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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 author shows how to associate to any polynomial \(P\), of degree \(d\), with non-negative integer coefficients and constant term equal to 1, a pair of elements \(y_P\) and \(w_P\) in the symmetric group \(S_n\) where \(n=d+P(1)+1\). Then he proves that \(P\) is indeed the Kazhdan-Lusztig polynomial of those two elements, by reducing the problem to the case when \(P-1\) is a monomial and by using intersection cohomology of Schubert varieties. Kazhdan-Lusztig polynomials; Schubert varieties; intersection cohomology P. Polo, Construction of arbitrary Kazhdan-Lusztig polynomials in symmetric groups, \textit{Repre-} \textit{sent. Theory}, 3 (1999), 90--104.Zbl 0968.14029 MR 1698201 Grassmannians, Schubert varieties, flag manifolds, Representation theory for linear algebraic groups, Representations of finite symmetric groups, Symmetric functions and generalizations, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Construction of arbitrary Kazhdan-Lusztig polynomials in symmetric groups | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(GL_ n\) be the group of \(n \times n\) invertible complex matrices, and \(P\) a parabolic subgroup of \(GL_ n\). In this paper we give a geometric description of the cohomology ring of a Schubert subvariety \(Y\) of \(Gl_ n/P\). Our main result (theorem 3.1) states that the coordinate ring \(A(Y \cap Z)\) of the scheme-theoretic intersection of \(Y\) and the zero scheme \(Z\) of the vector field \(V\) associated to a principal regular nilpotent element \({\mathfrak n}\) of \({\mathfrak g}{\mathfrak l}_ n\) is isomorphic to the cohomology algebra \(H^*(Y;\mathbb{C})\) of \(Y\). This theorem was conjectured for any reductive algebraic group \(G\) by \textit{E. Akyildiz}, \textit{J. B. Carell} and \textit{D. I. Liebermann} in Compos. Math. 57, 237- 248 (1986; Zbl 0613.14035), and it was proved for the Grassmannian manifolds by \textit{E. Akyildiz} and \textit{Y. Akyildiz} in J. Differ. Geom. 29, No. 1, 135-142 (1989; Zbl 0692.14031). We were recently informed that \textit{D. H. Peterson} has just proved that \(GL_ n\) is exactly the algebraic group \(G\) where the cohomology ring of any Schubert subvariety \(Y\) of the space \(G/B\) is isomorphic to \(A(Y \cap Z)\). Here \(B\) stands for a Borel subgroup of \(G\). It is also interesting to note that the cohomology ring of the union of two Schubert subvarieties in \(GL_ n/P\) may not admit such a description. This result is due to \textit{J. B. Carrell}. cohomology ring of a Schubert variety E. Akyıldız, A. Lascoux, and P. Pragacz, Cohomology of Schubert subvarieties of \?\?_{\?}/\?, J. Differential Geom. 35 (1992), no. 3, 511 -- 519. Grassmannians, Schubert varieties, flag manifolds, (Co)homology theory in algebraic geometry Cohomology of Schubert subvarieties of \(GL_ n/P\) | 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{J. Kajiwara} proved in J. Math. Soc. Japan 17, 36-46 (1965; Zbl 0146.110) that the limit of a monotonously increasing sequence of Cousin- II domains over a Stein manifold is a multiform Cousin-II domain. In this paper the author shows that a limit of a monotonously increasing sequence of Cousin-II domains over a Grassmann manifold is a Cousin-II domain if it is simply connected. envelope of holomorphy; limit of Cousin-II domains; Grassmann manifold Domains of holomorphy, Envelopes of holomorphy, Grassmannians, Schubert varieties, flag manifolds, Analytic continuation The limit of a sequence of Cousin-II domains over a Grassmann 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 We prove that if \(X\) is a Grassmannian of type A, then the Schubert basis of the (small) quantum cohomology ring \(\mathrm{QH}(X)\) is the only homogeneous deformation of the Schubert basis of the ordinary cohomology ring \(H^*(X)\) that multiplies with nonnegative structure constants. This implies that the (three point, genus zero) Gromov-Witten invariants of \(X\) are uniquely determined by Witten's presentation of \(\mathrm{QH}(X)\) and the fact that they are nonnegative. We conjecture that the same is true for any flag variety \(X=G/P\) of simply laced Lie type. For the variety of complete flags in \(\mathbb{C}^n\), this conjecture is equivalent to Fomin, Gelfand, and Postnikov's conjecture that the quantum Schubert polynomials of type A are uniquely determined by positivity properties. Our proof for Grassmannians answers a question of Fulton. quantum cohomology; Grassmannians; positivity; Gromov-Witten invariant; Schubert basis; quantum Schubert polynomials; flag varieties; symmetric functions; Seidel representation Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus Positivity determines the quantum cohomology of Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We call a flag variety admissible if its automorphism group is the projective general linear group. (This holds in most cases.)
Let \(K\) be a field of characteristic 0, containing all roots of unity. Let the \(K\)-variety \(X\) be a form of an admissible flag variety. We prove that \(X\) is either ruled, or the automorphism group of \(X\) is bounded, meaning that there exists a constant \(C \in \mathbb{N}\) such that if \(G\) is a finite subgroup of \(\Aut_K(X)\), then the cardinality of \(G\) is smaller than \(C\). Automorphisms of surfaces and higher-dimensional varieties, Grassmannians, Schubert varieties, flag manifolds Boundedness properties of automorphism groups of forms of 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 In this paper, and in characteristic \(0\), the authors describe the smooth locus of the moduli space of linear series with prescribed vanishing sequences in at most two marked points. In the particular case of no marked points, this result specializes to the Gieseker-Petri theorem, proved previously in [\textit{D. Gieseker}, Invent. Math. 66, 251--275 (1982; Zbl 0522.14015)] and [\textit{ D. Eisenbud} and \textit{J. Harris}, Invent. Math. 74, 269--280 (1983; Zbl 0533.14012)], and recently in [\textit{S. Payne } and \textit{D. Jensen}, Algebra Number Theory 8, No. 9, 2043-2066 (2014; Zbl 1317.14139)].
The main idea of the proof is based on degeneration to a chain of elliptic curves and studying the corresponding the moduli space of limit linear series, introduced by Eisenbud and Harris. In parallel, the smoothness conditions of a point impose that the point must lie on the smooth locus of Schubert cycles and the smooth locus of Schubert varieties is characterized. degeneration; Grassmannian; Schubert varieties; almost-transverse flags; linear series; ramification Special divisors on curves (gonality, Brill-Noether theory), Grassmannians, Schubert varieties, flag manifolds The Gieseker-Petri theorem and imposed ramification | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In the first half of this article, we review the Steinberg theory for double flag varieties for symmetric pairs. For a special case of the symmetric space of type AIII, we will consider \(X=\mathrm{GL}_{2n}/P_{(n,n)}\times\mathrm{GL}_n/B_n^+\times\mathrm{GL}_n/B_n^-\) on which \(K=\mathrm{GL}_n\times \mathrm{GL}_n\) acts diagonally. We give a classification of \(K\)-orbits in \(X\), and explicit combinatorial description of the Steinberg maps.
In the latter half, we develop the theory of embedding of a double flag variety into a larger one. This embedding is a powerful tool to study different types of double flag varieties in terms of the known ones. We prove an embedding theorem of orbits in full generality and give an example of type CI which is embedded into type AIII. Grassmannians, Schubert varieties, flag manifolds, Coadjoint orbits; nilpotent varieties, Differential geometry of symmetric spaces, Exact enumeration problems, generating functions Orbit embedding for double flag varieties and Steinberg maps | 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 To each polynomial \(P\) with integral non-negative coefficients and constant term equal to 1, of degree \(d\), we associate a pair of elements \((y,w)\) in the symmetric group \(S_n\), where \(n=1+d+P(1)\), for which we prove that the Kazhdan-Lusztig polynomial \(P_{y,w}\) equals \(P\). This pair satisfies \(\ell(w)-\ell(y)=2d+P(1)-1\), where \(\ell(w)\) denotes the number of inversions of \(w\). -- For details see the author's paper: \textit{P. Polo}, Represent. Theory 3, No. 4, 90-104 (1999; Zbl 0968.14029). Kazhdan-Lusztig polynomials; Schubert varieties; intersection cohomology Grassmannians, Schubert varieties, flag manifolds, Representation theory for linear algebraic groups, Representations of finite symmetric groups, Symmetric functions and generalizations, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Construction of arbitrary 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 What the authors call \textsl{The Main Theorem} of this enlightening paper can be conventionally split into two parts. The first is concerned with a certain symmetric structure of the \(n\)th exterior power of a polynomial ring; the second with a general determinantal formula evoking Giambelli's formula in classical Schubert Calculus on Grassmann Schemes or Jacobi-Trudy formula in the theory of symmetric polynomials. In spite of being strongly related with classical and widely investigated subjects, the result is new and shed further light on the beautiful algebraic properties of exterior powers of a module.
To describe the main theorem of the paper in a more precise way, here is a piece of notation. Let \(\bigotimes^n_AA[X]\) and \(\bigwedge^n_AA[X]\) be, respectively, the tensor and exterior \(n\)th-power of a polynomial algebra in one indeterminate with coefficients in \(A\), a commutative ring with unit. Also, let \(R:=A[X_1,\dots, X_n]\) be the ring of polynomials in \(n\) indeterminates. Because of the natural identification between \(\bigotimes^n_AA[X]\) and \(R\), the former is naturally an \(S:=R^{\text{sym}}\)-module, where \(R^{\text{sym}}\) is the \(A\)-algebra of the symmetric polynomials in \(R\). The first part of the main theorem then says that there exists a unique \(S\)-module structure on \(\bigwedge^n_AA[X]\) such that the canonical projection \(\bigotimes^nA[X]\rightarrow \bigwedge^nA[X]\) is \(S\)-linear. The \textsl{symmetric structure} of \(\bigwedge^n_AA[X]\) is described in a very explicit way: it turns out that \(\bigwedge^nA[X]\) is a free \(S\)-module of rank \(1\) generated by \(\phi:=X^{n-1}\wedge X^{n-2}\wedge\ldots\wedge X^0\) (\(X^0=1_A\)). As a consequence, if \(f_1,\dots, f_{n}\) are \(n\) arbitrary elements of \(A[X]\), the \(n\)-vector \(f_1(X)\wedge\ldots\wedge f_n(X)\) must be an \(S\) multiple of \(\phi\) and at this point the second part of the main theorem comes into play.
It shows that such a multiple can be computed by means of a very general and beautiful determinantal formula (that alluded to in the title) involving the coefficients of the polynomials \(f_i\)s only. We omit to write down the general determinantal formula, as it appear in the paper, which would require some additional explanations, but we mention a remarkable particular case: when \(f_i=X^{h_i+n-i}\) (\(1\leq i\leq n\)), one gets \( X^{h_1+n-1}\wedge X^{h_2+n-2}\wedge\ldots\wedge X^{h_{n}}=s_{h_{1},\ldots,h_n}\cdot\phi \) where, if \(s_h\) is the complete symmetric polynomial of degree \(h\), then \(s_{h_1,h_2,\ldots, h_n}\) is the usual Schur-polynomial \(\det(s_{h_i+j-1})\), which can be interpreted as the classical Giambelli's formula of Schubert calculus on Grassmann schemes. Hence Laksov and Thorup's determinantal formula can be seen as the ultimate and most natural generalization of it.
As one may expect from the nature itself of the results, the topic of this paper is related with many different subjects in mathematics, such as combinatorics, representation theory, geometry\dots To emphasize such a wide interplay, the authors care to prove the main theorem using different techniques within different frameworks. The most combinatorial in character is certainly that proposed in Section~2, based on a Pieri type formula enjoyed by the action of complete symmetric polynomials on the natural basis of \(\bigwedge^n_AA[X]\). That of Section~3, instead, relies on the isomorphism between \(\bigwedge^n_AA[X]\) and the ring of alternating polynomials. Section 4 proposes another proof based on symmetrization: let \(\xi\) be the residue class of \(T\) modulo \(P=\prod(T-X_i)\) in the ring \(S[\xi]=S[T]/(P)\). Then \(\bigwedge^nS[\xi]\) is naturally an \(S\)-module and remarkably such a module structure coincides with the symmetric structure defined in Section~1 of the paper, described in the first part of this review. Within the same framework, Section~5 proposes a very short proof of the main theorem which has the nice feature of implying Jacobi-Trudy formula. Finally, last two sections are devoted to look at the main theorem using the divided difference operators as well as the theory of universal splitting algebras, related with the work of Grothendieck on the homology of flag schemes. determinantal formula; Schubert calculus; exterior algebras; Giambelli's formula; Grassmann schemes; symmetric structures; symmetric functions; symmetrizing operators; divided difference operators; intersection theory; universal splitting algebras Laksov, D. and Thorup, A., A determinantal formula for the exterior powers of the polynomial ring, Indiana Univ. Math. J. 56 (2007), 825--845. Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations A determinantal formula for the exterior powers of the polynomial 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 The context of the research is as follows. Let \(G(l,\mathbb{C}^{k+l})\) be the Grassmannian of \(l\)-dimensional subspaces of \(\mathbb{C}^{l+k}\). Let \(R^{l,k}\) be the cohomology \(H^*(G(l,\mathbb{C}^{k+l}),\mathbb{Q})\). In [The fixed point property for Grassmann's manifold. Columbus, OH: Ohio State Univ. (PhD Thesis) (1974)], \textit{L. S. O'Neill} conjectured the form of all graded endomorphisms of \(R^{l,k}\). A special case of the conjecture is proved by \textit{M. Hoffman} [Trans. Am. Math. Soc. 281, 745--760 (1984; Zbl 0566.14022)] via complicated means. To simplify the proof of Hoffman, in [``Conjectures on the cohomology of the Grassmannian'', Preprint, \url{arXiv:math/0309281}], \textit{V. Reiner} and \textit{G. Tudose} made a series of weaker conjectures (1-4). The strongest conjecture described the Hilbert series \(\mathrm{Hilb}(R^{l,k,m},q)\) of \(R^{l,k}\)-the \(\mathbb{Q}\)-subalgebra of \(R^{l,k}\) generated by the homogeneous elements of degree at most \(m\). The strongest conjecture is verified for \(m=1\), \(m=\min(l,k)\) in Reiner and Tudose [loc. cit.]. For the case \(m=\min(l,k)\), a key idea of the proof is an interpretation of \(\mathrm{Hilb}(R^{l,k,m},q)\) [loc. cit., Proposition 8].
The main results of the paper are: A reformulation of [loc. cit., Proposition 8], and then the conjectures about the existence of two bases for \(R^{l,k}\) that would prove the strongest conjecture of Reiner and Tudose, see Conjectures 1.2, 1.3. Similarly, for Lagrangian Grassmannian \(LG(n,\mathbb{C}^{2n})\), the authors made an analogues conjecture to Conjectures 1.2, 1.3, see Conjecture 1.4. In particular, it is verified in extreme cases \(m=1\) and \(m=n\), see Proposition 5.2.
The strategy used to obtain the main results is the proof of [loc. cit., Proposition 8] for the extreme case \(m=\min(l,k)\).
The structure of the paper is as follows. Section 2 rephrases the proof of [loc. cit., Proposition 8]. Section 3 gives a proof of Theorem 1.1. Section 4 states Conjectures 1.2, 1.3. Section 5 states Conjecture 1.4 and verifies it in two extreme cases. Grassmannian; Lagrangian; Hilbert series; \(q\)-binomial; \(k\)-conjugation; \(k\)-Schur function Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations, Combinatorial aspects of algebraic geometry Filtering cohomology of ordinary and Lagrangian Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let $E$ be a vector bundle of rank $d$ on a complex projective manifold $X$ and let $s=(s_0, \dots, s_m)$ be a sequence of integers such that $0 = s_0 < s_1 < \dots < s_m=d$. For every $x \in X$ and the corresponding fiber $V=E_x$, consider the manifold $\mathcal F l_s(V)$ of incomplete flags $V=V_{s_0} \supset V_{s_1} \supset \dots \supset V_{s_m}=\{0\}$, where $V_{s_j}$ is a vector subspace of $V$ of codimension $s_j$. Their union as $x$ varies on $X$ form a manifold $\mathcal F l_s(E)$. To any partition $a=(a_1, a_2, \dots , a_l)$ of length $l \leq d$ such that $a_1=a_2= \dots =a_{s_1}$, $a_{s_1+1}=a_{s_1+2} = \dots = a_{s_2}$, $a_{s_2+1}=a_{s_2+2} = \dots = a_{s_3}$, etc. one can associate a line bundle $Q_a^s$ on $\mathcal F l_s(E)$ such that, letting $\pi:\mathcal F l_s(E) \to X$ denote the natural projection and $\mathcal S^a$ the Schur functor corresponding to $a$, for every $m \geq 0$ the following properties hold: $\pi_*\big((Q_a^s)^m\big)\cong \mathcal S_{ma}E$, and $R^q \pi_*\big((Q_a^s)^m\big) = 0$ if $q>0$.
