers of toric surfaces | 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 \(A=\mathbb{C} [[x_ 1,\dots,x_ n]]\) be the formal power series ring in \(n\) variables over the complex numbers, and \(G\) a finite abelian group acting linearly and faithfully on \(A\). The aim of this paper is to study the Grothendieck group \(K_ 0\pmod R\) of the category of finitely generated modules over the invariant ring \(R=A^ G\). \textit{M. Auslander} and \textit{I. Reiten} [J. Pure Appl. Algebra 39, 1-51 (1986; Zbl 0576.18008)] showed that \(K_ 0\pmod R\) is finitely generated by at most \(c(G)\) elements, where \(c(G)\) denotes the class number of \(G\). In particular, \(K_ 0\pmod R\) is a factor group of \(\mathbb{Z} [G^*]\) where \(G^*\) denotes the character group of \(G\). The authors prove that \(K_ 0\pmod R \simeq \mathbb{Z} [G^*]/K\), where \(K\) is defined as in the paper of \textit{J. Herzog} and \textit{H. Sanders} [Singularities, representation of algebras, and vector bundles, Proc. Symp., Lambrecht 1985, Lect. Notes Math. 1273, 134-149 (1987; Zbl 0652.14016)]. Grothendieck group; quotient singularity J. Herzog, E. Marcos and R. Waldi, On the Grothendieck group of a quotient singularity defined by a finite abelian group,J. Algebra 149 (1992), 122--138 \(K_0\) of group rings and orders, Grothendieck groups (category-theoretic aspects), Actions of groups on commutative rings; invariant theory, Singularities in algebraic geometry, Formal power series rings, Applications of methods of algebraic \(K\)-theory in algebraic geometry On the Grothendieck group of a quotient singularity defined by a finite abelian group | 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 is interested in a global version of the Lê-Ramanujam \(\mu \)-constant theorem for polynomials. He considers an analytic family \(\{f_s\}\), \(s \in [0, 1]\), of complex polynomials in two variables that are Newton non-degenerate. By assuming that the Euler characteristic of a generic fiber of \(f_s\) is constant, he then shows that the global monodromy fibrations of \(f_s\) are all isomorphic, and that the degree of \(f_s\) is constant (up to an algebraic C\(^{2}\)-automorphism). global monodromy fibration; family of polynomials; Lê-Ramanujam [25]T. S. Pha.m, Invariance of the global monodromies in families of nondegenerate polynomials in two variables, Kodai Math. J. 33 (2010), 294--309. Global theory of complex singularities; cohomological properties, Toric varieties, Newton polyhedra, Okounkov bodies Invariance of the global monodromies in families of nondegenerate polynomials in 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 The Severi variety parameterizes plane curves of degree \(d\) with \(\delta\) nodes. Its degree is called the Severi degree. For large enough \(d\), the Severi degrees coincide with the Gromov-Witten invariants of \(\mathbb{CP}^2\). \textit{S. Fomin} and \textit{G. Mikhalkin} [J. Eur. Math. Soc. (JEMS) 12, No. 6, 1453--1496 (2010; Zbl 1218.14047)] proved the 1995 conjecture that for fixed \(\delta\), Severi degrees are eventually polynomial in \(d\).
In this paper, we study the Severi varieties corresponding to a large family of toric surfaces. We prove the analogous result that the Severi degrees are eventually polynomial as a function of the multidegree. More surprisingly, we show that the Severi degrees are also eventually polynomial ``as a function of the surface''. We illustrate our theorems by explicitly computing, for a small number of nodes, the Severi degree of any large enough Hirzebruch surface and of a singular surface.
Our strategy is to use tropical geometry to express Severi degrees in terms of Brugallé and Mikhalkin's floor diagrams, and study those combinatorial objects in detail. An important ingredient in the proof is the polynomiality of the discrete volume of a variable facet-unimodular polytope.
[A preliminary version appeared in [Discrete Mathematics and Theoretical Computer Science. Proceedings, 863--874 (2012; Zbl 1440.14241)]. enumerative geometry; toric surfaces; Gromov-Witten theory; Severi degrees; node polynomials Enumerative problems (combinatorial problems) in algebraic geometry, Polyhedra and polytopes; regular figures, division of spaces, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Universal polynomials for Severi degrees of toric surfaces | 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 0614.00007.]
A geometrical description of the variety B of complete quadrics of an N- dimensional projective space (over an algebraically closed field of characteristic zero) is given. It is proved that the Schubert conditions of tangency to linear varieties give a basis for the cohomology ring of B. Although the results (as the author says) are contained in earlier work by \textit{C. De Concini} and \textit{C. Procesi} [in ''Invariant theory'', Proc. 1st 1982 Sess. C.I.M.E., Montecatini/Italy, Lect. Notes Math. 996, 1-44 (1983; Zbl 0581.14041)] and by \textit{J. Vaisencher} [in ''Enumerative geometry and classical algebraic geometry'', Proc. Math. 24, 199-235 (1982; Zbl 0501.14032)], they are obtained in a quite different way. Schubert calculus; complete quadrics Finat, J. A., A combinatorial presentation of the variety of complete quadrics. Preprint 1985. Enumerative problems (combinatorial problems) in algebraic geometry, Questions of classical algebraic geometry, Grassmannians, Schubert varieties, flag manifolds A combinatorial presentation of the variety of complete quadrics | 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 class of determinantal ideals (of mixed type) whose singular loci fail to be equidimensional (with arbitrarily large dimension-gap). Since these ideals are the defining ideals of a class of Schubert varieties, we get a family of Schubert varieties (as subvarieties of a variety of full flags or of a Grassmannian) possessing inequidimensional singular locus. Schubert varieties; singularities Grassmannians, Schubert varieties, flag manifolds, Homogeneous spaces and generalizations Schubert varieties with inequidimensional singular locus | 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 0716.00007.]
Let \(G\) be a semi-simple algebraic group over \(\mathbb{C}\), and \(B\) a Borel subgroup of \(G\). Then one knows that the cohomology ring \(H^*(G/B;\mathbb{C})\) is the coinvariant algebra \(A(h)/I^ W\) associated to a certain subalgebra \(h\) of the Lie algebra of \(G\) (here, \(A(h)\) is the coordinate ring of \(h\), and \(I^ W\) is the homogeneous ideal generated by the \(W\)-invariant functions \(f\) on \(h\) such that \(f(0)=0)\).
In this paper, the author proves a similar result for \(H^*(X,\mathbb{C})\), \(X\) being a Schubert variety in \(G/P\), for a parabolic subgroup \(P\supseteq B\). This paper makes a good contribution to the cohomology theory of Schubert varieties. flag manifold; cohomology theory of Schubert varieties Grassmannians, Schubert varieties, flag manifolds, Group actions on varieties or schemes (quotients), Cohomology theory for linear algebraic groups, Linear algebraic groups over the reals, the complexes, the quaternions, Classical real and complex (co)homology in algebraic geometry \(SL_ 2\) actions and cohomology 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 The aim of this paper is to study the existence of nontrivial, nonsingular zeros of a nonhomogeneous polynomial defined over a finite field. To accomplish this, we determine conditions that guarantee the existence of a prescribed number of nonsingular zeros of a homogeneous form \(f\) over a finite field \(k\) that are not zeros of a homogeneous form \(h\) when \(f,h\) are relatively prime. The cases of quadratic and cubic polynomials are considered in detail. This extends previous results that have usually considered only the homogeneous case. finite fields; forms in many variables; hypersurface; nonsingular zero; polynomials Varieties over finite and local fields, Forms of degree higher than two, Finite ground fields in algebraic geometry Nonsingular zeros of polynomials defined 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 explicit computation of the intersection cohomology IH(X) à la Goresky-MacPherson of a complex space X is usually difficult. Nevertheless, according to Goresky-MacPherson, if a small resolution of the singularities \(\tilde X\to X\) of X exists, then IH(X) is roughly speaking the same as the cohomology \(H(\tilde X)\) of \(\tilde X.\) The author proves by an explicit construction the existence of a small resolution for any Schubert cell and therefore obtains a combinatorial description of the intersection cohomology. intersection cohomology; small resolution for any Schubert cell Zelevinskiĭ, A. V.: Small resolutions of singularities of Schubert varieties. Funct. anal. Appl. 17, No. 2, 142-144 (1983) Global theory and resolution of singularities (algebro-geometric aspects), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, (Co)homology theory in algebraic geometry Small resolutions of singularities 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 We analyze the Point Decomposition problem (PDP) in binary elliptic curves. It is known that PDP in an elliptic curve group can be reduced to solving a particular system of multivariate non-linear equations derived from the so called Semaev summation polynomials. We modify the underlying system of equations by introducing some auxiliary variables. We argue that the trade-off between lowering the degree of Semaev polynomials and increasing the number of variables provides a significant speed-up. Semaev polynomials; elliptic curves; point decomposition problem; discrete logarithm problem Karabina, K., Point decomposition problem in binary elliptic curves, (International Conference on Information Security and Cryptology, (2015), Springer International Publishing) Cryptography, Applications to coding theory and cryptography of arithmetic geometry Point decomposition problem in binary elliptic 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 We discuss how the classification of finite simple groups is used and mention some specific applications to various other fields of mathematics as well as to group theory. finite groups; finite simple groups; applications of simple groups; Brauer groups; Riemann surfaces; polynomials; function fields Guralnick, Robert, Applications of the classification of finite simple groups.Proceedings of the International Congress of Mathematicians---Seoul 2014. Vol. II, 163-177, (2014), Kyung Moon Sa, Seoul Finite simple groups and their classification, Primitive groups, Coverings of curves, fundamental group, Algebraic field extensions Applications of the classification of finite simple 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 is a very well written paper which deals with an interesting problem, namely, the characterization of the so-called double fibrations. It is known that the language of double fibrations is convenient for describing a family of submanifolds and the construction of the dual family of submanifolds. It has also deep applications in integral geometry.