In the main result, the authors prove is that the line bundle $Q_a^s$ on $\mathcal F l_s(E)$ is ample if and only if the vector bundle $\mathcal S_aE$ on $X$ is ample. This can be regarded as a generalization to flag manifolds of the well known fact $E$ is ample if and only if its tautological line bundle $\mathcal O_{\mathbb P(E)}(1)$ is ample. Moreover, it gives as a corollary that the line bundle $\det Q$ on the Grassmann bundle $G_r(E)$ is ample if and only if the vector bundle $\bigwedge ^r E$ is ample on $X$. The proof of the ``only if'' part of the main result is deduced via the Ampleness Dominance Theorem, namely, if $a$ and $b$ are two partitions such that $b \preceq a$ in the partial dominance order, then $\mathcal S_a E$ ample implies $\mathcal S_b E$ ample. Here the authors provide a proof of this result avoiding the use of the saturation property of the Littlewood-Richardson semigroup unlike in their previous paper [Manuscr. Math. 113, No. 2, 165--189 (2004; Zbl 1048.14007)]. ampleness; vector bundles; flag variety; Schur functor Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Grassmannians, Schubert varieties, flag manifolds, Divisors, linear systems, invertible sheaves, Vanishing theorems in algebraic geometry Ampleness equivalence and dominance for 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 author gives a corrected revised version of the main theorem of his previous paper [ibid. 89, 583-586 (1983; Zbl 0572.14027)]. Counterexamples to the previous version appeared in a paper by \textit{N. Goldstein} [Compos. Math. 51, 189-192 (1984; Zbl 0543.14025)]. Grassmann variety; Chow ring; Castelnuovo's bound PAPANTONOPOULOU, A. , Corrigendum to Embeddings in G(1,3) , Proc. A.M.S. 95 ( 1985 ), 533-536. MR 87i:14033 | Zbl 0626.14036 Grassmannians, Schubert varieties, flag manifolds, Families, moduli, classification: algebraic theory, Embeddings in algebraic geometry Corrigendum to: ``Embeddings in G(1,3)'' | 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 [For the entire collection see Zbl 0723.00046.]
We examine the extended Riccati flow on a Grassmannian near the topological closure of the stable manifold of any invariant locus of dimension at least two. Such a closure is an example of a singular (in the sense of algebraic geometry) Schubert variety. The flow exhibits sensitive dependence on initial conditions near the singularities of this variety. The spectrum of Lyapunov exponents is derived from the spectrum of the infinitesimal generator of the flow. matrix Riccati differential equation; Riccati flow; Schubert variety Dynamics induced by flows and semiflows, Manifolds of solutions, Ergodic theorems, spectral theory, Markov operators, Grassmannians, Schubert varieties, flag manifolds, Structural stability and analogous concepts of solutions to ordinary differential equations The Riccati flow near the `edge' | 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 split element of a connected semisimple Lie group act on one of its flag manifolds. We prove that each connected set of fixed points of this action is itself a flag manifold. With this we can obtain a generalized Bruhat decomposition of a semisimple Lie group by entirely dynamical arguments. Bruhat decomposition; flag manifold L. Seco, A note on the Bruhat decomposition of semisimple Lie groups. \textit{J. Lie Theory}\textbf{18}(2008), 725-731. MR2493064 (2010c:22018) Zbl 1228.22016 Semisimple Lie groups and their representations, Grassmannians, Schubert varieties, flag manifolds A note on the Bruhat decomposition of semisimple Lie 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 By a diagram the authors mean a finite collection of cells in \({\mathbb Z}\times {\mathbb Z}\). They consider balanced labellings of diagrams. Special cases of these objects are the standard Young tableaux and the balanced tableaux introduced by \textit{P. Edelman} and \textit{C. Greene} [Adv. Math. 63, 42-99 (1987; Zbl 0616.05005)]. It turns out that for certain diagrams associated with permutations of the symmetric group \(\Sigma_n\), the set of balanced labellings has a remarkable rich structure. Balanced labellings of permutation diagrams yield the symmetric functions introduced by \textit{R. P. Stanley} [Eur. J. Comb. 5, 359-372 (1984; Zbl 0587.20002)] in the same way as the Schur functions can be constructed from column-strict tableaux. Balanced labelled diagrams can be also viewed as encodings of reduced decompositions of permutations. Imposing flag conditions, the authors obtain the Schubert polynomials of Lascoux and Schützenberger. Finally the authors construct an explicit basis for the Schubert module introduced in 1986 by W. Kraskiewicz and P. Pragacz (i.e. the representation of the upper triangular group with formal character equal to the corresponding Schubert polynomial). diagram; tableaux; Subert polynomials; symmetric functions; Schubert module S. Fomin, C. Greene, V. Reiner, and M. Shimozono, ''Balanced labellings and Schubert polynomials,'' European J. Combin. 18 (1997), no. 4, 373--389. the electronic journal of combinatorics 25(3) (2018), #P3.4622 Combinatorial aspects of representation theory, Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds Balanced labellings 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 We consider a certain class of Schubert varieties of the affine Grassmannian of type A. By embedding a Schubert variety into a finite-dimensional Grassmannian, we construct an explicit basis of sections of the basic line bundle by restricting certain Plücker co-ordinatea.
As a consequence, we write an explicit set of generators for the degree-one part of the ideal of the finite-dimensional embedding. This in turn gives a set of generators for the degree-one part of the ideal defining the affine Grassmannian inside the infinite Grassmannian which we conjecture to be a complete set of ideal generators.
We apply our results to the orbit closures of nilpotent matrices. We describe (in a characteristic-free way) a filtration for the coordinate ring of a nilpotent orbit closure and state a conjecture on the \(\text{SL}(n)\)-module structures of the constituents of this filtration. Kreiman, V., Lakshmibai, V., Magyar, P., Weyman, J.: On ideal generators for affine Schubert varieties. Algebraic groups and homogeneous spaces, Tata Inst. Fund. Res. Stud. Math., Tata Inst. Fund. Res., Mumbai 353-388 (2007) Grassmannians, Schubert varieties, flag manifolds On ideal generators for affine 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 use Hopf algebras to prove a version of the Littlewood-Richardson rule for skew Schur functions, which implies a conjecture of \textit{S. H. Assaf} and \textit{P. R. W. McNamara} [J. Comb. Theory, Ser. A 118, No. 1, 277-290 (2011; Zbl 1291.05205)]. We also establish skew Littlewood-Richardson rules for Schur \(P\)- and \(Q\)-functions and noncommutative ribbon Schur functions, as well as skew Pieri rules for \(k\)-Schur functions, dual \(k\)-Schur functions, and for the homology of the affine Grassmannian of the symplectic group. dual Hopf algebras; skew Schur functions; skew Littlewood-Richardson rules; ribbon Schur functions; skew Pieri rules; affine Grassmannians Lam, T., Lauve, A., Sottile, F.: Skew Littlewood--Richardson rules from Hopf algebras. Int. Math. Res. Not. 2011, 1205--1219 (2011). doi: 10.1093/imrn/rnq104 Hopf algebras and their applications, Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds Skew Littlewood-Richardson rules from Hopf 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 The author studies the quot functor of a quasi-coherent sheaf over a projective scheme. Let $X$ be a projective scheme over an algebraically closed field $\mathbf{k}$, $\mathcal{E}$ be a quasi-coherent sheaf over $X$, and $h$ be a Hilbert polynomial. Let $\mathrm{quot}_h^X(\mathcal{E}_{(-)})$ be the contravariant functor parametrizing isomorphism classes of coherent quotients $Q_T$ of $\mathcal{E}_T = \mathcal{E}\otimes_{\mathbf{k}}\mathcal{O}_T$, for every scheme $T$ over $\mathbf{k}$, such that $Q_T$ is flat over $T$ and has Hilbert polynomial $h$. Grothendieck's classical result states that if $\mathcal{E}$ is coherent, then the above functor is representable. The main theorem of the paper asserts that if $\mathcal{E}$ is a quasi-coherent $\mathcal{O}_X$-module, then there is a scheme $\text{Quot}^X_h(\mathcal{E})$ representing the functor $\mathrm{quot}_h^X(\mathcal{E}_{(-)})$. The main idea in the author's construction of $\text{Quot}^X_h(\mathcal{E})$ is a version of Grothendieck's Grassmannian embedding combined with a result of Deligne, realizing quasi-coherent sheaves as ind-objects in the category of quasi-coherent sheaves of finite presentation. \par Section 2 collects some background materials such as limits and quasi-compact schemes, ind-objects and the above theorem of Deligne, representable functors, and Castelnuovo-Mumford regularity. Section 3 is devoted to the filtering construction of the schematic Grassmannian. The author proves the main theorem in Section 4. quot scheme; quasi-coherent sheaf; sheaf of finite presentation; Grassmannian embedding Parametrization (Chow and Hilbert schemes), Grassmannians, Schubert varieties, flag manifolds, Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) The \textit{quot} functor of a quasi-coherent sheaf | 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 deals with invariants and canonical forms of proper rational functions acted on by the output feedback group. The group consists of nonsingular matrices \(\gamma =\left[ \begin{matrix} \alpha \quad 0\\ \kappa \quad \beta \end{matrix} \right]\in F\) and acts on \(g(s)=p(s)/q(s)\) according to the formula (p,q)\(\mapsto (\alpha p\), \(\beta q+\kappa p)\). The transfer functions are defined by polynomials \(p(s)=p_ 1s^{n- 1}+...+p_ n\), \(q(s)=s^ n+q_ 1s^{n-1}+...+q_ n\) and belong to a set \(R^*(n).\)
The first result of the paper states that the orbit space \(R^*(n)/F\) does not allow neither any finite set of global rational invariant functions nor global algebraic canonical forms. However, if \(Rat(n)\subset R^*(n)\) consists of those (p,q) which are relatively prime then it is established that the quotient Rat(n)/F does possess a complete set of rational invariant functions being actually homogeneous polynomials in the Plücker coordinates of \(Grass(2,n+1)\) divided by the resultant polynomial.
Eventually, it is proved that the orbit space Rat(n)/F does not have global continuous canonical forms for the transfer functions with the Cauchy index 0. The results are interpreted in terms of the root loci theory, compared with other existing results and illustrated with enlightening examples. invariants; canonical forms; proper rational functions; output feedback group; Cauchy index; root loci DOI: 10.1016/0167-6911(85)90074-X Algebraic methods, Group actions on varieties or schemes (quotients), Linear systems in control theory, Grassmannians, Schubert varieties, flag manifolds, Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral), Canonical structure Geometric methods for the classification of linear feedback systems | 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 a computation of the K-group of Grassmannians and flag varieties over an arbitrary Noetherian base scheme, and the K-group of forms of Grassmannians and flag varieties associated to a sheaf of Azumaya algebras. An extension to objects over \({\mathbb{Z}}\) of main points in the proof of the characteristic zero case is used in the computations. This include a weaker form of the Cauchy formula for \(Gl_ n\)- representations and the Bott theorem over \({\mathbb{Z}}\) in the appropriate Grothendieck group. K-group of Grassmannians; flag varieties; Azumaya algebras Marc Levine, V. Srinivas, and Jerzy Weyman, \?-theory of twisted Grassmannians, \?-Theory 3 (1989), no. 2, 99 -- 121. Applications of methods of algebraic \(K\)-theory in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, \(K\)-theory of schemes K-theory of twisted Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The invariant formulation of a differential operator Riccati equation as a vector field on an infinite dimensional Lagrangian Grassmannian is considered. In this connection some properties of infinite dimensional Grassmannians are studied. Applications are given such as the asymptotic behavior of the solutions of the Riccati equation and the stabilizability of linear control systems by unbounded feedbacks. operator Riccati equation; infinite dimensional Grassmannians; stabilizability of linear control systems Algebraic methods, Grassmannians, Schubert varieties, flag manifolds, Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems, Nonlinear differential equations in abstract spaces, Equations involving linear operators, with operator unknowns, Applications of dynamical systems, Linear systems in control theory, Stabilization of systems by feedback On the symplectic structure of the operator Riccati equation | 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 result of Borel states that the integer cohomology ring of a full flag variety \(Fl(n)=GL_n/B\) is isomorphic to \({\mathbb Z}[x_1,\dots,x_n]/I\), where \(I\) is the ideal generated by all symmetric polynomials without constant term. The generators \(x_1,\dots,x_n\) correspond to the first Chern classes of the line bundles \({\mathcal V}_i/{\mathcal V}_{i-1}\), where \({\mathcal V}_\bullet\) is the tautological vector bundle over \(Fl(n)\). On the other hand, this ring also has a natural additive basis consisting of the \textit{Schubert classes}, formed by the flags satisfying certain incidence conditions on their intersection with a given flag. A classical problem of Schubert calculus is to express these classes via the generators \(x_1,\dots,x_n\). This was done by Bernstein-Gelfand-Gelfand and Demazure. There are natural generalizations of this setting to full flag varieties for other classical groups, \(SO(n)\) and \(Sp(n)\), due to Harris-Tu, Fulton and the others.