Here the author introduces the notion of admissible double fibration in the sense that there exists a collection of densities for which the integration operator admits a local inversion formula ({\S} 1). A necessary condition for admissibility is given in terms of the degree of a certain covering map ({\S} 2). The author states the conjecture that the condition \(d(\Gamma)=1\) is also sufficient for the admissibility of n-parameter families of k-dimensional planes in \({\mathbb{C}}{\mathbb{P}}^ n\) ({\S} 3) and, in the remaining part of the paper verifies that the conjecture is true for linear complexes and also for n-parameter families of (n-2)-dimensional subspaces of \({\mathbb{C}}{\mathbb{P}}^ n\) ({\S} 7). dual family of submanifolds; integral geometry; admissible double fibration; covering map A. B. Goncharov, Admissible families of \(k\)-dimensional submanifolds , Dokl. Akad. SSSR 300 (1988), no. 3, 535-539. Structure of families (Picard-Lefschetz, monodromy, etc.), Analytic subsets and submanifolds, Integral geometry Admissible families of k-dimensional submanifolds | 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 propose efficient strategies for calculating point tripling on Hessian \((8M+5S)\), Jacobi-intersection \((7M+5S)\), Edwards \((8M+5S)\) and Huff \((10M+5S)\) curves, together with a fast quintupling formula on Edwards curves. \(M\) is the cost of a field multiplication and \(S\) is the cost of a field squaring. To get the best speeds for single-scalar multiplication without regarding perstored points, computational cost between different double-base representation algorithms with various forms of curves is analyzed. Generally speaking, tree-based approach achieves best timings on inverted Edwards curves; yet under exceptional environment, near optimal controlled approach also worths being considered. elliptic curves; scalar multiplication; point arithmetic; double-base number system Cryptography, Applications to coding theory and cryptography of arithmetic geometry, Elliptic curves Improved tripling on elliptic 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 \(K_0(\mathcal V_{\mathbb C})\) be the Grothendieck ring of all algebraic varieties over \(\mathbb C\). If \(X\) is a complex variety and \(n\geq 1\) is a natural number, denote by \(X^{(n)}\) the \(n\)-fold symmetric of \(X\). If \(\mu\colon K_0({\mathcal V}_{\mathbb C})\to A\) is an \(A\)-valued motivic measure (i.e. a ring homomorphism) then one can define the formal power series
\[
\zeta_{\mu}(X,t)=1+\sum_{n=1}^{\infty}\mu([X^{(n)}])t^n\in A[[t]],
\]
which is called the motivic zeta function of \(X\). \textit{M. Kapranov} [The elliptic curve in the \(S\)-duality theory and Eisenstein series for Kac-Moody groups, preprint, http://arxiv.org/math.AG/0001005] proved that if \(A\) is a field and \(X\) is a curve then \(\zeta_{\mu}(X,t)\) is a rational function. Moreover he asked whether this is a rational function for all varieties \(X\) of dimension \(\geq 2\) . In the paper under review the authors give a negative answer to Kapranov's question by proving the following.
Theorem: There is a field \(A\) and a motivic measure \(\mu\colon K_0({\mathcal V}_{\mathbb C})\to A\) such that for every smooth projective surface \(X\) with \(p_g(X)=h^{2,0}(X)\geq 2\), the motivic zeta function \(\zeta_{\mu}(X,t)\) is not rational. Grothendieck ring; motivic zeta function Larsen, M.; Lunts, V. A., \textit{motivic measures and stable birational geometry}, Mosc. Math. J., 3, 85-95, (2003) Rational and birational maps, Motivic cohomology; motivic homotopy theory Motivic measures and stable birational 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 Grassmannian has a natural action by an algebraic torus. In this survey we describe the diagonal action of this torus on subvarieties of products of Grassmannians. From this action we describe how to construct an associated moment polytope. The main varieties considered are Schubert varieties, Richardson varieties, and their desingularizations. The moment polytopes of these varieties include the permutahedron and associahedron. We also look at the Barbasch-Evens-Magyar varieties, which are desingularizations of symmetric orbit closures in the flag manifold. moment polytope; Grassmannian; flag manifold; Schubert variety; Bott-Samelson variety; Richardson variety; Brick variety; Barbasch-Evens-Magyar variety; permutahedron; associahedron Grassmannians, Schubert varieties, flag manifolds, Research exposition (monographs, survey articles) pertaining to algebraic geometry A brief survey about moment polytopes of subvarieties of products of Grassmanians | 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 establish numerically checkable sums of squares characterizations of containment of a convex semialgebraic set in another reverse convex semialgebraic set, described by SOS-convex polynomials. The significance of these characterizations is that they hold without any qualifications. In particular, when the semialgebraic sets are described by convex quadratic functions, we obtain a simple linear matrix inequality characterization for the containment.
We also present robust set containment characterizations for convex semialgebraic sets in the face of data uncertainty of the SOS-convex polynomials that define the convex semialgebraic sets. set containment; SOS-convex polynomials; sums of squares polynomials; convex semialgebraic sets Jeyakumar, V; Lee, GM; Lee, JH, Sums of squares characterizations of containment of convex semialgebraic sets, Pac. J. Optim., 12, 29-42, (2016) Semialgebraic sets and related spaces, Convex programming, Optimality conditions and duality in mathematical programming Sums of squares characterizations of containment of convex semialgebraic sets | 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 Two smooth, connected, proper varieties \(X\) and \(Y\) over an algebraically closed field \(k\) of characteristic 0 are \(K\)-\textit{equivalent} if there exists a smooth, connected, proper variety \(Z\) over \(k\) and birational morphisms \(X \overset{f}\longleftarrow Z \overset{g}\longrightarrow Y\) such that the relative canonical divisors \(K_{Z/X}\) and \(K_{Z/Y}\) coincide. A theorem of M. Kontsevich asserts that, in this case, \(X\) and \(Y\) have the same Hodge numbers. It is natural to ask whether \(X\) and \(Y\) have isomorphic Chow motives or, at least, whether they have isomorphic Chow groups.
In the paper under review, the authors propose a (conjectural) strategy for attacking this question by using the Grothendieck ring of varieties and piecewise algebraic geometry. Let \({\mathbb Z}[\text{Var}_k]\) be the free abelian group generated by the isomorphism classes \(\{X\}\) of \(k\)-varieties \(X\). The Grothendieck ring \(K_0(\text{Var}_k)\) is the quotient of \({\mathbb Z}[\text{Var}_k]\) by the subgroup generated by the elements of the form \(\{X\} - \{Z\} - \{X\setminus Z\}\), \(Z\) closed subvariety of \(X\), with the multiplicative structure defined by \([X]\cdot [Y] := [X\times_kY]\) (\([X]\) denoting the class of the variety \(X\) in \(K_0(\text{Var}_k)\)). The unit of \(K_0(\text{Var}_k)\) is \([\text{Spec}\, k]\). Let \(\mathbb L\) denote \([{\mathbb A}_k^1]\) and let \({\mathcal M}_k := K_0(\text{Var}_k)[{\mathbb L}^{-1}]\). A theorem of M. Kontsevich asserts that if \(X\) and \(Y\) are \(K\)-equivalent varieties as at the beginning of this review then \([X]\) and \([Y]\) have the same image into the completion \({\widehat {\mathcal M}}_k\) of \({\mathcal M}_k\) with respect to a certain filtration defined in terms of the dimension of the varieties.
On the other hand, two \(k\)-schemes of finite type \(X\) and \(Y\) are \textit{piecewise isomorphic} if they admit partitions \((X_i)_{0\leq i \leq n}\), \((Y_i)_{0\leq i \leq n}\) such that \((X_i)_{\text{red}} \simeq (Y_i)_{\text{red}}\), \(i = 0, \dots , n\). Two piecewise isomorphic \(k\)-varieties have the same class in \(K_0(\text{Var}_k)\). \textit{M. Larsen} and \textit{V. A. Lunts} [Moscow Math. J. 3, No. 1, 85--95 (2003; Zbl 1056.14015)] asked the difficult question whether the converse of this statement is true.