In this paper the author considers a similar problem for one of the five exceptional groups, namely, for \(G_2\). Geometrically, the full flag variety \(X\) of \(G_2\) is formed by the pairs of nested subspaces of dimensions 1 and 2 in a seven-dimensional vector space \(V\), that are isotropic with respect to a generic alternating trilinear form on \(V\). This variety also admits a Schubert decomposition. As in the classical case, one can consider two quotient line bundles associated with the tautological vector bundle on \(X\); the first Chern classes of these bundles generate \(H^*X\). The author expresses the classes of the Schubert varieties, corresponding to the 12 elements of the Weyl group of type \(G_2\), in terms of these generators. Some (apparently new) explicit geometric descriptions of the full flag variety of type \(G_2\) are also provided. degeneracy locus; equivariant cohomology; flag variety; Schubert variety; Schubert polynomial; exceptional Lie group; octonions Anderson, D., Chern class formulas for \(G_2\) Schubert loci, Trans. Amer. Math. Soc., 363, 12, 6615-6646, (2011) Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Exceptional groups, Symmetric functions and generalizations Chern class formulas for \(G_{2}\) Schubert loci | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We study the critical points of monomial functions over an algebraic subset of the probability simplex. The number of critical points on the Zariski closure is a topological invariant of that embedded projective variety, known as its maximum likelihood degree. We present an introduction to this theory and its statistical motivations. Many favorite objects from combinatorial algebraic geometry are featured: toric varieties, \(A\)-discriminants, hyperplane arrangements, Grassmannians, and determinantal varieties. Several new results are included, especially on the likelihood correspondence and its bidegree. This article represents the lectures given by the second author at the CIME-CIRM course on Combinatorial Algebraic Geometry at Levico Terme in June 2013. J. Huh and B. Sturmfels, \textit{Likelihood geometry}, in Combinatorial Algebraic Geometry, Lecture Notes in Math. 2108, Springer, Cham, 2014, pp. 63--117, . Research exposition (monographs, survey articles) pertaining to algebraic geometry, Projective techniques in algebraic geometry, Determinantal varieties, Grassmannians, Schubert varieties, flag manifolds, Toric varieties, Newton polyhedra, Okounkov bodies, Applications of commutative algebra (e.g., to statistics, control theory, optimization, etc.) Likelihood 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 This work arises from trying to answer Problem 4 presented in [\textit{B. Sturmfels}, IMA Vol. Math. Appl. 149, 351--363 (2009; Zbl 1158.13300)]: ``General problem: Study the geometry of conditional independence models for multivariate Gaussian random variables.'' Thus, this survey develops the geometric theory of gaussoids, based on the Lagrangian Grassmannian and its symmetries. A gaussoid is a combinatorial structure that encodes independence in probability and statistics, just like matroids encode independence in linear algebra. Throughout the 30 pages, the authors introduce and classify oriented gaussoids, connect valuated gaussoids to tropical geometry, addresses the realizability problem for gaussoids and oriented gaussoids and so on. The reader can find additional materials on the web \url{www.gaussoids.de}. gaussoid; matroid; Gaussian; Lagrangian Grassmannian; minor; symmetric matrix Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Determinants, permanents, traces, other special matrix functions, Probability distributions: general theory, Grassmannians, Schubert varieties, flag manifolds, Applications of commutative algebra (e.g., to statistics, control theory, optimization, etc.), Measures of association (correlation, canonical correlation, etc.), Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Combinatorial aspects of tropical varieties The geometry of gaussoids | 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 sign error in Lemma 6.4 of [Quantum Topol. 1, No. 1, 1--92 (2010; Zbl 1206.17015)], pointed out by M. Mackaay, M. Stošić and P. Vaz, is corrected. Khovanov, M.; Lauda, A., Erratum to: ``A categorification of quantum \(\mathfrak{sl}_n\)'', Quantum Topol., 2, 1, 97-99, (2011) Quantum groups (quantized enveloping algebras) and related deformations, Quantum groups and related algebraic methods applied to problems in quantum theory, Grassmannians, Schubert varieties, flag manifolds, Ring-theoretic aspects of quantum groups Erratum to: ``A categorification of quantum \(\text{sl}(n)\)'' | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We give a new operator formula for Grothendieck polynomials that generalizes Magyar's Demazure operator formula [\textit{P. Magyar}, Comment. Math. Helv. 73, No. 4, 603--636 (1998; Zbl 0951.14036)] for Schubert polynomials. Our proofs are purely combinatorial, contrasting with the geometric and representation theoretic tools used by Magyar. We apply our formula to prove a necessary divisibility condition for a monomial to appear in a given Grothendieck polynomial. Schubert polynomials; Demazure operator formula Symmetric functions and generalizations, Combinatorial aspects of representation theory, Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds An orthodontia formula for 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 An expanded version of the article can be found in the booklet of \textit{L. Gatto} [``Schubert calculus: an algebraic introduction''. Publicações Matemáticas do IMPA (2005; Zbl 1082.14054)]. We cite from the review of that book:\dots he presents a new point of view on Schubert calculus on a Grassmann manifold. In this interpretation the Chow ring of the Grassmann manifold of \(k\)-dimensional subspaces on an \(n\)-space is the \(k\)th exterior product of an \(n\)-dimensional vector space \(M\) with a fixed base \(e_1,\dots,e_n\), and the Chern classes are presented as certain-differential operators on the exterior product. These operators the author calls Schubert derivations, and are obtained from the operator \(D_1 :M\to M\) given by \(D_1(e_i) = e_{i+1}\) for \(i = 1,\dots,n-1\) and \(D_1(e_n) = 0\).
The treatment gives an easy and natural approach to Schubert calculus, that is well adopted to computations. In particular it gives a satisfactory explanation of the determinants appearing in Giambelli's formula.
It is refreshing and surprising that it is possible to take a new perspective on this classical part of geometry, particularly taken into account the massive amount of work in the field, coming from many different parts of mathematics like geometry, algebra and combinatorics. Chow ring; Schubert derivation; Schubert varieties; Pieri's formula; Giambelli's formula Gatto, L.: Schubert calculus via Hasse-Schmidt derivations. Asian J. Math. 3, 315-322 (2005) Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds Schubert calculus via Hasse-Schmidt derivations | 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 fundamental and beautiful article the author introduces universal Schubert polynomials that specialize to all previously known Schubert polynomials: those of Lascoux and Schützenberger, the quantum Schubert polynomials of Fomin, Gelfand, and Postnikov, and the quantum Schubert polynomials for partial flag varieties of Ciocan-Fontanine. Also double versions of these polynomials are given, that generalize the previously known double Schubert polynomials of Lascoux, MacDonald, Kirillov and Maeno, and those of Ciocan-Fontanine and Fulton.
The universal Schubert polynomials describe degeneracy loci of maps of vector bundles, in a more general setting than that of the author's beautiful earlier article [\textit{W. Fulton}, Duke Math. J. 65, 381-420 (1992; Zbl 0788.14044)].
The setting is a sequence of maps of locally free \(\mathcal O_X\)-modules
\[
F_1\to F_2\to \cdots \to F_n \to E_n \to \cdots \to E_2\to E_1
\]
on a scheme \(X\). In contrast to the mentioned article (loc. cit.) the maps \(F_i \to F_{i+1}\) do not have to be injective and the maps \(E_{i+1} \to E_i\) do not have to be surjective. For each \(w\) in the symmetric group \(S_{n+1}\), there is a degeneracy locus
\[
\Omega_w =\{x\in X\mid \text{rank}(F_q(x) \to E_p(x)) \leq r_w(p,q) \text{ for all } 1\leq p, q\leq n\},
\]
where \(r_w(p,q)\) is the number of \(i\leq p\) such that \(w(i)\leq q\). Such degeneracy loci are described by the double form \({\mathfrak S}_w(c,d)\) of universal Schubert polynomials evaluated at the Chern classes of all the bundles involved.
The classical approaches of \textit{Demazure}, or \textit{Bernstein, Gel'fand}, and \textit{Gel'fand} do not work in this case. Instead a locus in a flag bundle is found that maps to a given degeneracy locus \(\Omega_w\), such that one has injections and surjections of the bundles involved, and such that the results of the article mentioned above can be applied. Then the formula is pushed forward to get a formula for \(\Omega_w\). universal Schubert polynomials; quantum Schubert polynomials; partial flag varieties; double Schubert polynomials; degeneracy loci; Chern classes Fulton W. (1999). Universal Schubert polynomials. Duke Math. J. 96(3): 575--594 Grassmannians, Schubert varieties, flag manifolds, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Determinantal varieties, Characteristic classes and numbers in differential topology Universal 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 support of the tensor product decomposition of integrable irreducible highest weight representations of a symmetrizable Kac-Moody Lie algebra \(\mathfrak{g}\) defines a semigroup of triples of weights. Namely, given \(\lambda\) in the set \(P_+\) of dominant integral weights, \(V(\lambda )\) denotes the irreducible representation of \(\mathfrak{g}\) with highest weight \(\lambda\). We are interested in the tensor semigroup
\[
\Gamma_{\mathbb{N}}(\mathfrak{g}):=\{(\lambda_1,\lambda_2,\mu )\in P_+^3\,|\, V(\mu )\subset V(\lambda_1)\otimes V(\lambda_2)\},
\]
and in the tensor cone \(\Gamma (\mathfrak{g})\) it generates:
\[
\Gamma (\mathfrak{g}):=\{(\lambda_1,\lambda_2,\mu )\in P_{+,{\mathbb{Q}}}^3\,|\,\exists N\geq 1 \quad V(N\mu )\subset V(N\lambda_1)\otimes V(N\lambda_2)\}.
\]
Here, \(P_{+,{\mathbb{Q}}}\) denotes the rational convex cone generated by \(P_+\). In the special case when \(\mathfrak{g}\) is a finite-dimensional semisimple Lie algebra, the tensor semigroup is known to be finitely generated and hence the tensor cone to be convex polyhedral. Moreover, the cone \(\Gamma (\mathfrak{g})\) is described in [\textit{P. Belkale} and \textit{S. Kumar}, Invent. Math. 166, No. 1, 185--228 (2006; Zbl 1106.14037)] by an explicit finite list of inequalities.
In general, \(\Gamma (\mathfrak{g})\) is neither polyhedral, nor closed. In this article we describe the closure of \(\Gamma (\mathfrak{g})\) by an explicit countable family of linear inequalities for any untwisted affine Lie algebra, which is the most important class of infinite-dimensional Kac-Moody algebra. This solves a Brown-Kumar's conjecture in this case (see [\textit{M. Brown} and \textit{S. Kumar}, Math. Ann. 360, No. 3--4, 901--936 (2014; Zbl 1305.22027)]). The difference between the tensor cone and the tensor semigroup is measured by the saturation factors. Namely, a positive integer \(d\) is called a saturation factor, if \(V(N\lambda_1)\otimes V(N\lambda_2)\) contains \(V(N\mu )\) for some positive integer \(N\) then \(V(d\lambda_1)\otimes V(d\lambda_2)\) contains \(V(d\mu )\), assuming that \(\mu -\lambda_1-\lambda_2\) belongs to the root lattice. For \(\mathfrak{g}={\mathfrak{sl}}_n\), the famous Knutson-Tao theorem asserts that \(d=1\) is a saturation factor (see [\textit{A. Knutson} and \textit{T. Tao}, J. Am. Math. Soc. 12, No. 4, 1055--1090 (1999; Zbl 0944.05097)]). More generally, for any simple Lie algebra, explicit saturation factors are known. In the Kac-Moody case, \(\Gamma_{\mathbb{N}}(\mathfrak{g})\) is not necessarily finitely generated and hence the existence of such a factor is unclear a priori. Here, we obtain explicit saturation factors for any affine Kac-Moody Lie algebra. For example, in type \(\tilde A_n\), we prove that any integer \(d\geq 2\) is a saturation factor, generalizing the case \(\tilde A_1\) shown in [\textit{M. Brown} and \textit{S. Kumar}, Math. Ann. 360, No. 3--4, 901--936 (2014; Zbl 1305.22027)]. tensor cone; tensor semigroup; symmetrizable Kac-Moody Lie algebra Grassmannians, Schubert varieties, flag manifolds, Geometric invariant theory, Semisimple Lie groups and their representations, Representation theory for linear algebraic groups, Kac-Moody groups On the tensor semigroup of affine Kac-Moody 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 Let \(X\) be the toric variety associated to the Bruhat poset of Schubert varieties in the Grassmannian. In this paper the authors describe the singular locus of \(X\) in terms of faces of the cone associated to \(X\). It is proved that the singular locus is pure of codimension \(3\) in \(X\), and the generic singularities are of cone type. They also determine the tangent cones at the maximal singularities of \(X\). In the case of \(X\) being associated to the Bruhat poset of Schubert varieties in the Grassmannian of \(2\)-planes in a \(n\)-dimensional vector space (over the base field), they show a certain product formula relating the multiplicities at certain singular points. Brown, J.\!; Lakshmibai, V.\!, Singular loci of Grassmann--Hibi toric varieties, Michigan Math. J., 59, 243-267, (2010) Toric varieties, Newton polyhedra, Okounkov bodies, Grassmannians, Schubert varieties, flag manifolds Singular loci of Grassmann-Hibi toric 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 Let \(G\) be a connected reductive group over an algebraically closed field \(k\) of characteristic not 2; let \(\theta\in\Aut (C)\) be an involution and \(K=G^\theta\subseteq G\) the fixed point group of \(\theta\) and let \(P\subseteq G\) be a parabolic subgroup. The set \(K \setminus G/P\) of \((K,P)\)-double cosets in \(G\) plays an important role in the study of Harish Chandra modules. \textit{M. Brion} and the author [Can. J. Math. 52, 265--292 (2000; Zbl 0972.14039)] gave a description of the orbits of symmetric subgroups in a flag variety \(G/P\) mainly using geometric arguments. For general \(P\), it is difficult to describe the combinatorics of the decomposition of the closure of a double coset in terms of \(K\times P\) double cosets. However, in some special cases one can describe the combinatorics of the closures of the double cosets in more detail. This paper discusses the special case that \(P\) contains a \(\theta\)-stable Levi factor \(L\) and the set of roots of the connected center \(S\) of \(L\) is a root system with Weyl group \(W(S)=N_G (S)/Z_G(S)\). Here \(N_G(S)\) (resp. \(Z_G(S))\) is the normalizer (resp. centralizer) of \(S\) in \(G\). In this case the combinatorics of the Weyl Group can be used to describe the closures of the double cosets of a part of the double coset space which includes the open and closed orbits and we get a number of results similar to the case that \(P=B\) a Borel subgroup. This root system condition on \(P\) is satisfied in many cases. For example in the case that \(P\) is a minimal parabolic \(k_0\)-subgroup of \(G\) or a minimal \(\theta\)-split parabolic subgroup of \(G\) or a minimal \((\theta,k_0)\)-split parabolic \(k_0\)-subgroup of \(G\). Here \(k_0\subseteq k\) is a subfield of \(k\) and \(G,\theta\) are defined over \(k_0\). double cosets; Weyl group; root system; parabolic subgroup Helminck, A.: Combinatorics related to orbit closures of symmetric subgroups in flag varieties. CRM proc. Lecture notes 35, 71-90 (2004) Homogeneous spaces and generalizations, Grassmannians, Schubert varieties, flag manifolds, Representation theory for linear algebraic groups Combinatorics related to orbit closures of symmetric subgroups 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 Let \(k\) be an algebraically closed field of characteristic char \(k \neq 2\). Let \(G\) be a connected reductive algebraic group over \(k\) and let \(\mathcal B\) be the flag variety of \(G\). For any spherical subgroup \(H\) of \(G\) \textit{F. Knop} [Comment. Math. Helv. 70, No.2, 285-309 (1995; Zbl 0828.22016)] defined an action of the Weyl group \(W\) of \(G\) on the (finite) set of \(H\)-orbits in \(\mathcal B\). In his paper, F. Knop proved the existence of this action in a very technical and complicated way. The author of the paper under review constructs these \(W\)-orbits in an elementary fashion by using natural invariants. Weyl group; flag manifold; spherical subgroup; Knop action Ressayre, N, About knop's action of the Weyl group on the set of orbits of a spherical subgroup in the flag manifold, Transform. Groups, 10, 255-265, (2005) Semisimple Lie groups and their representations, Grassmannians, Schubert varieties, flag manifolds, Reflection and Coxeter groups (group-theoretic aspects), Linear algebraic groups over arbitrary fields About Knop's action of the Weyl group on the set of orbits of a spherical subgroup in the 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 Let \(X\) be a Schubert cycle of a Grassmannian \(G(r,n)\). Here the author gives a combinatorial proof of Hodge's postulation formula giving the Hilbert function of \(X\) with respect to the Plücker embedding of \(G(r,n)\). The main point of the paper is to introduce algebraists to some combinatorial techniques which seem to be important in this area. Hilbert polynomial; Schubert cycle; Grassmannian; Hilbert function Ghorpade, S. R.: A note on Hodge's postulation formula for Schubert varieties. Geometric and combinatorial aspects of commutative algebra (Messina, 1999), 211-220 (2001) Grassmannians, Schubert varieties, flag manifolds, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series A note on Hodge's postulation formula for 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 0671.00018.]
A statement in algebraic geometry over fields of arbitrary characteristic follows from the existence of matrices with integer entries of the type mentioned in the title. It is shown how these matrices can be built from a finite number of small matrices. It is reported how these small matrices, of which the largest is a 25 by 25 matrix, were found using computer algebra systems. symmetric matrices; alternating blocks; Grassmannian; matrices with integer entries; computer algebra Hermitian, skew-Hermitian, and related matrices, Matrices of integers, Grassmannians, Schubert varieties, flag manifolds, Symbolic computation and algebraic computation Symmetric matrices with alternating blocks | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper we construct free resolutions of a certain class of closed subvarieties of affine space of symmetric matrices (of a given size). Our class covers the symmetric determinantal varieties (i.e., determinantal varieties in the space of symmetric matrices), whose resolutions were first constructed by \textit{T. Jozefiak} et al. [Astérisque 87--88, 109--189 (1981; Zbl 0488.14012)]. Our approach follows the techniques developed by \textit{M. Kummini} et al. [Pac. J. Math. 279, No. 1--2, 299--328 (2015; Zbl 1342.14103)], and uses the geometry of Schubert varieties. Schubert varieties; Lagrangian Grassmannian; free resolutions Grassmannians, Schubert varieties, flag manifolds, Syzygies, resolutions, complexes and commutative rings, Singularities of surfaces or higher-dimensional varieties, Linear algebraic groups over the reals, the complexes, the quaternions, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Global theory and resolution of singularities (algebro-geometric aspects) Free resolutions of some Schubert singularities in the Lagrangian 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 This is an announcement of a description of loop Grassmannians of reductive groups in the setting of ``local spaces'' over a curve. The structure of a local space is a part of the fundamental structure of a factorisable space introduced and studied by \textit{A. Beilinson} and \textit{V. Drinfeld} [Chiral algebras. Colloquium Publications. American Mathematical Society 51. Providence, RI: American Mathematical Society, 375 p. (2004; Zbl 1138.17300)]. The weakening of the requirements formalizes some well-known examples of ``almost factorisable'' spaces and constructions with such spaces.