The strategy proposed by the authors of the paper under review for the question formulated at the beginning consists in splitting the question into two parts:
1) If \(X\) and \(Y\) are \(K\)-equivalent does it follow that \([X] = [Y]\) in \(K_0(\text{Var}_k)\)?;
2) If \([X] = [Y]\) in \(K_0(\text{Var}_k)\) do \(X\) and \(Y\) have isomorphic Chow motives?
Step 1 splits into two substeps:
1a) Is the morphism \({\mathcal M}_k \rightarrow {\widehat {\mathcal M}}_k\) injective?;
1b) Is the morphism \(K_0(\text{Var}_k) \rightarrow {\mathcal M}_k\) injective?
\textit{J. Sebag} [Proc. Am. Math. Soc. 138, No. 4, 1231--1242 (2010; Zbl 1192.14012)] related 1b) to the question of Larsen and Lunts. Using results of \textit{U. Jannsen} [Invent. Math. 197, No. 3, 447--452 (1992; Zbl 0762.14003)] and of \textit{S.-I. Kimura} [Math. Ann. 331, No. 1, 173--201 (2005; Zbl 1067.14006)], the authors notice that the question formulated in Step 2 would have a positive answer if the conjecture asserting that every Chow motive is ``finite dimensional'' were true.
The main contribution of the authors of the paper under review consists in answering some of the above questions for ``small dimensional'' varieties. They show that if two smooth, projective surfaces have the same class in \(K_0(\text{Var}_k)\) then they have isomorphic Chow motives. They also show that if two connected, smooth, projective varieties have the same class in \(K_0(\text{Var}_k)\) then they have isomorphic Néron-Severi groups. Finally, the authors show that if \(X\) and \(Y\) are two \(K\)-equivalent, smooth, projective varieties admitting a \(K\)-exceptional locus \((C_X,\, C_Y)\) with \(\dim C_X \leq 2\) then \(X\) and \(Y\) are piecewise isomorphic. As a corollary, two Calabi-Yau varieties of dimension \(\leq 4\) are piecewise isomorphic if and only if they are birationally equivalent. \(k\)-equivalence of algebraic varieties; Chow motives; Grothendieck ring of varieties; piecewise isomorphism of varieties; Calabi-Yau manifold Ivorra, F; Sebag, J, Géométrie algébrique par morceaux, \(K\)-équivalence et motifs, Enseign. Math. (2), 58, 375-403, (2012) (Equivariant) Chow groups and rings; motives, Rational and birational maps, Calabi-Yau manifolds (algebro-geometric aspects) Piecewise algebraic geometry, \(k\)-equivalence and motives | 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 \(C\) be a non-singular, projective, irreducible, algebraic curve defined over an algebraically closed field of characteristic zero, and let \(P\in C\). The Weierstrass semigroup \(H(P)\) of \(C\) at \(P\) is the set of poles of regular function on \(C\setminus\{P\}\). Thus \(H(P)\) is indeed a subsemigroup of the additive semigroup \(\mathbb N_0\) such that \(\#(\mathbb N_0\setminus H(P))\) equals the genus of \(C\) (The Weierstrass gap theorem). In general it is a difficult problem to compute Weierstrass semigroups.
In the paper under review, the author considers certain curves contained in \(K3\) surfaces over the complex numbers with Picard number one. In this way he continues the computation of Weierstrass semigroups for non-singular curves on non-singular surfaces such as those of plane curves of degree at most seven computed by \textit{S. J. Kim} and \textit{J. Komeda} and [J. Algebra 322, No. 1, 137--152 (2009; Zbl 1171.14020)], or those curves on non-singular toric surfaces computed by \textit{R. Kawaguchi} [Kodai Math. J. 33, No. 1, 63--86 (2010; Zbl 1221.14042)].
The main result in the paper under review is the following. Let \(X\) be a non-singular \(K3\) surface over the complex number with Picard number one and let \(X\) be defined by a double morphism \(\pi: X\to \mathbb P^2\). Let \(C\subseteq X\) be a non-singular, projective, irreducible algebraic curve of degree \(d\geq 4\) which is not the ramification divisor of \(\pi\) and such that \(\pi^{-1}(\pi(C))=C\). For \(R\in C\) let \(I(R)\) denote the intersection divisor of \(\pi(C)\) and the tangent line of \(\pi(C)\) at \(\pi(R)\). Let \(P\) be a ramification point of \(\pi|C: C\to\pi(C)\). If \(I(P)=d\pi(P)\), then \(H(P)=2H(\pi(P))+(6d-1){\mathbb N}_0\). If \(I(P)=(d-1)\pi(P)+Q\) with \(I(Q)=d\pi(Q)\), then \(H(P)=2H(\pi(p))+ \sum_{i=0}^{d-4}((8d-9)+2(d-2)i){\mathbb N}_0\). Weierstrass semigroups of pointed curves; double coverings of curves; curves on \(K3\) surfaces Watanabe, K, An example of the Weierstrass semigroup of a pointed curve on K3 surfaces, Semigroup Forum, 86, 395-403, (2013) Riemann surfaces; Weierstrass points; gap sequences, Coverings of curves, fundamental group, \(K3\) surfaces and Enriques surfaces An example of the Weierstrass semigroup of a pointed curve on \(K3\) surfaces | 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 \({\mathfrak G}\) be a Kac-Moody algebra defined by a Cartan matrix. The author studies the Kac-Moody group \(G\) associated with \({\mathfrak G}\). The group \(G\) is generated by automorphisms \(\exp (\text{ad} x)\), where \(x \in {\mathfrak G}_ r\) for \(r \in \Delta^{re}\), where \(\Delta^{re}\) denotes the set of real roots of \({\mathfrak G}\). We denote further by \(B\) the normalizer in \(G\) of the subgroup \(\langle \{\exp (\text{ad} x);\;x \in {\mathfrak G}_ r\), \(r \in (\Delta^{re})^ +\} \rangle\). For \(h \in {\mathfrak H}\) (the Cartan subalgebra), we define \(P_ h\) to be the normalizer in \(G\) of the subgroup \(\langle \{\exp (\text{ad} x);\;x \in {\mathfrak G}_ r, \langle h,r \rangle \geq 0\}\rangle\).
Let \(\Pi = \alpha_ 1,\dots,\alpha_ n\) be a system of simple real roots for \({\mathfrak G}\) of finite growth \(h=\sum \alpha_ i\). The first theorem asserts that \(G\) is a \(BN\)-pair with the Borel subgroup \(P=P_ h\) and normal standard parabolic subgroups \(P_{\alpha_ i}\). In his previous papers [see e.g., On embeddings of some geometries and flag systems in Lie algebras and superalgebras, Akad. Nauk Ukr. SSR Inst. Mat., Preprint 1990, No. 8, 3-17] the author showed that the incidence system of flags \(\Gamma_ P(G)\) in \(G\) can be considered also as a subset \(HG^ -_ \Pi ({\mathfrak G})\) in \({\mathfrak G}\). In the third theorem an isomorphism of incidence systems \(HG^ -_ \Pi ({\mathfrak G}) \to \Gamma_ P(G)\) is introduced, and an equivalence relation \(\sim_ c\) is defined such that each \(\sim_ c\)-equivalence class is mapped onto a small Schubert cell. The second and fourth theorems are only versions of the first and third theorems in the situation where a simple symmetry of \(\Pi\) is given. The proofs of the theorems are not included. geometry of flags; small Schubert cell; Kac-Moody algebra; Kac-Moody group Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Grassmannians, Schubert varieties, flag manifolds Small Schubert cells as subsets in Lie 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 paper deals with logarithmic geometry (or logarithmic spaces) in the sense of \textit{K. Kato} [in: Algebraic analysis, geometry, and number theory, Proc. JAMI Inaugur. Conf., Baltimore/MD (USA) 1988, 191--224 (1989; Zbl 0776.14004)]. Throughout the paper, in which the author assumes that the reader is familiar with the basics of this conceptual framework [\textit{L. Illusie}, in: Barsotti symposium in algebraic geometry (Abano Terme, 1991), Perspect. Math. 15, 183--204 (1994; Zbl 0832.14015)], a log structure on a scheme \(X\) means a log structure on the étale site \(X_{\text{ét}}\) of \(X\) in the sense of Kato.
In this context, the purpose of the present paper is to introduce a stack-theoretic approach to Kato's theory of logarithmic structures. More precisely, for any fine log scheme \(S\) with underlying scheme \(\mathring S\), the author constructs a fibred category \(\text{Log}_S\to (\mathring S\)-schemes). His main result consists in the proof of the fundamental theorem stating that \(\text{Log}_S\) is an algebraic stack locally of finite presentation over the underlying scheme \(\mathring S\). Moreover, it is shown that a morphism of fine log schemes \(f:X\to S\) defines tautologically a morphism \(\text{Log}(f) : \text{Log}_X\to \text{Log}_S\) of algebraic stacks, and that the association \(S\mapsto \text{Log}_S\) defines a 2-functor from the category of log schemes to the 2-category of algebraic stacks. It is then explained how this 2-functor can be used to reinterpret and study some original basic notions in Kato's logarithmic geometry.
The fine analysis carried out in this paper is enhanced by an appendix, in which the author compares the notions of log structure in the fppf, étale, and Zariski topology, respectively. Part of this comparison is used in the course of the main body of the paper, and the rest is included for the sake of completeness.
As the author points out, his main theorem on the structure of the stack \(\text{Log}_S\) has further applications which are not discussed in the present paper. Namely, one can develop the theory of log crystalline cohomology using a theory of crystalline cohomology over stacks [cf. \textit{M. C. Olsson}, Crystalline cohomology of schemes over algebraic stacks, Preprint 2002], and also the deformation theory of log schemes can be analyzed using the structure of \(\text{Log}_S\). In addition, the main theorem of the present paper has a natural place in the moduli theory of fine log schemes [cf. \textit{M. C. Olsson}, Tohoku Math. J., II. Ser. 55, No.~3, 397--438 (2003; Zbl 1069.14015)]. The author intends to discuss these subjects more thoroughly in forthcoming papers. Grothendieck topologies; étale topologies; sites; algebraic spaces; log structures; fibred categories Olsson, M. C., Logarithmic geometry and algebraic stacks, Ann. Sci. Éc. Norm. Supér. (4), 36, 5, 747-791, (2003) Algebraic moduli problems, moduli of vector bundles, Generalizations (algebraic spaces, stacks), Schemes and morphisms, Local structure of morphisms in algebraic geometry: étale, flat, etc., Fibered categories Logarithmic geometry and algebraic stacks | 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 Basset's upper bound for the double points of an algebraic surface is here extended to algebraic hypersurfaces in \(\mathbb{P}_r(\mathbb{C})\) [see \textit{A. B. Basset}, Nature 73, 246 (1906; JFM 37.0646.03)]. number of double points; hypersurfaces Singularities of surfaces or higher-dimensional varieties, Hypersurfaces and algebraic geometry, Singularities in algebraic geometry, Enumerative problems (combinatorial problems) in algebraic geometry The inequality of A. B. Basset for hypersurfaces in a hyperspace | 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 studies polynomial systems \(P_1,\dots, P_k\) of real polynomials in \(k\) variables with fixed set of Newton polytopes \(\Delta_1, \Delta_2,\dots, \Delta_k\) (\(\Delta_i\) is the convex hull of the exponent vectors of monomials in \(P_i\)) and a fixed set of signs in the coefficients. The main questions are: What is the number of real zeros of system in a given open orthant? What is the number of real zeros with nonzero coordinates? How large can these number be? The authors guarantee a combinatorial lower bound for these quantities. The construction is based on subdivisions of Minkowski sums of the Newton polytopes \(\Delta_1, \Delta_2,\dots, \Delta_k\), the so called Viro's method (more precisely a version proposed by Sturmfels). The authors conjectured the bounds are actually upper bounds, this motivated a lot of interst and active research.
A recent counterexample was announced by \textit{T. Y. Li} and \textit{X. Wang}. It is given by a 2-variable system
\[
-1-x_1+x_2, \qquad -2-9x_1^3+ x^3_2+ 0.01 x^3_1 x^3_2.
\]
real polynomials; Newton polytopes I. Itenberg and M. F. Roy, \textit{Multivariate Descartes' rule}, Beiträge zur Algebra und Geometrie \textbf{37} (1996), 337-346. Topology of real algebraic varieties, Toric varieties, Newton polyhedra, Okounkov bodies, Complete intersections Multivariate Descartes' rule | 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 an algebraically closed field of characteristic zero, let \(P=k\langle \langle u,v\rangle \rangle\) be the noncommutative power series ring in the two indeterminates \(u,v\), and \(r\in P\). Suppose that the leading term of \(r\) is quadratic with no linear factors. Then \(B=P/(r)\) is a regular ring of dimension two. In fact, it is a noncommutative analog of the power series ring \(k[[u,v]]\). The author studies the ring of invariants \(B^G\), where \(G\) is a finite subgroup of \(\text{SL}(V)\), \(V=ku+kv\). Such a ring is called a special quotient surface singularity and can be considered as a noncommutative analog of a rational double point. When \(G\) is cyclic an explicit description of such algebras in terms of generators and relations is given. It is also proved that these algebras are AS-Gorenstein of dimension two, they have finite representation type and, in many cases, are regular in codimension one. noncommutative regular rings; cyclic quotient singularities; rational double points; rings of invariants; local dualities; dualizing complexes; Gorenstein singularities; Cohen-Macaulay singularities Chan, D.: Noncommutative rational double points. J. algebra 232, 725-766 (2000) Rings arising from noncommutative algebraic geometry, Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting), Rational and ruled surfaces, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Actions of groups and semigroups; invariant theory (associative rings and algebras), Valuations, completions, formal power series and related constructions (associative rings and algebras), Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.), Representation type (finite, tame, wild, etc.) of associative algebras, Cohen-Macaulay modules in associative algebras Noncommutative rational double 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 See the review in Zbl 0647.13004. automorphisms of the ring of polynomials Polynomial rings and ideals; rings of integer-valued polynomials, Group actions on varieties or schemes (quotients) Algebraic generators of the automorphism group of polynomial ring over finitely generated algebra | 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 uses the theory of canonical bases of Kashiwara and Lusztig to generalize the notion of a totally positive matrix to the case of elements of an algebraic group. He also shows that the Kazhdan-Lusztig polynomials appear as matrix coefficients in the change of base between the canonical bases and other natural bases. canonical bases; totally positive matrix; Kazhdan-Lusztig polynomials P. Littelmann, ''Bases canoniques et applications,''Séminaire Bourbaki Vol. 1997/98. Astérisque No. 252 (1998), Exp. No. 847, 5, 287--306. Universal enveloping (super)algebras, Quantum groups (quantized enveloping algebras) and related deformations, Linear algebraic groups over the reals, the complexes, the quaternions, Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras, Grassmannians, Schubert varieties, flag manifolds Canonical bases and 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 We study variants of the local models constructed by the second author and and \textit{X. Zhu} [Invent. Math. 194, No. 1, 147--254 (2013; Zbl 1294.14012)] and consider corresponding integral models of Shimura varieties of abelian type. We determine all cases of good, resp. of semi-stable, reduction under tame ramification hypotheses. Shimura varieties; local models; Rapoport-Zink spaces; Schubert varieties Arithmetic aspects of modular and Shimura varieties, Modular and Shimura varieties Good and semi-stable reductions of Shimura 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 conjecture in question is that all continuous piecewise polynomial functions \(h\) on \(\mathbb R^ n\) (in finitely many pieces) are generated from polynomial functions by lattice operations. The author sketches a proof for the plane. It reduces to certain constructions around the common boundaries of pairs of the open sets on which \(h\) is equal to only one of the piecing polynomials. For any \(n\), common boundaries which are empty or everywhere \((n-1)\)-dimensional are taken care of; so for \(n=2\), only isolated points remain to be treated. Pierce-Birkhoff conjecture; piecewise polynomials; isolated points Mahé, Louis, On the Pierce-Birkhoff conjecture, Rocky Mountain J. Math., 14, 4, 983-985, (1984) Real algebra, Semialgebraic sets and related spaces, Local analytic geometry On the Pierce-Birkhoff 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 Let \(G\) be a semisimple algebraic group over an algebraically closed field \(k\) of characteristic zero. Further, let \(\sigma\) be a simple involution of \(G\) and \(H\) denote the corresponding fixed point subgroup of \(G\). Of interest here is the coordinate ring \(k[G/H]\) of the symmetric variety \(G/H\). The \(G\)-module structure of \(k[G/H]\) is known in terms of spherical \(G\)-modules (relative to \(H\)). The goal of this paper is to give a more precise description by identifying the relations among the generators. More precisely, the authors construct a standard monomial theory for \(k[G/H]\) and identify the defining relations when the associated \textit{restricted root system} is of type \(A\), \(C\), or \(BC\) for \(G\) simply connected or type \(B\) for \(G\) of adjoint type. For the root system associated to \(G\), the positive roots can be chosen so that for a positive root \(\alpha\) either \(\sigma(\alpha) = \alpha\) or \(\sigma(\alpha)\) is a negative root. With such a choice, the restricted root system then consists of all non-zero roots of the form \(\alpha - \sigma(\alpha)\) for a root \(\alpha\).
The construction of the standard monomial theory for \(k[G/H]\) is accomplished by showing that \(k[G/H]\) is isomorphic as a \(G\)-module to the coordinate ring of a Richardson variety inside a Grassmann variety associated to \(G\). The Grassmann variety is constructed by creating an extended Lie algebra by adding a node to the Dynkin diagram for the Lie algebra of \(G\). The standard monomial theory for the Grassmann variety can then be translated to \(k[G/H]\). When the restricted root system is of type \(A\) an even more specific description of \(k[G/H]\) is obtained.