The main observation of the paper is that the point of view of local spaces produces a generalization of loop Grassmannians corresponding to central extensions of loop groups of tori. The last section advertises local spaces as a setting for the conjecture of \textit{P. Baumann} and \textit{J. Kamnitzer} [Represent. Theory 16, 152--188 (2012; Zbl 1242.05273)] and Allen Knutson on a topological reconstruction of certain pieces of the loop Grassmannian (the MV-cycles) in terms of representations of quivers. Most of the content in this paper comes from a joint work with Kamnitzer and Baumann (loc. cit.) and Knutson. Grassmannians, Schubert varieties, flag manifolds, Loop groups and related constructions, group-theoretic treatment, Families, moduli of curves (algebraic) Loop Grassmannians in the framework of local spaces over a curve | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Finding examples of compact hyperkähler manifolds is a difficult problem in algebraic geometry. In this paper, the authors propose a new conjectural method to associate to each Lie algebra of type \(G_2, D_4, F_4, E_6, E_7, E_8\) a family of polarized hyperkähler fourfolds.
It is known that to each simple complex Lie algebra one can associate a projective homogeneous variety \(X_{\textrm{ad}}\), called the adjoint variety; this adjoint variety is covered by a family of special subvarieties, called Legendrian cycles.
The authors conjecture that for the considered Lie algebras, it is possible to construct a projective variety \(X\) such that
\begin{itemize}
\item \(\dim X = \dim X_{\textrm{ad}}\),
\item \(X\) is also covered by Legendrian cycles,
\item there is a suitable cycle space \(P\) for these Legendrian cycles,
\item for a generic hyperplane section \(X_H\) of \(X\), the subspace \(P_H\subset P\) parametrizing the Legendrian cycles contained in \(X_H\) is birational to a hyperkähaler fourfold.
\end{itemize}
The conjecture is proven and \(P_H\) is explicitly described for the Lie algebras \(G_2, D_4, F_4, E_6\); the cases \(E_7, E_8\) stay open. Fano varieties; hyperkähler varieties; Legendrian varieties; Tits-Freudenthal square \(K3\) surfaces and Enriques surfaces, Fano varieties, Grassmannians, Schubert varieties, flag manifolds Hyperkähler manifolds from the Tits -- Freudenthal magic square | 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 maps of the Grassmannian manifold \(\text{Gr}(k,n)\) which preserve the class of irregular subsets are studied. It is shown that in the case \(n\neq2k\) any map of this class is induced by a linear automorphism of \(\mathbb{R}^n\). Pankov M., Irregular subsets of the Grassmannian manifolds and their mappings, Mat. Fiz., Anal., Geom., 7(2000), 3, 331--344. Grassmannians, Schubert varieties, flag manifolds, Varieties and morphisms Irregular subsets of the Grassmannian manifolds and their maps. | 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 Toric degenerations are a particularly useful tool for describing algebraic properties of varieties in terms of combinatorics of polytopes and polyhedral fans. The goal of the paper under review is to construct a family of toric degenerations for Richardson varieties inside the Grassmannian. To do this, the authors consider a family of matching fields, which were originally introduced by Sturmfels and Zelevinsky for studying certain Newton polytopes. They associate a weight vector to each block diagonal matching field and characterise when the corresponding initial ideal is toric, thus providing a family of toric degenerations for Richardson varieties.
Given a Richardson variety \(X_{w}^v\) and a weight vector \(\mathbf{w}_\ell\) arising from a matching field, they consider two ideals: an ideal \(G_{k,n,l}|_{w}^{v}\) obtained by restricting the initial of the Plücker ideal to a smaller polynomial ring, and a toric ideal defined as the kernel of a monomial map \(\phi_{l}|_{w}^{v}\). First they characterise the monomial-free ideals of form \(G_{k,n,l}|_{w}^{v}\). Next they construct a family of tableaux in bijection with semi-standard Young tableaux which leads to a monomial basis for the corresponding quotient ring. Finally, they prove that when \(G_{k,n,l}|_{w}^{v}\) is monomial-free and the initial ideal \(\mathrm{in}_{\mathbf{w}_l}(I(X_{w}^v))\) is generated by degree two polynomials, then the ideals \(\mathrm{in}_{\mathbf{w}_l}(I(X_{w}^v))\), \(G_{k,n,l}|_{w}^{v}\) and \(\ker(\phi_{l}|_{w}^{v})\) are all equal, and provide a toric degeneration for the Richardson variety \(X_{w}^v\) . Gröbner and toric degenerations; Grassmannians; semi-standard Young tableaux; Schubert varieties; Richardson varieties; standard monomial theory Grassmannians, Schubert varieties, flag manifolds, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Fibrations, degenerations in algebraic geometry Standard monomial theory and toric degenerations of Richardson varieties in 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 The author determines a factorization of a double specialization of Schubert polynomials from which he derives a factorization of a specialization of \(q\)-factorial Schur functions. Schubert polynomials; factorial and \(q\)-factorial Schur functions; factorization Prosper, V.: Factorization properties of the q-specialization of Schubert polynomials, Ann. comb. 4, 91-107 (2000) Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds Factorization properties of the \(q\)-specialization of 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 This monograph includes much of the author's previous work on Schur and Schubert polynomials and generalizations, and considerable background material. \textit{G. P. Thomas} [Frames, Young tableaux, and Baxter sequences; Adv. Math. 26, 275-289 (1977; Zbl 0375.05005) and Further results on Baxter sequences and generalized Schur functions; Lect. Notes Math. 579, 155-167 (1977; Zbl 0364.05007)] constructed the Schur functions \(S_\lambda\) combinatorially from the set of standard Young tableaux of shape \(\lambda\), using algebraic ``mixed Baxter-multiplication operators'' on the algebra of polynomials. The author [Sequences of symmetric polynomials and combinatorial properties of tableaux; Adv. Math. 134, No. 1, 46-89 (1998; Zbl 0902.05078)] generalizes this construction, providing an effective construction of Q-Schur, Hall-Littlewood, Jack, and Macdonald polynomials, all of which are generalizations of the Schur functions. The Baxter construction can also be applied to the Schubert polynomials; see \textit{R. Winkel} [A combinatorial bijection between standard Young tableaux and reduced words of Grassmannian permutations; Sémin. Lothar. Comb. 36, B36h (1996; Zbl 0886.05115) and Schubert functions and the number of reduced words of permutations; Sémin. Lothar. Comb. 39, B39a (1997; Zbl 0886.05119)]. Recursive methods for construction of the Schubert polynomials are given, and used to prove their basic properties; see \textit{R. Winkel} [Recursive and combinatorial properties of Schubert polynomials; Sémin. Lothar. Comb. 38, B38c (1996; Zbl 0886.05111)]. The original construction of the Schubert polynomials of type \(A_n\) by \textit{A. Lascoux} and \textit{M. P. Schützenberger} [Polynômes de Schubert, C. R. Acad. Sci. Paris, Sér. I 294, 447-450 (1982; Zbl 0495.14031)] by divided differences, and also the construction by recursive structures, are generalized to give constructions of the cases \(B_n\), \(C_n\), and \(D_n\); see \textit{R. Winkel} [Schubert polynomials of types A--D; Manuscr. Math. 100, No. 1, 55-79 (1999; Zbl 0936.05088)]. The weak Bruhat order on Coxeter groups gives a partial order which can be partitioned into poset-isomorphic parts. This is used to give a combintorial computation of the Poincaré polynomials of the finite and some affine Coxeter groups, and a non-recursive computation of standard reduced words for signed and unsigned permutations; see \textit{R. Winkel} [A combinatorial derivation of the Poincaré polynomials of the finite irreducible Coxeter groups; Discrete Math., to appear]. Expansions of Schubert polynomials into standard elementary monomials are constructed combinatorially; see \textit{R. Winkel} [On the expansion of Schur and Schubert polynomials into standard elementary monomials; Adv. Math. 136, No. 2, 224-250 (1998; Zbl 0920.05069)]. Baxter operator; Lehmer code; Schubert polynomials; Schur functions; Young tableaux; Macdonald polynomials; weak Brunat order on Coxeter groups; Poincaré polynomials; reduced words Research exposition (monographs, survey articles) pertaining to combinatorics, Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of representation theory, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] On algebraic and combinatorial properties of Schur 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 Let \(G(b^+, b^-)\) be the Grassmannian of \(B^+\)-dimensional positive definite subspaces of the inner product space \(\mathbb R^{b^+, b^-}\) of signature \((b^+, b^-)\). This paper concerns the construction of automorphic forms on \(G(b^+, b^-)\) which have singularities along smaller sub-Grassmannians. The main tool used in the paper is the extension of the usual theta correspondence to automorphic forms with singularities developed by \textit{J. Harvey} and \textit{G. Moore} [Nucl. Phys. B 463, 315--368 (1996; Zbl 0912.53056)]. It is used to construct families of holomorphic automorphic forms which can be written as infinite products. This extends the previous results for \(G(2, b^-)\) by the author [\textit{R. E. Borcherds}, Invent. Math. 120, 161--213 (1995; Zbl 0932.11028)], and such automorphic forms provide many new examples of generalized Kac-Moody superalgebras.
The paper gives a common generalization of several well-known correspondences, including the Shimura and Maass-Gritsenko correspondences, to modular forms with poles at cusps. It also contains proofs of some congruences satisfied by the theta functions of positive definite lattices and provides a sufficient condition for a Lorentzian lattice to have a reflection group with a fundamental domain of finite volume. Finally, the paper discusses some examples suggesting that these automorphic forms with singularities are related to Donaldson polynomials and to mirror symmetry for \(K3\) surfaces. automorphic forms; Grassmannians; theta functions; Kac-Moody algebras; correspondences R. E. Borcherds, Automorphic forms with singularities on Grassmannians, Invent. Math. 132 (1998), no. 3, 491-562. Other groups and their modular and automorphic forms (several variables), Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Relationship to Lie algebras and finite simple groups, Modular correspondences, etc., Grassmannians, Schubert varieties, flag manifolds Automorphic forms with singularities on Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Given a semisimple complex linear algebraic group \(G\) and a lower ideal \(I\) in positive roots of \(G\), three objects arise: the ideal arrangement \(\mathcal{A}_I \), the regular nilpotent Hessenberg variety \(\operatorname{Hess}(N,I)\), and the regular semisimple Hessenberg variety \(\operatorname{Hess}(S,I)\). We show that a certain graded ring derived from the logarithmic derivation module of \(\mathcal{A}_I\) is isomorphic to \(H^*(\operatorname{Hess}(N,I))\) and \(H^*(\operatorname{Hess}(S,I))^W \), the invariants in \(H^*(\operatorname{Hess}(S,I))\) under an action of the Weyl group \(W\) of \(G\). This isomorphism is shown for general Lie type, and generalizes Borel's celebrated theorem showing that the coinvariant algebra of \(W\) is isomorphic to the cohomology ring of the flag variety \(G/B\).
This surprising connection between Hessenberg varieties and hyperplane arrangements enables us to produce a number of interesting consequences. For instance, the surjectivity of the restriction map \(H^*(G/B)\to H^*(\operatorname{Hess}(N,I))\) announced by Dale Peterson and an affirmative answer to a conjecture of Sommers and Tymoczko are immediate consequences. We also give an explicit ring presentation of \(H^*(\operatorname{Hess}(N,I))\) in types \(B, C\), and \(G\). Such a presentation was already known in type \(A\) and when \(\operatorname{Hess}(N,I)\) is the Peterson variety. Moreover, we find the volume polynomial of \(\operatorname{Hess}(N,I)\) and see that the hard Lefschetz property and the Hodge-Riemann relations hold for \(\operatorname{Hess}(N,I)\), despite the fact that it is a singular variety in general. Hessenberg varieties Grassmannians, Schubert varieties, flag manifolds, Configurations and arrangements of linear subspaces Hessenberg varieties and hyperplane arrangements | 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. Let \({\mathcal B}\) be the variety of Borel subgroups. Consider the projective embedding \({\mathcal B}\to \mathbb{P}(H^0 ({\mathcal B}, L_\rho)^*)\), where \(L_\rho\) is the line bundle associated to the Steinberg weight \(\rho\), which is the half sum of positive roots. In this paper we show that any positive dimensional projective space contained in the image of \({\mathcal B}\) is necessarily of dimension one.
Furthermore we determine exactly these lines. Indeed we show that if \(\ell\subset {\mathcal B}\) is such a line, then there exists a minimal parabolic subgroup \(P\subset G\) such that \(\ell\) is the set of Borel subgroups which are contained in \(P\). In particular this implies that the possible homology classes of such lines correspond under the usual identification of the root lattice with \(H_2({\mathcal B}, \mathbb{Z})\) to the set of simple roots.
The result is obtained as an application of some properties of the intersection of Schubert cycles in the cohomology ring of \({\mathcal B}\). flag varieties; Borel subgroups; Schubert cycles Grassmannians, Schubert varieties, flag manifolds, Projective techniques in algebraic geometry Projective spaces 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 Conjugacy classes of nilpotent \(n\times n\) matrices are labeled by partitions of \(n\). Given a partition \(\lambda\) of \(n\) denote by \(C_{\lambda}\) the Zariski closure of the corresponding conjugacy class, and let \(J_{\lambda}\) be the ideal of polynomial functions on the space of \(n\times n\) matrices over a field of characteristic zero vanishing on \(C_{\lambda}\). Denote by \(I_{\lambda}\) the defining ideal of the scheme theoretic intersection of \(C_{\lambda}\) and the space of diagonal matrices. A generating set of \(I_{\lambda}\) was given by \textit{C. De Concini} and \textit{C. Procesi} [Invent. Math. 64, 203--219 (1981; Zbl 0475.14041)], and was simplified by \textit{T. Tanisaki} [Tohoku Math. J., II. Ser. 34, 575--585 (1982; Zbl 0544.14030)]. This generating set is simplified further in the present paper, and a minimal generating set is obtained in special cases. A conjecture of \textit{J. Weyman} [Invent. Math. 98, No. 2, 229--245 (1989; Zbl 0717.20033)] on a minimal generating set of \(J_{\lambda}\) is disproved. partition; conjugacy class of nilpotent matrices; defining ideal Biagioli, Riccardo; Faridi, Sara; Rosas, Mercedes: The defining ideals of conjugacy classes of nilpotent matrices and a conjecture of Weyman. Int. math. Res. not. IMRN, 33 (2008) Group actions on varieties or schemes (quotients), Grassmannians, Schubert varieties, flag manifolds The defining ideals of conjugacy classes of nilpotent matrices and a conjecture of Weyman | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(X=G/P\) be a rational homogeneous space of Picard number one. A subdiagram of the marked Dynkin diagram of \(X\) induces naturally an embedding of a rational homogeneous space \(Z=G_0/P_0\) into \(X\). An embedding is called standard if it is a composition of the embedding just described and an automorphism from the connected component of the group \(\text{Aut}(X)\). The authors obtain a characterization of standard embeddings of \(Z\) into \(X\) in terms of varieties of minimal rational tangents provided \(Z\) is nonlinear. Also a characterization of maximal linear subspaces in \(X\) is given. These results extend the results of [\textit{J.~Hong} and \textit{N.~Mok}, J. Differ. Geom. 86, No. 3, 539--567 (2010; Zbl 1230.14057)]. homogeneous space; Dynkin diagram; minimal rational tangent Hong, Jaehyun; Park, Kyeong-Dong, Characterization of standard embeddings between rational homogeneous manifolds of Picard number 1, Int. Math. Res. Not. IMRN, 10, 2351-2373, (2011) Grassmannians, Schubert varieties, flag manifolds, Group actions on varieties or schemes (quotients), Questions of classical algebraic geometry, Homogeneous complex manifolds, Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations) Characterization of standard embeddings between rational homogeneous manifolds of Picard number 1 | 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 height defined by a hermitian line bundle is the main new ingredient in Arakelov theory, i.e. intersection theory of arithmetic schemes, compared with the classical (geometric) intersection theory. For an arithmetic variety \(\mathcal X\) (given as an algebraic variety \(X/\mathbb{Q}\) with a canonical regular model over \(\text{Spec} \mathbb{Z}\)) with a canonical hermitian line bundle, one has in particular the notion of the height
\(h(X)\), generalizing the classical notion of a height of a rational point known from Diophantine approximation.
There are several approaches to compute the height of an arithmetic variety. Although the computation via arithmetic intersection numbers
as in the intrinsic definition of \textit{G. Faltings} [Ann. Math. (2) 133, 549--576 (1991; Zbl 0734.14007); see also \textit{H. Gillet} and \textit{C. Soulé}, Publ. Math., Inst. Hautes Étud. Sci. Publ. Math. 72, 93--174 (1990; Zbl 0741.14012); \textit{J.-B. Bost}, \textit{H. Gillet} and \textit{C. Soulé}, J. Am. Math. Soc. 7, 903--1027 (1994; Zbl 0973.14013)] seems to be natural, other alternatives can be applied to the direct computation of \(h(X)\).
The approach of \textit{P. Philippon} [Math. Ann. 289, No. 2, 255--283 (1991; Zbl 0726.14017)] used explicit equations and was successfully applied to the case of projective hypersurfaces. Several other families of varieties were treated, most notably \(\text{SL}_n\)- and Lagrangian Grassmannians by the author in a series of papers [Duke Math. J. 98, No.3, 421--443 (1999; Zbl 0989.14007); J. Reine Angew. Math. 516, 207--223 (1999; Zbl 0934.14018); Math. Ann. 314, No.4, 641--665 (1999; Zbl 0955.14037)] were an arithmetic analogue of the Schubert calculus was developed.
A third approach relies on the powerful method of fixed point formulae, which had already proved to be effective in the geometric case.
Developed by \textit{K. Köhler} and \textit{D. Roessler} [Invent. Math. 145, 333--396 (2001; Zbl 0999.14002); Ann. Inst. Fourier 52, 81--103 (2002; Zbl 1001.14006); Invent. Math. 147, No.3, 633--669 (2002; Zbl 1023.14008); J. Reine Angew. Math. 556, 127--148 (2003; Zbl 1032.14004)],
it allows to avoid computations in the arithmetic Chow ring, using the fact that the height is the leading term of an arithmetic
Hilbert-Samuel function. In the case of a flag variety \(X = G / P\), this leads to a particularly nice expression of \(h(X)\) as a purely cohomological formula in terms of integrals of closed differential forms, without using currents.
In the paper under review, this is applied to the cases of the \(\text{SL}_n\)-Grassmannian, the Lagrangian and orthogonal Grassmannians and the complete \(\text{SL}_n\)-flag variety. It turns out that using the Köhler-Roessler formula, all computations can be done in terms of the classical Schubert calculus, giving explicit formulae for the heights in completely combinatorial terms.
Moreover, the results confirm for the considered cases the conjecture of Köhler and Roessler (initially based on computer calculations) that \(h(G/P)\) is a rational number with a certain denominator. Tamvakis H.: Height formulas for homogeneous varieties. Mich. Math. J. 48, 593--610 (2000) Arithmetic varieties and schemes; Arakelov theory; heights, Heights, Grassmannians, Schubert varieties, flag manifolds Height formulas for homogeneous 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 consider the action of a semisimple subgroup \(\hat{G}\) of a semisimple complex group \(G\) on the flag variety \(X=G/B\) and the linearizations of this action by line bundles \(\mathcal{L}\) on \(X\). We give an explicit description of the associated \textit{unstable locus} in dependence of \(\mathcal{L}\), as well as a formula for its (co)dimension. We observe that the codimension is equal to 1 on the regular boundary of the \(\hat{G}\)-ample cone and grows towards the interior in steps by 1, in a way that the line bundles with unstable locus of codimension at least \(q\) form a convex polyhedral cone. We also give a description and a recursive algorithm for determining all GIT-classes in the \(\hat{G}\)-ample cone of \(X\). As an application, we give conditions ensuring the existence of GIT-classes \(C\) with an unstable locus of codimension at least two and which moreover yield geometric GIT quotients. Such quotients \(Y_C\) reflect global information on \(\hat{G}\)-invariants. They are always Mori dream spaces, and the Mori chambers of the pseudoeffective cone \(\overline{\mathrm{Eff}}(Y_C)\) correspond to the GIT chambers of the \(\hat{G}\)-ample cone of \(X\). Moreover, all rational contractions \(f: Y_C\dashrightarrow Y'\) to normal projective varieties \(Y'\) are induced by GIT from linearizations of the action of \(\hat{G}\) on \(X\). In particular, this is shown to hold for a diagonal embedding \(\hat{G}\hookrightarrow(\hat{G})^k\), with sufficiently large \(k\). Grassmannians, Schubert varieties, flag manifolds, Divisors, linear systems, invertible sheaves, Geometric invariant theory, Group actions on varieties or schemes (quotients) Unstable loci in flag varieties and variation of quotients | 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 book grew out of a series of advanced undergraduate lectures given by the author at the Park City Mathematics Institute (PCMI) during its summer program of 2001. As the PCMI (founded in 1991) is to foster interaction between research and education in mathematics by yearly three-week summer programs for researchers and postdoctoral scholars, graduate students, undergraduate students, high school teachers, mathematics education researchers, and undergraduate faculty, and as one of its main goals is to make all of the participants aware of current trends in both mathematics education and research, each summer a different topic is chosen as the focus of discussion. According to this policy, the summer program of 2001 was devoted to recent developments in the interplay of complex geometry and mathematical physics, typified by the spectacular topic ``Enumerative Geometry and String Theory'', and the author's charge was to give students exposure to some of the central ideas, concepts, methods, and results in this current field of research, prudently on an adequate level. In view of this challenging task, the author's main goals were to provide first an introduction to some basic ideas of classical enumerative geometry, to lead over from there to the rudiments of both modern enumerative algebraic geometry and Gromov-Witten theory, and to explain finally some connections to current topological quantum field theory in physics. However, assuming no specific background knowledge beyond the standard undergraduate courses in mathematics and physics, he has also included some necessary introductory material on abstract algebra, geometry, analysis, and topology, thereby simultaneously broadened the range of areas in undergraduate mathematics for a less seasoned audience of students. As for the incorporation of the relevant physics, the author has skilfully chosen a merely example-driven approach, with emphasizing connections to enumerative geometry throughout. Finally, for the sake of expediency, and in order to streamline the process of this kind of sophisticated undergraduate teaching, sometimes very nonstandard treatments of advanced topics are given, in particular with regard to the involved concepts and methods of modern algebraic geometry. No doubt, this strategy bespeaks the author's pedagogical passion and mastery, and it is very much to the advantage of non-specialists (or beginners) in the field. The fourteen chapters of the book under review reflect reasonably faithfully both the spirit and the content of the author's PCMI course on the subject, as he points out in the preface, and the informal classroom style has been retained throughout. Chapter 1 provides a first warming up to enumerative geometry by studying the projective line. Chapter 2 turns then to projective hypersurfaces, with several concrete examples of enumerative problems in the projective plane and an illustration of Bezout's theorem. Chapter 3 is entitled ``Stable maps and enumerative geometry''. By means of the problem of the enumeration of rational curves on a quintic threefold, the author explains the famous Clemens conjecture, the link to mirror symmetry and Gromov-Witten invariants, the notion of stable maps to \(\mathbb{P}^n\) and their naive (compactified) moduli spaces, and the space of plane conics within this framework. In order to delve deeper into this topic, Chapters 4 and 5 are inserted as intermediate crash courses in topology, differentiable manifolds, differential forms, and singular cohomology. Chapter 6 provides more material on cellular homology and cohomology of compact complex manifolds, de Rham cohomology, and line bundles on manifolds.
Chapter 7 returns to enumerative geometry of lines in projective space by discussing Grassmannians, some Schubert calculus, universal bundles on Grassmannians, and vector bundles in general. The example of lines on a quintic threefold is taken up again and analyzed via Chern classes of vector bundles. Chapter 8 extends the explanation of intersection theory (touched upon in Chapter 2) by introducing excess intersection calculations. This is done concretely in the case of plane conics, thereby effectively comparing classical counts and advanced methods in enumerative geometry. Chapter 9 sets up the computation of the number of rational curves on the quintic threefold in the mathematical way, that is by using integration on the moduli spaces \(\overline M(\mathbb{P}^4,d)\) of degree \(d\) and genus 0 stable maps to \(\mathbb{P}^4\). In this chapter, the author also explains how this computation was inspired by intuitive reasonings from topological string theory in physics. This connection serves as the driving motivation for the remaining five chapters, in which the increasing cross-fertilization of enumerative algebraic geometry and theoretical physics is elucidated from a more physical point of view. Chapter 10 gives a brief introduction to classical mechanics and quantum mechanics, touching upon conformal symmetry and Feynman path integrals, whereas Chapter 11 discusses the concept of supersymmetry by means of some examples from 0-dimensional supersymmetric quantum field theories. This is used to illustrate how ideas from physics can be applied to solve enumerative problems in projective geometry. Chapter 12 briefly describes the ideas of bosonic string theory, the fermionic symmetry of the \(A\)-model, and the related BRST cohomology, again illustrated by the example of a quintic threefold as underlying spacetime. The relations to topological quantum field theory are explained in Chapter 13, with a special emphasis on Calabi-Yau threefolds, the B-model, the phenomenon of mirror symmetry, and the axiomatic definition of a \((1+1)\)-dimensional topological quantum field theory. The initial sketch of quantum cohomology given in this context is made more rigorous in the final Chapter 14, where the algebro-geometric approach to Gromov-Witten invariants via integrals on moduli spaces of stable maps is described. Various examples are then given to show how the quantum cohomology of \(\mathbb{P}^2\) can be calculated, and how Gromov-Witten invariants can be applied to the enumerative geometry of of the projective plane. At this point, the author has come full circle, returning to plane enumerative geometry after quite a tour through modern complex geometry and string theory.
Each chapter comes with a number of related exercises, most of which are however quite challenging. It is understood that the ambitious reader will follow the hints for further reading, on the one hand, or seek professional help from teachers or graduate students, on the other. As the author indicates already in the preface, this book will be quite demanding for an undergraduate student, especially with regard to the physics-related chapters at the end. Nevertheless, this text is perfectly suited for giving students a first research experience in a current field of mathematical activity, and it is certainly a lovely invitation to the subject. The author has managed to provide the first mathematically profound down-to-earth introduction to this fascinating area in contemporary mathematics and physics for beginners, and that with admirable didactic mastery. A steadfast student can profit a great deal from working through this text, be it by an impetus for a possible academic career in the future orby getting just a flavour of what is going on in current mathematics and physics. However, this book is by far not self-contained, and it is not meant to be a substitute for a more thorough and more systematic treatment of any of the topics panoramically reviewed here. But it surely is an excellent introduction to the more advanced monographs in the field, among those being [``Mirror symmetry and algebraic geometry'' by \textit{D. A. Cox} and \textit{S. Katz} (Mathematical Surveys and Monographs, 68, AMS, Providence, RI) (1999; Zbl 0951.14026)] and [``Mirror symmetry'' by \textit{K. Hori}, \textit{S. Katz}, \textit{A. Klemm}, \textit{R. Pandharipande} \textit{R.Thomas}, \textit{C. Vafa}, \textit{R. Vakil} and \textit{E. Zaslow} (Clay Mathematics Monographs 1, AMS, Providence, RI) (2003; Zbl 1044.14018)]. undergraduate lecture notes; projective geometry; Grassmannians; intersection theory; Gromov-Witten theory; topological quantum field theory; strings Sheldon Katz, \textit{Enumerative Geometry and String Theory}. Student Mathematical Library, IAS/Park City Mathematical Subseries 32, AMS, Providence 2006. Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Enumerative problems (combinatorial problems) in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Topological field theories in quantum mechanics, String and superstring theories; other extended objects (e.g., branes) in quantum field theory Enumerative geometry and string 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 Let \(P_i(x)\) for \(1\leq i\leq n-1\) be the \(n\times n\)-matrix obtained from the \(n\times n\) identity matrix by placing the block \(\left( \begin{smallmatrix} x&1\\0&1\end{smallmatrix}\right)\) with \(x\) at the \((i,i)\)'th coordinate. Then the matrices \(P_i(x)\) satisfy the Coxeter relations \(P_i(x) P_j(y) =P_j(y) P_i(x)\) if \(|i-j|\geq 2\) and \(P_i(x) P_{i+1}(y) P_i(z) =P_{i+1}(z)P_i(y+xz) P_{i+1}(x)\). It is shown that, for any reduced decomposition \(i=(i_1, i_2, \dots , i_N)\) of a permutation \(w\) and any ring \(R\), there is a bijection \(P_i:(x_1,x_2, \dots , x_N) \to P_{i_1}(x_1) P_{i_2}(x_2) \cdots P_{i_N}(x_N)\) from \(\mathbb{R}^N\) to the Schubert cell of \(w\). Moreover, it is shown how to factor explicitly any element of the Schubert cell corresponding to \(w\) into a product of such matrices. Thus one obtains a parametrization of the Schubert cell.