While the work here is done over a field of characteristic zero, the authors discuss the extent to which the results could be extended to arbitrary characteristic. symmetric varieties; spherical modules; Grassmannians; standard monomial theory; wonderful compactification; Schubert variety; Richardson variety Rocco Chirivì, Peter Littelmann, and Andrea Maffei, Equations defining symmetric varieties and affine Grassmannians, Int. Math. Res. Not. IMRN 2 (2009), 291 -- 347. Homogeneous spaces and generalizations, Group varieties, Group actions on varieties or schemes (quotients), Grassmannians, Schubert varieties, flag manifolds, Representation theory for linear algebraic groups Equations defining symmetric varieties and affine 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 Authors' abstract: Period domains, the classifying spaces for (pure, polarized) Hodge structures, and more generally Mumford-Tate domains, arise as open \(G_{\mathbb{R}}\)-orbits in flag varieties \(G / P\). We investigate Hodge-theoretic aspects of the geometry and representation theory associated with these flag varieties. In particular, we relate the Griffiths-Yukawa coupling to the variety of lines on \(G / P\) (under a minimal homogeneous embedding), construct a large class of polarized \(G_{\mathbb{R}}\)-orbits in \(G / P\), and compute the associated Hodge-theoretic boundary components. An emphasis is placed throughout on adjoint flag varieties and the corresponding families of Hodge structures of levels two and four. Mumford-Tate domain; variation of Hodge structure; boundary component; Schubert variety M. Kerr and C. Robles, Hodge theory and real orbits in flag varieties , preprint, [math.AG]. arXiv:1407.4507v1 Variation of Hodge structures (algebro-geometric aspects), Period matrices, variation of Hodge structure; degenerations, Grassmannians, Schubert varieties, flag manifolds, Transcendental methods, Hodge theory (algebro-geometric aspects) Variations of Hodge structure and orbits in flag 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 [For the entire collection see Zbl 0665.00004.]
The paper is an overview of the author's generalization of the Kazhdan- Lusztig polynomials introduced for the study of Bruhat orderings on Coxeter groups. Details will be found in the author's article [J. Algebra 111, 483-506 (1987; Zbl 0656.22007)]. If (W,S) is a Coxeter group and J any subset of generators in S, consider the Bruhat ordering and let \(W_ J\) denote the subgroup generated by J and \(W^ J\) the set of minimal coset representatives of \(W/W_ J\). If \({\mathcal H}\) denotes the Hecke algebra of (W,S) with coefficients in the ring \({\mathcal R}={\mathbb{Z}}[\sqrt{q}, 1/\sqrt{q}]\) then the author shows the existence of an \({\mathcal H}\)-module \(M^ J\) with an \({\mathcal R}\)-basis \(\{m_{\sigma}|\) \(\sigma \in W^ J\}\), and of an involution \(m\mapsto \bar m\) on \(M^ J\) compatible with the action of \({\mathcal H}\), and, finally, of two families of polynomials \(\{R_{\tau \sigma}|\) \((\tau,\sigma)\in \Gamma \}\), \(\{P_{\tau \sigma}|\) \((\tau,\sigma)\in \Gamma \}\), where \(\Gamma =\{(\tau,\sigma)\in W^ J\times W^ J|\) \(\tau\leq \sigma \}\) which satisfy a number of properties. For instance, with the length \(\ell (\sigma)\) of an element of the Coxeter group one may express the involution on the basis elements as follows
\[
\overline{m_{\sigma}}=\sum_{(\tau,\sigma)\in \Gamma}(- 1)^{\ell (\sigma)+\ell (\tau)}\cdot q^{-\ell (\sigma)}*R_{\tau \sigma}\cdot m_{\tau}.
\]
The R- and P-polynomials are linked by a formula involving the involution, and the P-polynomials allow the explicit description of a basis for the fixed elements of the involution. There is a uniqueness statement for the P-polynomials. For \(J=\emptyset\) one obtains the Kazhdan-Lusztig polynomials as a special case. One of the author's results specifies that for the element w of maximum length in \(W^ J\) and for \((\tau,\sigma)\in \Gamma\), his polynomial \(P_{\tau \sigma}\) is the Kazhdan-Lusztig polynomial \(p_{\tau w,\sigma w}\). The modules \(M^ J\) are used for the construction of a complex giving a (essentially) resolution of \({\mathcal H}\) which is investigated.
The entire set-up is applied to Kac-Moody algebras and the associated groups G for the study of the geometry of the varieties \(G/P_ J\) where \(P_ J\) are generalizations of a Borel subgroup \(P_{\emptyset}\). Kazhdan-Lusztig polynomials; Bruhat orderings on Coxeter groups; Kac- Moody algebras Infinite-dimensional Lie (super)algebras, Other algebraic groups (geometric aspects), Representation theory for linear algebraic groups, Infinite-dimensional Lie groups and their Lie algebras: general properties, Linear algebraic groups over arbitrary fields, Lie algebras of linear algebraic groups, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) An extension of Kazhdan-Lusztig theory | 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 By definition, the enhancement to the Milnor number is an invariant of the homotopy classes of fibred links in the sphere \(S^{2n-1}\) containing either in \(\mathbb Z\) if \(n=2\) or in \(\mathbb Z/2\,\mathbb Z\) if \(n>2\) (see [\textit{W. D. Neumann} and \textit{L. Rudolph}, Lect. Notes Math. 1350, 109--121 (1988; Zbl 0655.57015)]). The author proves that any element of \(\mathbb Z\) and \(\mathbb Z/2\,\mathbb Z\) is realized by the enhancement to the Milnor number of the corresponding fibred link associated with a certain class of mixed polynomials, that is, convergent power series with complex coefficients of the form \(f(z,\bar z) = \sum c_{I,J}z^I{\bar z}^J,\) where \(z=(z_1,\dots, z_n),\) \(\bar z\) is the complex conjugate of \(z,\) and \(I=(i_1,\dots, i_n),\) \(J=(j_1,\dots, j_n)\) for \(n\geq 2.\) enhanced Milnor number; fibered link, open book decomposition; mixed polar weighted homogeneous polynomials; radial Newton polygon; Newton boundary Inaba, K., On the enhancement to the Milnor number of a class of mixed polynomials, J. Math. Soc. Jpn., 66, 25-36, (2014) Singularities of surfaces or higher-dimensional varieties, Periodic orbits of vector fields and flows, Singularities of vector fields, topological aspects On the enhancement to the Milnor number of a class of mixed 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 For a Grothendieck category having a noetherian generator, we prove that there are only finitely many minimal atoms. This is a noncommutative analogue of the fact that every noetherian scheme has only finitely many irreducible components. It is also shown that each minimal atom is represented by a compressible object. minimal atom; Grothendieck category; Noetherian generator; compressible object Abelian categories, Grothendieck categories, Noetherian rings and modules (associative rings and algebras), Module categories in associative algebras, Noncommutative algebraic geometry, Module categories and commutative rings Finiteness of the number of minimal atoms in Grothendieck 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 This paper studies the analogue of the Grothendieck-Ogg-Shafarevich formula for curves over local fields. This formula was originally proved for curves over algebraically closed fields [\textit{A. Grothendieck}, SGA 5, Lect. Notes Math. 589, Exposé No. X, 372-406 (1977; Zbl 0356.14005)].
The main theorem in this paper is the following.
Let \(R\) be a complete discrete valuation ring with algebraically closed residue field \(k\); let \(S=\text{Spec}R\); let \(\eta\) be the generic point of \(S\); let \(X\) be a connected normal scheme, proper and flat over \(S\), of relative dimension one, with smooth generic fiber; let \(U\) be an open dense subscheme of \(X\) contained in \(X_\eta\); let \((F_i)_{i\in I}\) be the one-dimensional irreducible components of \(F:=X\setminus U\); let \(\mathbb F\) be a finite field with \(\text{char} \mathbb F\neq\text{char} k\); and let \(\mathcal F\) be a locally constant constructible étale sheaf on \(U\) of \(\mathbb F\)-vector spaces. Assume that the residue field of any point in \(X_\eta\setminus U\) is separable over \(\eta\), and that \(\mathcal F\) has no fierce ramification (see the paper for the definition). Then the main theorem asserts that, for each \(i\in I\), there is an open dense subscheme \(F_i^\circ\) of \(F_i\) such that:
(i) \(F\) is regular along \(F^\circ:=\bigcup_{i\in I}F_i^\circ\);
(ii) if \(x\in F_i^\circ\) for some \(i\in I\), then \(\text{sw}_x^{V/U}(\mathcal F)=\text{sw}_i(\mathcal F)\); and
\[
\begin{multlined}\text{totdim}_{\mathbb F}(R\Gamma_c(U_{\bar\eta},\mathcal F)) =\\ =\text{totdim}_{\mathbb F}(R\Gamma_c(U_{\bar\eta},\mathbb F)) \text{rk}_{\mathbb F}(\mathcal F) +\sum_{i\in I}\chi_c(F_i^\circ)\text{sw}_i(\mathcal F) +\sum_{x\in F\setminus F^\circ}\text{sw}_x^{V/U}(\mathcal F).\end{multlined}\tag{iii}
\]
Here \(\chi_c\) is the Euler-Poincaré characteristic if \(F_i^\circ\) is vertical, or minus the discriminant over \(S\) if it is horizontal; \(\text{totdim}_{\mathbb F}\) is the sum of the dimension over \(\mathbb F\) and the dimension over \(\mathbb F\) of the Swan conductor; and \(R\Gamma_c\) refers to the alternating sum of the groups \(H^i_c\). See the paper for the definition of \(\text{sw}_i(\mathcal F)\).