The formulas use planar configurations naturally associated to reduced decompositions. It is shown that the linear parts of these parametrizations give exactly all injective balanced labelings of the diagram of \(w\) [as defined by \textit{S. Fomin, C. Greene, V. Reiner} and \textit{M. Shimozono}, Eur. J. Comb. 18, No. 4, 373-389 (1997; Zbl 0871.05059)], and that the quadratic part characterizes the commutation classes of reduced decompositions. elementary matrices; Coxeter relations; Schubert cells; reduced decompositions; labelings of diagrams; planar configurations; factorization of matrices; commutation classes C. Kassel, A. Lascoux, and C. Reutenauer, ''Factorizations in Schubert cells,'' Adv. Math. 150 (2000), no. 1, 1--35. Factorization of matrices, Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of representation theory, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Factorizations in Schubert cells | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let C be a smooth curve, \(L\in Pic(C),V\subseteq H^ 0(C,L)\) defining a morphism into \({\mathbb{P}}^ n\); there is a bundle \(E_ L\) on the d-th symmetric product \(C^{(d)}\) and a map \(\sigma:\;V\otimes {\mathcal O}_{C^{(d)}}\to E_ L\) whose degeneracy locus gives the secant subschemes \(V^ n_ d\) of \(C^{(d)}\). In part I of this paper by \textit{H. Huibregtse} and the author (cf. the preceding review) it is given a local matrix description of \(\sigma\). Here the results and methods of part I are applied to compute several examples: local tangent space dimensions of \(V^ 1_ d\), tangent cones of \(V^ 1_ n\), stationary bisecants for space curves. In a few cases the painstaking explicit calculations are omitted. local singularity; secant subschemes; local tangent space dimensions; tangent cones; stationary bisecants Johnsen, T., Local properties of secant varieties in symmetric product II, Trans. AMS, 313, 205-220, (1989) Grassmannians, Schubert varieties, flag manifolds, Special algebraic curves and curves of low genus, Local deformation theory, Artin approximation, etc., Projective techniques in algebraic geometry Local properties of secant varieties in symmetric products. II | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The subgroup \(K=\mathrm{GL}_p \times \mathrm{GL}_q\) of \(\mathrm{GL}_{p+q}\) acts on the (complex) flag variety \(\mathrm{GL}_{p+q}/B\) with finitely many orbits. We introduce a family of polynomials specializing representatives for cohomology classes of the orbit closures in the Borel model. We define and study \(K\)-orbit determinantal ideals to support the geometric naturality of these representatives. Using a modification of these ideals, we describe an analogy between two local singularity measures: the \(H\)-polynomials and the Kazhdan-Lusztig-Vogan polynomials. flag variety; symmetric pair; cohomology class representative; Kazhdan-Luztig-Vogan polynomials Wyser, BJ; Yong, A, Polynomials for \(\text{GL}_p\times \text{ GL }_q\) orbit closures in the flag variety, Sel. Math., 20, 1083-1110, (2014) Group actions on combinatorial structures, Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds Polynomials for \(\mathrm{GL}_p\times \mathrm{GL}_q\) orbit closures in the flag variety | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let $G$ be a connected reductive group over $\mathcal{O}_F$ with a Borel pair $B \supset T$, where $F = \mathbb{F}_q (\!(\epsilon)\!)$. Put $L := \widehat{F^{\mathrm{nr}}}$ with residue field $k = \overline{\mathbb{F}_q}$ and Frobenius automorphism $\sigma$. Then $G(L)$ decomposes into the disjoint union of $K\epsilon^\mu K$ where $\mu$ ranges over the dominant cocharacters in $X_*(T)$ and $K = G(\mathcal{O}_L)$. \par The set $B(G)$ of $\sigma$-conjugacy classes in $G(L)$ plays a fundamental role in arithmetic geometry. It carries a partial order $\preceq$ and is stratified into $B(G, \mu)$ consisting of the classes intersecting $K\epsilon^\mu K$. For any $[b] \in B(G, \mu)$, the corresponding Newton stratum $\mathcal{N}_{[b], \mu}$ is $[b] \cap K\epsilon^\mu K$; for every $y$ therein, the central leaf $\mathcal{C}_y$ is defined as $\{ g^{-1} y \sigma(g): g \in K\}$. It carries the structure of smooth locally-closed subscheme of the loop group $LG$ over $k$, which is closed in $\mathcal{N}_{[b],\mu}$. Although these schemes are infinite-dimensional, one can pass to quotients by congruence subgroups to define their dimensions as a finite number; see Section 2 of the paper. The computation of $\dim \mathcal{C}_y$ is given in Theorem 2.11; the dimension depends solely on the class $[y] \in B(G)$. \par The closure relation between central leaves are also explored in this paper. Lemma 1.2 asserts that if the affine Deligne-Lusztig variety $X_\mu(b)$ is zero-dimensional, then $\mathcal{N}_{[b], \mu}$ consists of a single central leaf, whose closure is thus $\overline{\mathcal{N}_{[b], \mu}}$. On the other hand, Proposition 1.3 says that when $\mu$ is minuscule, $x_b$ is a fundamental alcove of $[b]$ in $W\mu W$ where $W$ is the finite Weyl group and $[b'] \preceq [b]$, then there exsts a fundamental alcove $x_{b'}$ in $W\mu W$ such that $\mathcal{C}_{x_{b'}} \subset \overline{\mathcal{C}_{x_b}}$. For the notion of fundamental alcoves in the affine Weyl group $\widetilde{W}$ of a $\sigma$-conjugacy class $[b]$, see [\textit{S. Nie}, Math. Ann. 362, No. 1--2, 485--499 (2015; Zbl 1367.20039)]. \par Finally, an example for $\mathrm{GL}(5)$ in Theorem 1.4 (see also Section 3) shows that not all closure relations between central leaves arise in this way, even though $\mu$ is still minuscule. affine Deligne-Lusztig varieties Formal groups, \(p\)-divisible groups, Grassmannians, Schubert varieties, flag manifolds, Linear algebraic groups over local fields and their integers Central leaves in loop groups | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This article studies questions of extrinsic rigidity of certain homogeneous projective algebraic varieties, in particular of the adjoint variety of a simple Lie algebra and of the Segre varieties. Let us denote by \(X\) such a variety and by \(Y\) some smooth subvariety of the same projective space such that \(\dim(Y)=\dim(X)\). Then one looks at the Fubini forms up to some order \(k\) in a general point \(y\in Y\) and in some point \(x\in X\) up to conjugation by linear isomorphisms of the tangent and normal spaces in the two points. Then \(X\) is called rigid at order \(k\) if coincidence of these Fubini forms up to order \(k\) implies that there is a projective transformation of the ambient projective space mapping \(Y\) to \(X\). In contrast, \(X\) is called flexible at order \(k\) if the space of varieties \(Y\) having the same Fubini forms up to order \(k\) is infinite dimensional. (The intermediate concept of quasi--rigidity does not occur in any of the cases studied in the article.)
The article contains rigidity results for the adjoint varieties (at order three), Veronese varieties, quadrics in the Veronese embedding, and Segre varieties. Moreover, some flexibility results are proved, which imply sharpness of the rigidity results. In particular, it is shown that the adjoint variety of \(\mathfrak{sl}(3,\mathbb C)\) is flexible at order two, with some of the varieties \(Y\) showing up even being non-flat as path geometries (so even the intrinsic geometries inherited from the Fubini forms are not locally isomorphic).
The main tool used in the article are exterior differential systems (EDS). The crucial idea is to formulate the rigidity questions in such a way that in the EDS problems, one can use the information on Lie algebra cohomology groups provided by Kostant's theorem instead of Spencer cohomology groups, which are notoriously difficult to compute. In order to do this, the authors develop a general concept of filtered EDS and their prolongations, which should be interesting in its own right. homogeneous variety; adjoint variety; generalized flag variety; rigidity; flexibility; Fubini forms; filtered exterior differential system; Lie algebra cohomology Landsberg, JM; Robles, C, Fubini-Griffiths-harris rigidity and Lie algebra cohomology, Asian J. Math., 16, 561-586, (2012) Rigidity results, Exterior differential systems (Cartan theory), Grassmannians, Schubert varieties, flag manifolds, Coadjoint orbits; nilpotent varieties, Cohomology of Lie (super)algebras Fubini-Griffiths-Harris rigidity and Lie algebra cohomology | 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 connected reductive complex algebraic group and \(H\) its spherical subgroup, i.e., the Borel subgroup \(B\subset G\) acts on \(G/H\) with finitely many orbits. The author considers \((B\times H)\)-orbits in \(G\) and studies the geometry of their closures in \((G\times G)\)-equivariant embeddings \(X\supset G\). This work is inspired by a paper of \textit{M. Brion} [Comment. Math. Helv. 76, No.2, 263--299 (2001; Zbl 1043.14012)], where the \(B\)-orbit closures in \(G\)-equivariant embeddings \(Y\supset G/H\) are studied. A direct link between these two papers is provided by an observation that there is a natural bijection between the sets of \(B\)-orbits in \(G/H\) and of \((B\times H)\)-orbits in \(G\) and furthermore, certain \(Y\) (and \(B\)-orbit closures therein) are obtained as geometric quotients by \(H\) of (\((B\times H)\)-orbit closures in) open subsets in certain \(X\) [\textit{N. Ressayre}, J. Algebra 265, No.1, 1--44 (2003; Zbl 1052.14061)]. First, the author proves that there are finitely many \((B\times H)\)-orbits in \(X\). It suffices to prove it for toroidal \(X\), where it is possible to describe the \((B\times H)\)-orbits explicitly using the structure of toroidal varieties. The other results concern toroidal embeddings. The author describes the local structure of \((B\times H)\)-orbit closures and their intersections with \((G\times G)\)-orbits in \(X\) (they appear to be proper). For smooth \(X\), the intersection multiplicities are determined (they are powers of 2). For smooth complete \(X\), the expression of the cohomology classes of \((B\times H)\)-orbit closures in terms of the classes of \((B\times B)\)-orbit closures (which span the cohomology of \(X\)) is found using \(B\)-equivariant cohomology. Finally, the author constructs a smooth toroidal embedding \(Y\) of \(G/H\) (namely, the wonderful embedding of \(\text{PGL}_4/\text{PO}_4\)) containing a \(G\)-orbit \(\mathcal{O}\) (namely, the unique closed orbit) and a \(B\)-orbit \(V\) such that there exists an irreducible component of \(\overline{V}\cap\overline{\mathcal{O}}\) consisting of singular points of \(\overline{V}\). This answers negatively a question of M.~Brion. The main tools used in the paper are the structure of toroidal embeddings of \(G\) (orbits, isotropy groups, transversal slices, see [\textit{M. Brion}, Comment. Math. Helv. 73, No.1, 137--174 (1998; Zbl 0935.14029)] and the oriented graph of \((B\times H)\)-orbits in \(G\) [see \textit{M. Brion}, Comment. Math. Helv. 76, No.2, 263--299 (2001; Zbl 1043.14012)]. group embedding; spherical variety; orbit closure; flag variety; equivariant cohomology N. Ressayre, Surs les orbites d'un sous-groupe sphérique dans la variété des drapeaux, Bull. Soc. Math. France 132 (2004), 543-567. Homogeneous spaces and generalizations, Grassmannians, Schubert varieties, flag manifolds, Equivariant homology and cohomology in algebraic topology Orbits of a spherical subgroup in 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 We study the geometry underlying the Wilson loop diagram approach to calculating scattering amplitudes in Supersymmetric Yang Mills (SYM) \( N = 4\). In particular, we study the smallest non-trivial multi-propagator case, consisting of 2 propagators on 6 vertices. We do this by translating the integrals of the theory to the combinatorics of the positive geometry each diagram represents, specifically identifying the positroid cells defined by each diagram and the homology of the subcomplex they collectively generate in \(\mathbb{G}_{\mathbb{R} , \geq 0} ( 2 , 6 )\). We verify the conjecture that the spurious singularities of the volume functional doall cancel on the codimension 1 boundaries of these cells, in this case. We also show that how the spurious singularities cancel is actually much more complicated than previously understood. The direct calculation laid out in this paper identifies many intricacies and artifacts of the geometry of Wilson loop diagram that need further study. SYM \(N=4\); positive Grassmannians; Seodhar decomposition Supersymmetry and quantum mechanics, Grassmannians, Schubert varieties, flag manifolds, Cluster algebras A study in \(\mathbb{G}_{\mathbb{R} , \geq 0} ( 2 , 6 )\): from the geometric case book of Wilson loop diagrams and SYM \(N =4\) | 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 should be seen as part of the study by I. M. Gelfand and his school of generalized hypergeometric functions. A rather detailed study of the notion of strata on compact homogeneous spaces of complex semisimple Lie groups is given. Several equivalent definitions are described, and the relations to matroids and to convex polytopes are explained. Schubert cells; Grassmannian; generalized hypergeometric functions; strata; compact homogeneous spaces; complex semisimple Lie groups Semisimple Lie groups and their representations, Harmonic analysis on homogeneous spaces, Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions), Grassmannians, Schubert varieties, flag manifolds Strata of a maximal torus in a compact homogeneous 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 We introduce inversions for classical Weyl group elements and relate them, by counting, to the length function, root systems and Schubert cells in flag manifolds. Special inversions are those that only change signs in the Weyl groups of types \(B_{n}, C_{n}\) and \(D_{n}\). Their counting is related to the (only) generator of the Weyl group that changes signs, to the corresponding roots, and to a special subvariety in the flag manifold fixed by a finite group. inversion set; flag manifold Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds Inversions in classical Weyl groups | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The classical Thom-Porteous formula expresses the homology class of the degeneracy locus of a generic map between two vector bundles as an alternating sum of Schur polynomials. A proof of this formula was given by Pragacz by expressing this alternating sum as the Euler characteristic of a Schur complex, which gives an explanation for the signs. Fulton later generalized this formula to the situation of flags of vector bundles by using alternating sums of Schubert polynomials. Building on the Schubert functors of Kraśkiewicz and Pragacz, we introduce Schubert complexes and show that Fulton's alternating sum can be realized as the Euler characteristic of this complex, thereby providing a conceptual proof for why an alternating sum appears.
See also the review of the journal version [\textit{S. V. Sam}, J. Algebra 337, No. 1, 103--125 (2011; Zbl 1242.13017)]. Schubert polynomials; Schubert complexes; degeneracy loci; balanced labelings; Thom-Porteous formula Syzygies, resolutions, complexes and commutative rings, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Determinantal varieties, Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations Schubert complexes and degeneracy loci | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(\mathbb X\subset\mathbb P(V)\) be a projective variety, which is not contained in a hyperplane. Then every vector \(v\) in \(V\) may be written as a sum of vectors from the affine cone over \(\mathbb X\). The minimal number of summands in such a sum is called the rank of \(v\). In this paper, we classify all equivariantly embedded homogeneous projective varieties \(\mathbb X\subset\mathbb P(V)\) whose rank function is lower semi-continuous. Classical examples are: the variety of rank one matrices (Segre variety with two factors) and the variety of rank one quadratic forms (quadratic Veronese variety). In the general setting, \(\mathbb X\) is the orbit in \(\mathbb P(V)\) of a highest weight line in an irreducible representation \(V\) of a reductive algebraic group \(G\). Thus, our result is a list of all irreducible representations of reductive groups, for which the corresponding rank function is lower semi-continuous. Grassmannians, Schubert varieties, flag manifolds, Group actions on varieties or schemes (quotients), Representation theory for linear algebraic groups Homogeneous projective varieties with semi-continuous rank function | 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 a subclass of Hessenberg varieties called abelian Hessenberg varieties, inspired by the theory of abelian ideals in a Lie algebra developed by Kostant and Peterson. We prove that the cohomology of an abelian regular semisimple Hessenberg variety, with respect to the symmetric group action defined by \textit{J. S. Tymoczko} [Contemp. Math. 460, 365--384 (2008; Zbl 1147.14024)], is a non-negative combination of tabloid representations. Our result implies that a graded version of the Stanley-Stembridge conjecture holds in the abelian case. As part of our arguments, we obtain inductive formulas for the Betti numbers of regular Hessenberg varieties. Stanley-Stembridge conjecture; symmetric functions; Hessenberg varieties Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations The cohomology of abelian Hessenberg varieties and the Stanley-Stembridge 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 The paper under review concerns the dimensions of certain affine Deligne-Lusztig varieties. We first set up the necessary notations.
Let \(L = k((t))\) and \(F = \bar{k}((t))\) for a finite field \(k\). Let \(\sigma\) denote the Frobenius automorphism of \(\bar{k}/k\) and extend it in a natural way to an automorphism of \(F/L\) by demanding \(\sigma(t) = t\). We denote the valuation ring of \(L\) by \({\mathcal O}_L\). Let \(G\) be a split connected reductive \(k\)-group, \(A\) a split maximal torus of \(G\) and \(K = G({\mathcal O}_L) \subseteq G(L)\). For \(b \in G(L)\) and for a dominant \(\mu \in X_*(A)\), the \textit{affine Deligne-Lusztig variety} is the locally closed subscheme of the affine Grassmannian, \(X := G(L)/K\), defined by
\[
X_{\mu}(b) := \{x \in G(L)/K: x^{-1}b\sigma(x) \in K\mu(t)K\} .
\]
\textit{M. Rapoport} [Astérisque 298, 271--318 (2005; Zbl 1084.11029)] conjectured a formula for the dimensions of these affine Deligne-Lusztig varieties which was modified later by \textit{R. Kottwitz} [Pure Appl. Math. Q. 2, No.~3, 817--836 (2006; Zbl 1109.11033)]. Let \(\mathbb D\) be a torus over \(F\) with character group \(\mathbb Q\). The element \(b\) determines a map \(\nu_b : {\mathbb D}\to G\) and the associated coweight \(\bar{\nu}_b \in X_*(A)_{\mathbb Q}\) is dominant. Let \(M_b\) denote the centraliser of \(\nu_b({\mathbb D})\) in \(G\). There exists an inner form \(J\) of \(M_b\) such that \(J(R) = \{g \in G(R \otimes_F L): g^{-1}b\sigma(g) = b\}\) for any \(F\)-algebra \(R\). If \(X_{\mu}(b)\) is nonempty, then the conjectured dimension formula is
\[
\dim X_{\mu}(b) = \langle \rho, \mu-\bar{\nu}_b \rangle-\frac{1}{2}(\text{rk}_F(G)-\text{rk}_F(J))
\]
where \(\rho \in X^*(A)_{\mathbb Q}\) denotes the half sum of the positive roots. It is proved in this paper that if this dimension formula holds for ``superbasic'' elements \(b \in\text{GL}_n(L)\), then it holds in general. This particular case is recently proved by \textit{E. Viehmann} [Ann. Sci. Éc. Norm. Supér. (4) 39, No.~3, 513--526 (2006; Zbl 1108.14036)] so now the conjecture is completely solved.
The paper also investigates the dimensions of the affine Deligne-Lusztig varieties in the affine flag manifold \(G(L)/I\) where \(I\) is the Iwahori subgroup of \(G(L)\) obtained from an alcove \({\mathbf a}_1\) in the apartment associated to \(A\). Deligne-Lusztig varieties; reductive groups Görtz, U.; Haines, T. J.; Kottwitz, R. E.; Reuman, D. C., Dimension of certain affine Deligne-Lusztig varieties, Ann. Sci. Ecole Norm. Sup. (4), 39, 467-511, (2006) Group actions on varieties or schemes (quotients), Grassmannians, Schubert varieties, flag manifolds, Linear algebraic groups over local fields and their integers, Modular and Shimura varieties, Arithmetic aspects of modular and Shimura varieties Dimensions of some affine Deligne-Lusztig 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 authors provide an overview of the Gelfand-Zeitlin integrable system on the Lie algebra of \(n\times n\) complex matrices \(\mathfrak{gl}(n,{\mathbb C})\) introduced by \textit{B. Kostant} and \textit{N. Wallach} [Prog. Math. 243, 319--364 (2006; Zbl 1099.14037); ibid. 244, 387--420 (2006; Zbl 1099.14038)]. They discuss results concerning the geometry of the set of strongly regular elements, which consists of the points where the Gelfand-Zeitlin flow is Lagrangian. They use the theory of \(K\)-orbits on the flag variety \({\mathcal B}_n\) of \(\mathrm{GL}(n,{\mathbb C})\), where \(K = K_n = \mathrm{GL}(n-1,{\mathbb C}) \times \mathrm{GL}(1,{\mathbb C})\), to describe the strongly regular elements in the nilfiber of the moment map of the system. They review the general theory of orbits of a symmetric subgroup of a reductive algebraic group acting on its flag variety, and illustrate how the general theory can be applied to understand the specific example of \(K_n\) and \(\mathrm{GL}(n,{\mathbb C})\). flag variety; symmetric subgroup; nilpotent matrices; integrable systems; Gelfand-Zeitlin theory Colarusso, M.; Evens, S., The Gelfand-Zeitlin integrable system and K-orbits on the flag variety, (Symmetry: Representation Theory and Its Applications. Symmetry: Representation Theory and Its Applications, Progress. Math., (2014), Birkhäuser: Birkhäuser Boston), 36 pages Grassmannians, Schubert varieties, flag manifolds The Gelfand-Zeitlin integrable system and \(K \)-orbits on the flag variety | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors construct examples of indecomposable vector bundles on Grassmannians \(\mathrm{Gr}(r,k)\), whose rank is smaller than the dimension of \(\mathrm{Gr}(r,k)\). These bundles are obtained as a quotient of globally generated bundles, by generalizing the construction of H. Tango for rank \(n-1\) bundles on \(\mathbb P^n\). The proof of the existence is based on computations in Schubert calculus, which are performed by means of a method introduced by \textit{L. Gatto} [Asian J. Math. 9, No. 3, 315--322 (2005; Zbl 1099.14045)]. The authors describe the structure of Tango bundles on Grassmannians and prove that they are stable (in the sense of Mumford-Takemoto). Finally, the authors describe the component of Tango bundles in the Maruyama moduli scheme. vector bundles; Grassmannians Grassmannians, Schubert varieties, flag manifolds, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Tango bundles on Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The aim of this paper is to give first a formula for the Maslov index of a gradient holomorphic disk, which is an analogue of a gradient sphere in [\textit{Y. Karshon}, Periodic Hamiltonian flows on four-dimensional manifolds. Providence, RI: American Mathematical Society (AMS) (1999; Zbl 0982.70011)], bounded by an \(S^1\)-invariant Lagrangian submanifold in a symplectic manifold admitting a Hamiltonian \(S^1\)-action. Using the formula, the authors classify all monotone Lagrangian fibers of Gelfand-Cetlin systems on partial flag manifolds. This paper is organized as follows: Section 1 is an introduction to the subject and a description of the results. In Section 2, the authors define a gradient holomorphic disk generated by a Hamiltonian \(S^1\)-action. Section 3 discusses the Maslov index for gradient disks and proves the first main result. In Section 4, they review Gelfand-Cetlin systems and recall some results in [\textit{Y. Cho}, ``Lagrangian fibers of Gelfand-Cetlin systems'', Preprint, \url{arXiv:1704.07213}]. Section 5 is devoted to classifying the monotone Lagrangian Gelfand-Cetlin fibers and to proving the second main result. Lagrangians; flag varieties; Gelfand-Cetlin systems; Maslov index Mirror symmetry (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Nonautonomous Hamiltonian dynamical systems (Painlevé equations, etc.), Lagrangian submanifolds; Maslov index Monotone Lagrangians 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 Using results from some previous papers of theirs, the authors classify the congruences in \(G(1,4)\), the Grassmann variety of lines in the complex projective 4-space, with degree less or equal than 10. An explicit construction as degeneracy loci of suitable vector bundles on \(G(1,4)\) of most of these congruences is given.
See also the appendix to this paper \textit{E. Arrondo}, Commun. Algebra 26, No. 10, 3267-3274 (1998; see the following review Zbl 0963.14029). Grassmann variety; degeneracy loci Arrondo, E.; Bertolini, M.; Turrini, C.: Congruences of small degree in \(G(1,4)\). Comm. algebra 26, No. 10, 3249-3266 (1998) Grassmannians, Schubert varieties, flag manifolds, Families, moduli of curves (algebraic) Congruences of small degree in \(G(1,4)\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a simply connected complex semisimple Lie group of rank \(r\) with a fixed Borel subgroup \(B\) and a maximal torus \(H\subset B\). Let \(W=\text{Norm}_G(H)/H\) be the Weyl group of \(G\). The generalized flag manifold \(G/B\) can be decomposed into the disjoint union of Schubert cells \(X^\circ_w=(BwB)/B\), for \(w\in W\). To any weight \(\gamma\) that is \(W\)-conjugate to some fundamental weight of \(G\), one can associate a generalized Plücker coordinate \(p_\gamma\) on \(G/B\). In the case of type \(A_{n-1}\) (i.e., \(G=SL_n)\), the \(p_\gamma\) are the usual Plücker coordinates on the flag manifold. The closure of a Schubert cell \(X^\circ_w\) is the Schubert variety \(X_w\), an irreducible projective subvariety of \(G/B\) that can be described as the set of common zeroes of some collection of generalized Plücker coordinates \(p_\gamma\). It is also known that every Schubert cell \(X^\circ_w\) can be defined by specifying vanishing and/or non-vanishing of some collection of Plücker coordinates. The main two problems studied in this paper are the following.
(1) Describe a given Schubert cell by as small as possible number of equations of the form \(p_\gamma=0\) and inequalities of the form \(p_\gamma\neq 0\).
(2) Suppose a point \(x\in G/B\) is unknown to us, but we have access to an oracle that answers questions of the form: ``\(p_\gamma(x)=0\), true or false?'' How many such questions are needed to determine the Schubert cell \(x\) is in?
The number of equations of the form \(p_\gamma=0\) needed to define a Schubert variety is generally much larger than its codimension. We show that for a certain Schubert variety \(X_w\) in the flag manifold of type \(A_{n-1}\), one needs exponentially many such equations to define it, even though \(\text{codim}(X_w) \leq\dim (G/B)={n\choose 2}\). Given this kind of ``complexity'' of Schubert varieties, it may appear surprising that for the types \(A_r,B_r,C_r\), and \(G_2\), we provide a description of an arbitrary Schubert cell \(X^\circ_w\) that only uses \(\text{codim} (X_w)\) equations of the form \(p_\gamma=0\) and at most \(r\) inequalities of the form \(p_\gamma\neq 0\). For the type \(D\), a description of Schubert cells is slightly more complicated. Our main result regarding (2) is an algorithm that recognizes a Schubert cell \(X^\circ_w\) containing an element \(x\). For the types \(A_r,B_r,C_r\), and \(G_2\), our algorithm ends up examining precisely the same Pücker coordinates of \(x\) that appear in the previous result. In the case of type \(A_{n-1}\), recognizing a cell requires testing the vanishing of at most \({n\choose 2}\) Plücker coordinates. Finally, we discuss the problem of presenting a subset of Plücker coordinates whose vanishing pattern determines which cell a point is in. Schubert variety; flag manifold; Plücker coordinate; Bruhat cell; vanishing pattern S. Fomin and A. Zelevinsky, ''Recognizing Schubert cells,'' preprint math. CO/9807079, July 1998. Grassmannians, Schubert varieties, flag manifolds Recognizing Schubert cells. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials A geometric extension algebra is an extension algebra of a semi-simple perverse sheaf (allowing shifts), e.g., a push-forward of the constant sheaf under a projective map. Particular nice situations arise for collapsings of homogeneous vector bundles over homogeneous spaces. In this paper, we study the relationship between partial flag and complete flag cases. Our main result is that the locally finite modules over the geometric extension algebras are related by a recollement. As examples, we investigate parabolic affine nil Hecke algebras, geometric extension algebras associated with parabolic Springer maps and an example of Reineke of a parabolic quiver-graded Hecke algebra. Hecke algebras and their representations, Grassmannians, Schubert varieties, flag manifolds, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] From complete to partial flags in geometric extension 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 In [Compos. Math. 154, No. 11, 2403--2425 (2018; Zbl 1490.20032)], we have established a Springer theory for the symmetric pair \((\operatorname{SL}(N), \operatorname{SO}(N))\). In this setting we obtain representations of (the Tits extension) of the braid group rather than just Weyl group representations. These representations arise from cohomology of families of certain (Hessenberg) varieties. In this paper we determine the Springer correspondence explicitly for IC sheaves supported on order \(2\) nilpotent orbits. In this process we encounter universal families of hyperelliptic curves. As an application we calculate the cohomology of Fano varieties of \(k\)-planes in the smooth intersection of two quadrics in an even dimensional projective space. Grassmannians, Schubert varieties, flag manifolds, Fano varieties Springer correspondence, hyperelliptic curves, and cohomology of Fano 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 Denote by \(W_{\vec{a}}\) Schubert varieties associated to \(\vec{a}\) and by \(\sigma_{\vec{a}}\in\text{H}^{2|\vec{a}|}(G,\mathbb{C})\) the corresponding elements in cohomology where \(G\) is the Grassmannian. The symbol \(W_a\) stands for a special Schubert variety associated to \((a,0,\dots,0)\). Choose general points \(p_1,\dots,p_N\in\mathbb{P}^1\) and general translates of the \(W_{\vec{a}}\). The Gromov-Witten intersection number \(\langle W_{\vec{a}_1},\dots,W_{\vec{a}_N}\rangle_d\) is, by a naive definition, the number of holomorphic maps \(f:\mathbb{P}^1\to G\) of degree \(d\) with the property that \(f(p_i)\in W_{\vec{a}_i}\) for all \(i=1,\dots,N\). For \(d=0\) one gets the original intersection number. The small quantum ring is the vector space \(\text{H}^\ast(G,\mathbb{C})[q]\) over \(C[q]\) with an associative product which obeys
\[
\sigma_{\vec{a}_1}\ast\dots\ast\sigma_{\vec{a}_N}= \sum_{d\geq 0}q^d\left(\sum_{\vec{a}} \langle W_{\vec{a}},W_{\vec{a}_1},\dots,W_{\vec{a}_N}\rangle_d \sigma_{\vec{a}}\right).
\]
The paper generalizes Giambelli's formula and Pieri's formula to the small quantum ring. In the quantum Giambelli formula that reads \(\sigma_{\vec{a}}=\Delta_{\vec{a}}(\sigma_\ast)\) no higher terms in \(q\) arise. The Giambelli determinant in cohomology classes corresponding to special Schubert varieties is evaluated in \(\text{H}^\ast(G,\mathbb{C})[q]\). On the other hand, the quantum Pieri formula has a correction term,
\[
\sigma_a\ast\sigma_{\vec{a}}= p_{a,\vec{a}}(\sigma_{\vec\ast})+ q\left(\sum_{\vec c}\sigma_{\vec c}\right),
\]
with an appropriate range of \(\vec c\). Before giving the proofs the Gromov-Witten number is defined rigorously by considering intersections of Schubert varieties on the moduli space \(\mathcal M_d\) of holomorphic maps of degree \(d\) from \(\mathbb{P}^1\) to \(G\) with \(\mathcal M_d\) being an open subscheme in the Grothendieck quoted scheme. As a corollary of the quantum Giambelli's formula the author also shows a Vafa and Intriligator formula for the Gromov-Witten intersection number of special Schubert varieties. Gromov-Witten intersection number; small quantum ring; Giambelli's formula; Pieri's formula A. Bertram. ''Quantum Schubert calculus''. Adv. Math. 128 (1997), pp. 289--305.DOI. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Two-dimensional field theories, conformal field theories, etc. in quantum mechanics, Grassmannians, Schubert varieties, flag manifolds, Relationships between surfaces, higher-dimensional varieties, and physics, Quantum field theory on curved space or space-time backgrounds Quantum Schubert calculus | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a connected semisimple algebraic group of adjoint type over an algebraically closed field of characteristic zero, and \(\mathrm{g}\) be its Lie algebra. Recall that a subalgebra \(\mathrm{h}\subseteq\mathrm{g}\) is said to be spherical if there is a Borel subalgebra \(\mathrm{b}\subseteq\mathrm{g}\) such that \(\mathrm{h}+\mathrm{b}=\mathrm{g}\). Suppose that the spherical subalgebra \(\mathrm{h}\) coincides with its normalizer in \(\mathrm{g}\). The closure \(\overline{G\mathrm{h}}\) of the orbit \(G\mathrm{h}\) in the corresponding Grassmann variety is called the Demazure embedding of \(G\mathrm{h}\).
\textit{M.~Brion} [J. Algebra 134, No. 1, 115--143 (1990; Zbl 0729.14038)] conjectured that the Demazure embedding is smooth. Positive results in this direction are contained in the works of M.~Demazure, C.~De Concini, C.~Procesi, D.~Luna, P.~Bravi and G.~Pezzini. The main theorem of the article states that the Demazure embedding is indeed smooth. In the proof, the results of \textit{D.~Luna} [J. Algebra 258, No. 1, 205--215 (2002; Zbl 1014.17009)] and \textit{G.~Pezzini} [Math. Z. 255, No. 4, 793--812 (2007; Zbl 1122.14036)] are used. spherical subalgebra; Grassmannian; wonderful variety I. Losev, \textit{Demazure embeddings are smooth}, Int. Math. Res. Not. IMRN (2009), no. 14, 2588-2596. Compactifications; symmetric and spherical varieties, Grassmannians, Schubert varieties, flag manifolds, Group actions on varieties or schemes (quotients) Demazure embeddings are smooth | 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 an algorithm for deciding automatically whether a given polynomial relation among program variables is a loop invariant for loops whose assignment statements are affine. The algorithm, based on techniques from symbolic summation and computer algebra, works equally well in the presence of pre- and post conditioning. The correctness and termination of the algorithm are established by previous work of the first author [Proc. of TACAS, vol. 4963 of LNCS, 249--264 (2008; Zbl 1134.68600)], [Proc of PSI, 184--194 (2009)].
The method presented here has been implemented in the \texttt{Aligator} software package. program verification; loop invariants; polynomial relations; symbolic summation; \texttt{Aligator} Algebraic combinatorics, Computational aspects of field theory and polynomials, Computational aspects in algebraic geometry, Mathematical aspects of software engineering (specification, verification, metrics, requirements, etc.), Specification and verification (program logics, model checking, etc.) Deciding properties of affine loops | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(F\) be a general cubic fourfold in the projective space \({\mathbb P}^5\) over the complex field and let \(f\) be an equation for \(F\). One can write \(f\) as a linear combination of 10 cubic powers of linear forms (and not less), as an immediate dimension count suggests. The subsets \(Z=\{L_0,\dots,L_9\}\) of the dual space \(({\mathbb P}^5)^\vee\) such that \(F=a_0L_0^3+\dots+a_9L_9^3\) are precisely the elements of \(\text{Hilb}_{10}({\mathbb P}^5)^\vee\) which are ''apolar'' to \(F\), in the sense that if one reads homogeneous polynomials over \(({\mathbb P}^5)^\vee\) as differential operators on \(f\), then \(g(f)=0\) for all \(g\) in the ideal of \(Z\).
The authors prove that the variety \(VPS =\{Z\in \text{Hilb}_{10}({\mathbb P}^5)^\vee: Z\) is apolar to \(F\}\) is isomorphic to the Fano variety of lines contained in some other cubic \(F'\subset{\mathbb P}^5\).