The first part of the paper is devoted to defining the Swan conductor \(\text{sw}_x^{V/U}(\mathcal F)\) in codimension 2, and the second part gives the proof of the above theorem. A key tool in that proof is the Lefschetz fixed point formula for arithmetic surfaces proved by the author [\textit{A. Abbes}, Compos. Math. 122, 23-111 (2000; see the preceding review Zbl 0986.14014)].
The paper also formulates a conjecture that, for all closed points \(x\in X\), \(\text{sw}_x^{V/U}(\mathcal F)\) is (a) an integer, and (b) independent of the choice of \(V\). Part (a) of this conjecture extends a conjecture of Serre on the existence of Artin representations. Grothendieck-Ogg-Shafarevich formula; Swan conductor; Lefschetz fixed point formula; Artin representation A. Abbès, The Grothendieck-Ogg-Shafarevich formula for arithmetic surfaces, Journal of Algebraic Geometry, 9 (2000), 529-576. Arithmetic varieties and schemes; Arakelov theory; heights, Topological properties in algebraic geometry, Étale and other Grothendieck topologies and (co)homologies The Grothendieck-Ogg-Shafarevich formula for arithmetic surfaces | 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 classical problem in invariant theory is to find a precise description of the subring of the coordinate ring for the \(n\times r\) matrices \(k[M_{nr}]\), \(r<n\), consisting of all the invariant elements with respect to the natural right action of the special linear group \(\text{SL}_r(k)\), where \(k\) is an algebraically closed field of characteristic \(0\). In this paper the authors present a solution for the analogous problem in the quantum case. Here \(k[M_{nr}]\) is replaced by the quantum matrix bialgebra and \(\text{SL}_r(k)\) by the quantum special linear group. This leads to the formulation of the first and second fundamental theorem for the quantum special linear group. Similarly to what happens for the commutative case, the first theorem of quantum coinvariant theory states that the ring of quantum coinvariants coincides with the ring generated by certain quantum minors in the quantum matrix bialgebra. This is precisely the ring of the so-called quantum Grassmannian. Using the results in the first author's earlier paper they are able to give a presentation of the ring of quantum coinvariants in terms of generators and relations. This is the content of the second fundamental theorem of quantum coinvariant theory. Both the first and the second theorem of quantum coinvariant theory reduce to the corresponding classical results when the indeterminate \(q\) is specialized to \(1\). At the end they use the given presentation of the quantum Grassmannian to define quantum Schubert varieties and to show that they are quantum homogeneous spaces; that is, they admit a coaction by a suitable quantum group. quantum special linear groups; quantum matrix bialgebras; rings of quantum coinvariants; quantum Grassmannians; presentations; quantum Schubert varieties; coactions R. Fioresi and C. Hacon, Quantum coinvariant theory for the quantum special linear group and quantum Schubert varieties. J. Algebra 242 (2001), 433-446. Quantum groups (quantized function algebras) and their representations, Geometric invariant theory, Grassmannians, Schubert varieties, flag manifolds, Actions of groups and semigroups; invariant theory (associative rings and algebras), Coalgebras, bialgebras, Hopf algebras [See also 57T05, 16S30--16S40]; rings, modules, etc. on which these act, Quantum groups (quantized enveloping algebras) and related deformations Quantum coinvariant theory for the quantum special linear group and quantum 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 [For the entire collection see Zbl 0546.00024.]
This excellent survey article addresses the question of viewing a topos as a ''generalized space''. Since its birth in SGA 4 and the subsequent pioneering work of Lawvere and Tierney, the scope of topos theory has greatly expanded and it is distinguished by its generality and the diversity of its examples and applications. However, as a result of the represenation theorems outlined in this article, the notion of Grothendieck topos can be described in terms of locales and localic groups and groupoids.
After an outline of the necessary preliminaries on geometric morphisms and geometric theories, the article proceeds to the two main open covering theorems. The first one is an improvement by André Joyal of Barr's theorem. It states that if \({\mathcal E}\) is a Grothendieck topos, there exists a connected, locally connected geometric morphism \({\mathcal F}\to^{f}{\mathcal E}\), where \({\mathcal F}\) is localic over \({\mathcal S}\) (the base topos of sets). The second major result is Peter Freyd's theorem that given a Grothendieck topos \({\mathcal E}\), there is a connected, atomic geometric morphism \({\mathcal F}\to^{f}{\mathcal E}\), where \({\mathcal F}\) is localic over \({\mathcal C}(G)\), the topos of continuous G-sets, where G is a topological group. The author also briefly discusses the result of the reviewer, which states that every Grothendieck topos \({\mathcal E}\) admits a hyperconnected geometric morphism \({\mathcal F}\to^{f}{\mathcal E}\) where \({\mathcal F}\) is an étendue, i.e. \({\mathcal F}\) is ''locally localic''.
The covering theorems of Joyal and Freyd lead to representation theorems for Grothendieck topoi. Joyal and Tierney have shown that every Grothendieck topos is equivalent to a topos of \({\mathfrak G}\)-sheaves, where \({\mathfrak G}\) is a localic groupoid. On the other hand, from Freyd's result it follows that a Grothendieck topos is equivalent to an exponential variety (full subcategory closed under limits, colimits, and power objects) in a topos localic over continuous G-sets, where G is a topological group. These results provide some justification to the view of toposes as ''generalized spaces''.
The author is to be commended for compiling these results about the basic nature of Grothendieck toposes into this highly readable article. Survey articles such as this one serve the valuable function of making results from diverse sources more readily accessible as well as providing a perspective on the interrelationships of these results. survey; topos; generalized space; represenation theorems; Grothendieck topos; locales; localic groups; groupoids; geometric morphisms; Barr's theorem; Freyd's theorem; étendue; covering theorems; exponential variety; topological group Peter T. Johnstone, How general is a generalized space?, Aspects of topology, London Math. Soc. Lecture Note Ser., vol. 93, Cambridge Univ. Press, Cambridge, 1985, pp. 77 -- 111. Topoi, Abstract manifolds and fiber bundles (category-theoretic aspects), Étale and other Grothendieck topologies and (co)homologies, Research exposition (monographs, survey articles) pertaining to category theory, Groupoids, semigroupoids, semigroups, groups (viewed as categories), Topological and differentiable algebraic systems, Varieties How general is a generalized space? | 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 Surfaces of degree s and with a line of multiplicity s-2 are easily seen to be rational. In this note the author shows that they can be obtained as blow-ups of Hirzebruch surfaces of degree at most s-1 blowing up at most 3s-4 points. - Special attention is given to quartics with a double line. Curves on such quartics were exploited by Peskine and Gruson to classify all pairs of degree and genus that can occur for space curves.
The author studies the k-normality of smooth irreducible curves on such quartics, and gives a nice bound on k. blow-ups; Hirzebruch surfaces; quartics with a double line; normality Special surfaces, Projective techniques in algebraic geometry, Families, moduli of curves (algebraic) Courbes tracées sur les surfaces quartiques à droite double. (Curves on quartics with double line) | 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 conchoid of a plane curve \(C\) is constructed using a fixed circle \(B\) in the affine plane. We generalize the classical definition so that we obtain a conchoid from any pair of curves \(B\) and \(C\) in the projective plane. We present two definitions, one purely algebraic through resultants and a more geometric one using an incidence correspondence in \(\mathbb{P}^{2} \times \mathbb{P}^{2}\). We prove, among other things, that the conchoid of a generic curve of fixed degree is irreducible, we determine its singularities and give a formula for its degree and genus. In the final section we return to the classical case: for any given curve \(C\) we give a criterion for its conchoid to be irreducible and we give a procedure to determine when a curve is the conchoid of another. conchoid; resultant; double planes Albano, A.; Roggero, M., Conchoidal transform of two plane curves, Appl. Algebra Eng. Commun. Comput., 21, 309-328, (2010) Special algebraic curves and curves of low genus, Plane and space curves, Computational aspects of algebraic curves Conchoidal transform of two plane 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 The aim of the book is to make an attempt to expose a new unifying approach to the study of numerous quantized algebras. This approach is based on investigations of noncommutative schemes in categories. The point of departure in the first chapter of the book is the notion of a left spectrum \(\text{Spec}_l R\) of an associatve ring \(R\) with unity. It consists of all left ideals \(P\) in \(R\) with the following property. For any element \(x\in R\setminus P\) there exists a finitely generated additive subgroup \(H\) in \(R\) such that \(zHx\subseteq P\) where \(z\in R\) implies \(z\in P\). The set \(\text{Spec}_l R\) is nonempty since it contains the set \(\text{Max}_l R\) of all maximal left ideals and the set \(\text{Spec}_l' R\) of all completely prime left ideals. If \(P,P'\in \text{Spec}_l R\) then \(P\leq P'\) if there exists a finitely generated additive subgroup \(H\) in \(R\) such that \(zH\subseteq P\) implies \(z\in P'\). There exists a topology \(\tau_*\) on \(\text{Spec}_l\) with the base \(\{\bigcup_{P'\geq P} P'\mid P\in \text{Spec}_l R\}\). There exists a topology \(\tau^*\) on \(\text{Spec}_l R\) determined by the base of closed subsets of the form \(\{p\in \text{Spec}_l R\mid p\leq M\}\) where \(M\) runs through the set of all proper ideals in \(R\). Flat localizations of Abelian categories are discussed. A stability theorem with respect to localizations is proved for the left spectrum. Special categories of rings are selected. (Pre)images of morphisms in these categories preserve preorder \(\leq\). It is shown that the intersection of all elements of \(\text{Spec}_l R\) is the Levitzki radical of \(R\). In particular the topological space \(\text{Spec}_l R\) has a base of quasi-compact open sets. Structure presheaves of modules over rings, noncommutative quasi-affine schemes and projective spectra are introduced.
In the second chapter all main `small' quantized rings are introduced: the quantum plane \(k_q[X, Y]\), the algebra of \(q\)-differential operators \(D_{q,h}\), quantum Heisenberg and Weyl algebra \(W_{q, 1}\), the quantum envelope \(U_q(sl(2))\), the coordinate ring \(M_q(2)\) of \(2\times 2\) matrices, the coordinate ring \(A(SL_q(2))\), twisted \(SL(2)\) group, \(W_v(sl(2))\) of Woronovicz. The goal of chapter 2 is to develop the representation theory of these algebras. It is shown that all these algebras are specializations of the hyperbolic ring \(A\{\theta, \zeta\}\) which is generated by a commutative ring \(A\) with fixed \(\theta\in \Aut A\), \(\zeta\in A\), and by elements \(x\), \(y\) with defining relations
\[
xa= \theta(a) x,\quad ay= y\theta(a),\quad a\in A;\quad xy= \zeta,\quad yx= \theta^{- 1}(\zeta).
\]
It is worth to mention that the theory of these and even more general classes of algebras under the name of generalized Weyl algebras is developed by \textit{V. V. Bavula} [Generalized Weyl algebras (Bielefeld, preprint, 1994)]. Unfortunately, it is not mentioned in the book.
In section 1 of chapter 2 the left spectrum of a skew polynomial ring \(A[X, \theta]\) is studied. These results are applied to the quantum plane. Section 3 contains an almost complete description of \(\text{Spec}_l R\) of a hyperbolic ring \(R\). This description is applied to the rings mentioned above.
Chapter 3 is devoted to the categorical point of view on geometrical objects. The author introduces the notion of the spectrum of an Abelian category, studies its behaviour with respect to localizations, Serre subcategories, Grothendieck categories, local Abelian categories, localizations at points, topologies on categories.
Chapter 4 contains generalization of the notion of a hyperbolic ring. Namely the author introduces the notion of a hyperbolic category over an Abelian category. The main result of the chapter is the description of a hyperbolic category which is naturally related to the spectrum of the underlying Abelian category. This result generalizes similar results from chapter 2.
Chapter 5 is concerned with skew PBW monads in a monoidal category \(A\). A skew PBW monad is a generalization to monads of the notion of a hyperbolic ring. Other examples of skew PBW monads are related to Kac-Moody and Virasoro Lie algebras. The author studies semigroup-graded monads \(F\) and their spectra. The main result of the chapter shows that all points of \(F\)-modules which `grow up' over a given point of \(\text{Spec } A\) can be represented by a graded module with the grading associated to this point. The chapter ends with considerations of quasi-holonomic modules, \(F\)-comodules, spectra of Weyl algebra and their quantizations, representations of Kac-Moody algebra, two-parameter deformations of the algebras \(M(2)\), \(GL(2)\).
In chapter 6 the author surveys major approaches to noncommutative spectral theory such as injective spectrum by Gabriel, Goldman's spectrum of a ring, affine scheme by F. Van Oystaeyen and A. Verschoren, Cohn's affine scheme. It is shown that these spectra can be deduced from the case considered in the book. The chapter ends with exposing properties of Gabriel-Krull dimension and a calculations of dimensions of some spectra.
The last chapter 7 is concerned with the projective spectrum of graded monads. Affine and projective fibres, blowing up and related topics are considered. flat localizations of Abelian categories; structure presheaves of modules; quantized algebras; noncommutative schemes in categories; left spectrum; maximal left ideals; completely prime left ideals; categories of rings; Levitzki radical; quasi-affine schemes; projective spectra; quantized rings; quantum planes; algebra of \(q\)-differential operators; Weyl algebras; quantum envelopes; coordinate rings; generalized Weyl algebras; skew polynomial rings; Serre subcategories; Grothendieck categories; hyperbolic rings; skew PBW monads; monoidal category; Kac-Moody and Virasoro Lie algebras; semigroup-graded monads; Gabriel-Krull dimension Rosenberg, A.L.: Algebraic Geometry Representations of Quantized Algebras. Kluwer Academic Publishers, Dordrecht, Boston London (1995) Research exposition (monographs, survey articles) pertaining to associative rings and algebras, Coalgebras, bialgebras, Hopf algebras [See also 57T05, 16S30--16S40]; rings, modules, etc. on which these act, Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras, Quantum groups (quantized enveloping algebras) and related deformations, Noncommutative algebraic geometry, Torsion theories; radicals on module categories (associative algebraic aspects), Rings of differential operators (associative algebraic aspects), Local categories and functors, Abelian categories, Grothendieck categories, Graded rings and modules (associative rings and algebras), Associative rings of functions, subdirect products, sheaves of rings, ``Super'' (or ``skew'') structure, Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects), Abstract manifolds and fiber bundles (category-theoretic aspects) Noncommutative algebraic geometry and representations of quantized 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 We present a relationship between discriminantly separable polynomials and quad-graphs. We start from a classification of strongly discriminantly separable polynomials in three variables of degree two in each variable. We provide their geometric interpretation, connecting discriminantly separable polynomials with the equations of pencils of conics written in the Darboux coordinates. Then we give a construction of integrable quad-graphs associated with representatives of strongly discriminantly separable polynomials.
See also the authors [J. Geom. Mech. 6, No. 3, 319--333 (2014; Zbl 1309.37067)]. discriminantly separable polynomials; integrable quad-equations; pencils of conics; Möbius transformations; two-valued groups Dragović, V. and Kukić, K., Discriminantly Separable Polynomials and Quad-Graphs, arXiv:1303.6534v1 (26 Mar 2013). Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions, Projective techniques in algebraic geometry, Relationships between algebraic curves and integrable systems Quad-graphs and discriminantly separable polynomials of type \({\mathcal P}^2_3\) | 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 four lectures delivered in 2001, the author gives a beautiful introduction to the theory of
pure motives, together with a survey over the standard conjectures, the most important related results, and
recent developments for elliptic modular varieties.
In the first chapter, the concept of pure motives is introduced together with a motivation and first results and
examples. The second chapter continues with Grothendieck's standard conjectures, Jannsen's results and
the Chow-Künneth decomposition of the diagonal. Both parts form a comfortable and straightforward guide into the framework of the theory [for more details, the reader may look into the papers of the
1991 Seattle conference, especially \textit{A. Scholl}, in: Motives, Proc. Summer Res. Conf. Motives, Seattle 1991, Proc. Symp. Pure Math. 55, Part 1,
163--187 (1994; Zbl 0814.14001); \textit{S. Kleiman}, ibid., 3--20 (1994; Zbl 0820.14006) and
\textit{U. Jannsen}, ibid., 245--302 (1994; Zbl 0811.14004)].
The third chapter surveys the author's results on surfaces
[J. Reine Angew. Math. 409, 190--204 (1990; Zbl 0698.14032)], his conjectures on a filtration of the Chow group
and its relation with the Bloch-Beilinson filtration and the Beilinson conjectures, and the work of
\textit{A. M. Shermenev} [Funct. Anal. Appl. 8, 47--53 (1974; Zbl 0294.14003)], \textit{A. Beauville} [in:
Algebraic geometry, Proc. Jap.-Fr. Conf., Lect. Notes Math. 1016, 238--260 (1983; Zbl 0526.14001)],
\textit{C. Deninger} and \textit{J. Murre} [J. Reine Angew. Math. 422, 201--219 (1991; Zbl 0745.14003)]
on the Chow-Künneth decomposition for abelian varieties. The last chapter deals with even more recent results
of \textit{B. B. Gordon, M. Hanamura} and the author [J. Reine Angew. Math. 514, 145--164 (1999; Zbl 0942.14019); ibid., 558, 1--14 (2003; Zbl 1038.14002)] on this decomposition for modular varieties, including the advanced example of the elliptic modular threefold.
The paper is very clearly written and gives an impressive guideline from the fundamentals of the theory to very recent achievements. Despite A. Beilinson stated once ``the motivic papers often hopelessly increase
their volume while you write them; they have a tendency to lead the reader astray
from the central ideas and simple computations to the burdock thicket of generalities'',
fortunately, this is not the case with these lecture notes. Grothendieck's motives; standard conjectures; Chow-Künneth decomposition; Bloch-Beilinson filtration; elliptic modular varieties Murre, J.: Lectures on motives. In: Müller-Stach, S., Peters, C. (eds.) Trancendental Aspects of Algebraic Cycles, London Math. Soc. Lecture Note Series, No: 133, pp. 123--170. Cambridge University Press, Cambridge (2004) (Equivariant) Chow groups and rings; motives, Algebraic cycles, Motivic cohomology; motivic homotopy theory, Modular and Shimura varieties, Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects) Lecture on motives | 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 a well-known configuration of 9 points and 12 lines in \(\mathbb P^2(k)\), the Hesse configuration, in which each point lies on 4 lines and each line contains 3 points. Such points can be chosen as the nine inflection points of a nonsingular plane cubic curve, and they can be as well taken as common inflection points of the Hesse pencil
\[
\lambda (x^3+y^3+z^3)+\mu xyz=0.
\]
The group of plane automorphisms preserving the Hesse pencil has order 216 and it is isomorphic to \((\mathbb Z/3\mathbb Z)SL)^2 \rtimes SL(2, \mathbb F_3)\). The algebra of invariant polynomials of one of its extensions to a subgroup of \(GL(3,\mathbb C)\) has a generator of degree 6 defining a plane sextic \(C_6\).
The double covers of \(\mathbb P^2\) branched over \(C_6\) and over the singular sextic \(C'_6\) with 8 cuspidal singularities are both \(K3\) surfaces and they are singular in the sense of Shioda, i.e. the subgroup of algebraic cycles in the second cohomology group is of maximal rank.
The authors compute the intersection form defined by the cup-product on these subgroups and describe the geometrical meaning of the set of intersection points \(C_6\) cuts each curve of the Hesse pencil at. configuration; pencil; double covers M. Artebani; I. Dolgachev, The Hesse pencil of plane cubic curves, L'Enseignement Mathématique. Revue Internationale. 2e Série, 55, 235-273, (2009) Families, moduli of curves (algebraic), Plane and space curves, Families, moduli, classification: algebraic theory The Hesse pencil of plane cubic 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 The paper under review is concerned with certain nice properties enjoyed by semisimple simply connected algebraic groups. If \(G\) is any one such, and \(Z\) is its center, the latter can be identified with the fundamental group of \(G/Z\). Others important interpretations of the center of \(G\), for instance in terms of coroots and coweight lattices, are quickly listed in the introduction. The paper however focuses on some properties studied in an important paper by \textit{P.~Seidel} [Geom. Funct. Anal. 7, No. 6, 1046--1095 (1997; Zbl 0928.53042)], who proved that the fundamental group of the group of hamiltonian symplectomorphisms of a symplectic variety \(X\) can be mapped to the group of invertible elements of the quantum cohomology ring of \(X\) localised in the quantum parameters.
The point is that if \(X\) is a rational homogeneous space, there is a symplectic structure on \(X\) induced by its natural projective structure, and one so gets what the authors baptize as the \textsl{Seidel's representation}, i.e. a map from the fundamental group of \(G/Z\) and the subgroup of the invertibles elements of the quantum cohomology ring of \(X\) localized at the quantum parameter. In a previous paper [Int. Math. Res. Not. 2007, No. 22, Article ID rnm107, 29 p. (2007; Zbl 1142.14033)] the authors study this map in the case when \(X\) a \textsl{minuscule} or \textsl{cominuscule} homogeneous space (see the review of that paper, by the same reviewer, for glossary and terminology).
The main achievement of this work consists in extending the description of the Seidel's representation for all homogeneous spaces, proving in particular that it is faithful. The product structure in \(QH^*_T(X)_{loc}\) is described in the first main theorem of the paper, while the second main theorem focuses on the explicit description of the representation map. The results rely on two previous results gotten by Peterson [Quantum Cohomology of \(G/P\), 1997, unpublished] and P.~Magyar (Notes on Schubert classes of a loop group (2007), \url{arXiv:0705.3826}).
The detailed description of the Seidel's representation is performed in Section 4, the last of the paper, while the second section, after the general introduction, is devoted to describe the crucial results by Peterson and Magjar. A key formula by Magyar is extended in section 3 in a more general situation. This paper is very well written, like some others previous papers on similar subjects by the same authors. The interested and motivated reader is advised to read all of them at once, for the best profit. equivariant quantum cohomology; homogeneous spaces; Schubert Calculus; Gromov-Witten invariant DOI: 10.4310/MRL.2009.v16.n1.a2 Grassmannians, Schubert varieties, flag manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Affine symmetries of the equivariant quantum cohomology ring of rational 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 In this paper the authors develop methodology for estimating the number of \(\mathbb{F}_q\)-rational solutions of equations of the type \[f(P_{m_1},P_{m_2},\ldots,P_{m_d})+X_1^e+\cdots+X_n^e+g_1=0,\] where \(P_{m_i}\) are power-sum polynomials in \(\mathbb{F}_q[X_1,\ldots,X_n]\), \(g_1\in \mathbb{F}_q[X_1,\ldots,X_n]\) with \(\deg g_1<e.\) The methodology relies on the study of the geometry of the set of common zeros of symmetric polynomials over the algebraic closure of \(\mathbb{F}_q\).
In the last section the authors apply their methodology to obtain estimates and existence results for deformed diagonal equations: \[c_1X_1^m+\ldots+c_nX_n^m=g,\quad c_i\in \mathbb{F}_q, c_i\neq 0, \deg g<m,\] generalized Markoff-Hurwitz-type equations: \[(a_1X_1^{m_1}+\cdots+a_nX_n^{m_n})^k=bX_1^{k_1}\ldots X_n^{k_k}, \quad a_i\in \mathbb{F}_q\setminus \{0\},\] and Carlitz's equations: \[h_1(X_1)+\ldots+h_n(X_n)=g,\quad g\in \mathbb{F}_q[X_1,\ldots,X_n], \deg h_i =d, \deg g<d.\] The obtained estimates are of the form \(q^{n-1}+\mathcal{O}(q^{n/2}).\) finite fields; symmetric polynomials; singular locus; rational solutions; diagonal equations Polynomials over finite fields, Symmetric functions and generalizations, Rational points, Finite ground fields in algebraic geometry, Varieties over finite and local fields Estimates on the number of \(\mathbb{F}_q\)-rational solutions of variants of diagonal equations 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 An elliptic curve \(E\) over a field \(K\) of characteristic \(p>0\) is called supersingular if the group \(E(\overline{K})\) has no \(p\)-torsion. This condition depends only on the \(j\)-invariant of \(E\) and it is well known that there are only finitely many supersingular \(j\)-invariants in \(\mathbb{F}_p\).
The authors of the paper under review describe several different ways of constructing canonical polynomials in \(\mathbb{Q} [j]\) whose reduction modulo \(p\) gives the supersingular polynomial
\[
ss_p(j):= \prod_{\substack{ E/\overline{\mathbb{F}}_p\\ E\text{ supersingular }}} (j-j(E))\in \mathbb{F}_p[j].
\]
These polynomials are of three kinds:
A. Polynomials coming from modular forms of weight \(p-1\).
Four special modular forms of weight \(p-1\) are defined and, if \(f\) is one of these four forms, the coefficients of the associated polynomial \(\widetilde{f}\) are \(p\)-integral and
\[
ss_p(j)= \pm j^\delta (j-1728)^\varepsilon \widetilde{f}(j) \bmod p\qquad (\delta\in \{0,1,2\},\;\varepsilon\in \{0,1\}).
\]
B. The Atkin orthogonal polynomials.
This description was found by Atkin more than ten years ago but proofs have never been published. Atkin has defined a sequence of polynomials \(A_n(j)\in \mathbb{Q}[j]\), one in each degree \(n\), as the orthogonal polynomials with respect to a special scalar product. The coefficients of \(A_n\) are rational numbers in general but they are \(p\)-integral for primes \(p> 2n\). In particular if \(n_p\) is the degree of the supersingular polynomial \(ss_p\), then \(A_{n_p}\) has \(p\)-integral coefficients and we have the congruence
\[
ss_p(j) \equiv A_{n_p}(j) \pmod p
\]
as well as recursion relation, closed formula and differential equation of \(A_n\).
The proofs here are simpler than those of Atkin.
C. Other orthogonal polynomials coming from hyperelliptic series.
This is a partially expository paper. supersingular elliptic curve; orthogonal polynomials from hyperelliptic series; supersingular polynomial; modular forms; Atkin orthogonal polynomials; congruence; recursion relation Kaneko, M., Zagier, D.: Supersingular \(j\)-invariants, hypergeometric series, and Atkin's orthogonal polynomials. In: Computational Perspectives on Number Theory (Chicago, IL, 1995). AMS/IP Stud. Adv. Math., 7, pp. 97-126. American Mathematical Society, Providence, RI (1998) Curves over finite and local fields, Congruences for modular and \(p\)-adic modular forms, Local ground fields in algebraic geometry, Holomorphic modular forms of integral weight Supersingular \(j\)-invariants, hypergeometric series and Atkin's orthogonal polynomials | 0 |
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