This correspondence is particularly interesting when \(F\subset{\mathbb P}^5\) is the apolar cubic fourfold of a general K3 surface \(S\) of genus \(8\), naturally embedded in \({\mathbb P}^8\) by a generator of the Picard group. In this case it turns out that the cubic \(F'\) corresponding to \(F\) in the previous construction is the pfaffian cubic such that \(\text{Hilb}_2(S)\) is isomorphic to the set of lines in \(F'\). K3 surfaces; apolar elements; general cubic fourfold; Fano variety; Picard group; pfaffian cubic Iliev, Atanas; Ranestad, Kristian, \textit{K}3 surfaces of genus 8 and varieties of sums of powers of cubic fourfolds, Trans. Amer. Math. Soc., 353, 4, 1455-1468, (2001) \(4\)-folds, \(K3\) surfaces and Enriques surfaces, Hypersurfaces and algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Projective techniques in algebraic geometry, Fano varieties, Complete intersections \(K3\) surfaces of genus 8 and varieties of sums of powers of cubic fourfolds | 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 formulate a conjecture for an arbitrary simple Lie algebra \(\mathfrak g\) in terms of the geometry of principal nilpotent pairs. When \(\mathfrak g\) is specialized to \({\mathfrak {sl}}_n\), this conjecture readily implies the \(n!\) result and it is very likely that, in fact, it is equivalent to the \(n!\) result in this case. In addition, this conjecture can be thought of as generalizing an old result of Kostant. In another direction, we show that to prove the validity of the \(n!\) result for an arbitrary \(n\) and an arbitrary partition of \(n\), it suffices to show its validity only for the staircase partitions''. Simple, semisimple, reductive (super)algebras, Symmetric functions and generalizations, Representation theory for linear algebraic groups, Grassmannians, Schubert varieties, flag manifolds A conjectural generalization of the \(n!\) result to arbitrary groups | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author computes the Euler characteristics of quiver Grassmannians and quiver flag varieties of tree and band modules and proves their positivity. As an application, the author considers the Ringel-Hall algebra \(\mathcal C(A)\) of a string algebra \(A\) and computes, in combinatorial terms, the product of arbitrary functions in \(\mathcal C(A)\). Finally, it is shown that the Euler characteristics of a quiver Grassmannian for some finite dimensional representation of a locally finite quiver is determined by the Euler characteristics of the quiver Grassmannian for a free covering of the corresponding quiver and module. Euler characteristics; coverings; quiver Grassmannians; flag varieties; representations; Ringel-Hall algebras; string algebras; band modules; string modules; tree modules Haupt, N., Euler characteristics of quiver Grassmannians and ringel-Hall algebras of string algebras, Algebr. Represent. Theory, 15, 755-793, (2012) Representations of quivers and partially ordered sets, Grassmannians, Schubert varieties, flag manifolds, Quantum groups (quantized enveloping algebras) and related deformations, Group actions on affine varieties, Representation theory of lattices Euler characteristics of quiver Grassmannians and Ringel-Hall algebras of string 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 Consider the projective space \(\mathrm{PG}(rt-1,q)\) and let \(S\) be a Desarguesian \((t-1)\) spread of the space, \(\Pi\) an \(m\)-dimensional subspace and \(\Lambda\) the linear set consisting of the elements of the spread with non-empty intersection with \(\Pi\), where a linear set is a set of points defined by an additive subgroup of the ambient vector space. The Plücker map is an embedding of the set of subspaces of a vector space with a given dimension into the projective space. The authors describe the image under this embedding of the elements of \(\Lambda\) and they show that it is an \(m\)-dimensional variety, a projection of a Veronese variety of dimension \(m\) and degree \(t\) and that it is a suitable linear section of the Plücker embedding of the elements of \(S\) . Grassmannian; linear set; Desarguesian spread; Schubert variety Desarguesian and Pappian geometries, Combinatorial aspects of finite geometries, Grassmannians, Schubert varieties, flag manifolds On some subvarieties of the Grassmann variety | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(X\) be a smooth complete complex algebraic variety. A connective \(K\)-theory \(K(X,\beta)\) of \(X\) is, roughly speaking, an interpolation of the cohomology theory and the \(K\)-theory of \(X\), in the sense that \(K(X,0)=H^*(X)\), the ordinary singular cohomology ring of \(X\), and \(K(X,1)=K(X)\), the ordinary Grothendieck ring of \(X\).
To say it with a slogan, the paper under review is concerned with the quantum equivariant connective \(K\)-theory, \(qh^*_n:=qh^*(X;\beta)\), of the Grassmann variety \(X:=G(n,N)\), parametrizing \(n\)-dimensional subspaces of \(\mathbb{C}^N\).
The ring \(qh^*_n\) can be seen as a multiparameter deformation of the classical cohomology ring of \(X\). The involved deformation parameters \((t_1,\ldots,t_n)\), \(q\) and \(\beta\) play different roles. The first are the equivariant parameters related with the action of the algebraic torus \(\mathbb{T}:=(\mathbb{C}^*)^n\), induced by the diagonal action on \(\mathbb{P}^{N-1}\), \(q\) is the quantum deformation parameter and \(\beta\) is a parameter that connects the generalized (i.e. quantum, equivariant) \(K\)-theory of \(X\) (for \(\beta=1\)) to the quantum, equivariant cohomology ring \(QH^*_{\mathbb{T}}(X)\) (for \(\beta=0\)).
The main result of this paper is without doubt the description of the ring \(qh^*_n\). Its impact is described in another main result, named Theorem 1.1. in the introduction, where three different specializations of \(qh^*_n\), obtained by setting to zero some of the deformation parameters, are considered. It is so shown that \(qh^*_n\) generalizes all the presentations known so far, relying on one hand on the classical Schubert Calculus, ruled by Giambelli's and Pieri's formula and, on the other, on important work appeared along the last couple of decades, due to \textit{D. Peterson} [``Quantum Cohomology of \(G/P\)'', Lecture Notes, M.I.T. (1997)], \textit{B. Kostant} and \textit{S. Kumar} [J. Differ. Geom. 32, No. 2, 549--603 (1990; Zbl 0731.55005)] and more recently to \textit{A. S. Buch} and \textit{L. C. Mihalcea} [Duke Math. J. 156, No. 3, 501--538 (2011; Zbl 1213.14103)].
In particular, Theorem 1.1. shows that i) setting \(\beta=0\) one recovers the presentation due to \textit{L. C. Mihalcea} [Adv. Math. 203, No. 1, 1--33 (2006; Zbl 1100.14045)] of the quantum cohomology ring \(QH^*_{\mathbb{T}}(X)\); ii) setting \(\beta=1\) and \((t_1,\ldots, t_N)=(0,\ldots,0)\), one obtains Buch's quantum \(K\) theory \(KQ(X)\) and that iii) for \(\beta=-1\), \(q=0\) and \(t_j\) equal to certain expressions involving generators of the character ring of \({\mathfrak gl}(N)\), recovers \(K_{\mathbb{T}}(X)\), the equivariant \(K\)-functor. The proof of iii) above is certainly the most intriguing, as it involves a generalization of the celebrated Goresky-Kottwitz-MacPherson theory [\textit{M. Goresky} et al., Invent. Math. 131, No. 1, 25--83 (1998; Zbl 0897.22009)] and the localized Schubert classes are identified with certan polynomials related to the Bethe ansatz of quantum integrable models, which is another part of the story told in this impressive article.
As a matter of fact, the topic faced in the paper is so wide and important that it is hard to bound it within the narrow borders of conventional subject classifications. Indeed, the ring \(qh^*_n\), the main character of the paper, allows the authors to dig up a breath taking relationship between Schubert Calculus and certain quantum integrable systems that in statistical mechanics are known as \textsl{asymmetric six-vertex model}, invented to describe the physics of anti-ferroelectric materials.
This very well written paper is rather long and dense but the authors put a special effort not to loose the readers by clearly segmenting it in sections, with the aim to provide pre-requisites with graduality. Although combinatorial tools are inspired by the Yang Baxter equations as well as the six vertex models in statistical mechanics, a preliminary knowledge of the latter is not necessary to follow the mathematical content of the paper, which candidates itself to be a must for all mathematicians interested in the cohomological theories of homogeneous varieties. quantum cohomology; quantum \(K\)-theory; enumerative combinatorics; exactly solvable models; Bethe ansatz; Yang Baxter equations; statistical mechanics V. Gorbounov and C. Korff. ''Quantum integrability and generalised quantum Schubert calculus''. Adv. Math. 313 (2017), pp. 282--356.DOI. Grassmannians, Schubert varieties, flag manifolds, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Bordism and cobordism theories and formal group laws in algebraic topology, Symmetric functions and generalizations, Exactly solvable models; Bethe ansatz, Equivariant \(K\)-theory Quantum integrability and generalised quantum Schubert calculus | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The set of row reduced matrices (and of echelon form matrices) is closed under multiplication. We show that any system of representatives for the \(\mathrm{Gl}_n(\mathbb{K})\) action on the \(n\times n\) matrices, which is closed under multiplication, is necessarily conjugate to one that is in simultaneous echelon form. We call such closed representative systems Grassmannian semigroups. We study internal properties of such Grassmannian semigroups and show that they are algebraic semigroups and admit gradings by the finite semigroup of partial order preserving permutations, with components that are naturally in one-one correspondence with the Schubert cells of the total Grassmannian. We show that there are infinitely many isomorphism types of such semigroups in general, and two such semigroups are isomorphic exactly when they are semiconjugate in \(M_n(\mathbb{K})\). We also investigate their representation theory over an arbitrary field, and other connections with multiplicative structures on Grassmannians and Young diagrams. semigroup; Grassmannian; simultaneous form; echelon form; representations Algebraic systems of matrices, Semigroups of transformations, relations, partitions, etc., Grassmannians, Schubert varieties, flag manifolds, Representation of semigroups; actions of semigroups on sets Grassmannian semigroups and their representations | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Algebraic and geometric structures associated with Birkhoff strata of Sato Grassmannian are analyzed. It is shown that each Birkhoff stratum \(\Sigma_S\) contains a subset \(W_{\hat{S}}\) of points for which each fiber of the corresponding tautological subbundle \(TB_{W_S}\) is closed with respect to multiplication. Algebraically \(TB_{W_S}\) is an infinite family of infinite-dimensional commutative associative algebras and geometrically it is an infinite tower of families of algebraic curves. For the big cell the subbundle \(TBW_{\emptyset}\) represents the tower of families of normal rational (Veronese) curves of all degrees. For \(W_1\) such tautological subbundle is the family of coordinate rings for elliptic curves. For higher strata, the subbundles \(TB_{W_{1,2,\dots,n}}\) represent families of plane (\(n + 1, n + 2\)) curves (trigonal curves at \(n=2\)) and space curves of genus \(n\). Two methods of regularization of singular curves contained in \(TBW_{\hat{s}}\), namely, the standard
blowing-up and transition to higher strata with the change of genus are discussed. Sato Grassmannian; algebraic varieties; associative algebras; desingularization Grassmannians, Schubert varieties, flag manifolds, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) Birkhoff strata of Sato Grassmannian and 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 The aim of the paper is to study the eigenscheme of order three partially symmetric and symmetric tensors. They also show that a subvariety of the Grassmannian \(Gr(3,\mathbb{P}^{14})\) parametrizes the eigenscheme of \(4 \times 4 \times 4\) symmetric tensors.
The spectral theory of tensors is a multi-linear generalization of the study of eigenvalues and eigenvectors in the case of matrices. The eigenscheme \(E(\mathcal{T})\) of a tensor \(\mathcal{T}\) can be roughly thought as the set of eigenpoints of the tensor, i.e. eigenvectors of \(\mathbb{C}^{n+1}\) of a particular contraction of the tensor. In the case of partial symmetric or symmetric tensors of order \(3\) a contraction may be the following. A partial symmetric tensor \(\mathcal{T} \in \operatorname{Sym}^2 \mathbb{C}^{n+1} \otimes \mathbb{C}^{n+1}\) can be seen as an \((n+1)\)-tuple of quadratic forms \((q_0,\dots,q_n)\) in the variables \(x_i\). Analogously, given a a symmetric tensor \(f \in \operatorname{Sym}^3 \mathbb{C}^{n+1}\), i.e. a homogeneous cubic polynomial, one can associate to it an \((n+1)\)-tuple of quadratic forms given by its derivatives \(\frac{\partial f}{\partial x_i}\). The authors investigates the eigenscheme and some its particular subschemes of the aforementioned tensors with those contractions.
At first they recall some basic notions regarding the theory. In particular they introduce the irregular eigenscheme \(\operatorname{Irr}(\mathcal{T})\) and the regular eigenscheme \(\operatorname{Reg}(\mathcal{T})\). The first can be thought as the subscheme of \(E(\mathcal{T})\) given by points with zero eigenvalue, while the second is the residue of \(E(\mathcal{T})\) with respect to \(\operatorname{Irr}(\mathcal{T})\). After that they focus on the case of order \(3\) symmetric tensors providing bounds on the dimensions and geometric properties of the irregular and regular eigenschemes. Numerous examples of symmetric tensors satisfying all the described properties are provided. As they observe, if the regular eigenscheme of a cubic polynomial is \(0\) dimensional, then it consists of at most \(2^{n+1}-1\) points. Therefore they investigate in the ternary and quaternary case whether there exists a cubic polynomial with a prescribed number of regular eigenpoints. Eventually they show that a open subvariety of a linear subspace of the Grassmannian \(Gr(3,\mathbb{P}^{14})\) parametrizes the eigenschemes of order \(3\) quaternary symmetric tensors. eigenpoints of tensors; cubic surfaces; Grassmannians Rational and ruled surfaces, Grassmannians, Schubert varieties, flag manifolds, Eigenvalues, singular values, and eigenvectors, Multilinear algebra, tensor calculus On the eigenpoints of cubic 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 0539.00031.]
The problem of determining the minimum order \(q=\delta (K)\) of a compensator K(s) which stabilizes in closed-loop a \(p\times m\) generic transfer function G(s) of fixed McMillan degree \(\delta (G)=n\) is considered. It is shown in the paper that q satisfies the inequality \((q+1)\max (m,p)+\min (m,p)-1\geq n\) and \(mp\geq n\) represents a necessary condition for generic stabilization by constant gain output feedback. Using the Lyusternick and Shnirel'mann category, the following sufficient criterion for generic stabilizability is obtained:
\[
L-S cat(Grass(p,m+p))-1\geq n,
\]
\(Grass(p,m+p)\) denoting the space of all p- planes in \(R^{m+p}\). Some applications to generic stabilizability by constant gain feedback are also presented. minimum order; compensator; generic stabilization; constant gain feedback Stabilization of systems by feedback, Linear systems in control theory, Multivariable systems, multidimensional control systems, Grassmannians, Schubert varieties, flag manifolds, Abstract critical point theory (Morse theory, Lyusternik-Shnirel'man theory, etc.) in infinite-dimensional spaces High gain feedback and the stabilizability of multivariable systems | 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 construct projective embeddings of horospherical varieties of Picard number one by means of Fano varieties of cones over rational homogeneous varieties. Then we use them to give embeddings of smooth horospherical varieties of Picard number one as linear sections of rational homogeneous varieties. horospherical varieties; rational homogeneous varieties; varieties of minimal tangents Hong, J., Smooth horospherical varieties of Picard number one as linear sections of rational homogeneous varieties. J, Korean Math. Soc., 53, 433-446, (2016) Grassmannians, Schubert varieties, flag manifolds, Homogeneous spaces and generalizations, Group actions on varieties or schemes (quotients) Smooth horospherical varieties of Picard number one as linear sections of rational homogeneous varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials A family of \(N\)-qubit entanglement monotones invariant under stochastic local operations and classical communication (SLOCC) is defined. This class of entanglement monotones includes the well-known examples of the concurrence, the 3-tangle and some of the four-, five- and \(N\)-qubit SLOCC invariants introduced recently. The construction of these invariants is based on bipartite partitions of the Hilbert space in the form \(\mathbb C^{2^N} \simeq \mathbb C^L \otimes C^l\) with \(L = 2^{N-n} \geq l = 2^n\). Such partitions can be given a nice geometrical interpretation in terms of Grassmannians \(Gr(L, l)\) of \(l\)-planes in \(\mathbb {C}^L\) that can be realized as the zero locus of quadratic polynomials in the complex projective space of suitable dimension via the Plücker embedding. The invariants are neatly expressed in terms of the Plücker coordinates of the Grassmannians. Levay, P, On the geometry of a class of \(N\)-qubit entanglement monotones, J. Phys. A Math. Gen., 38, 9075, (2005) Quantum computation, Grassmannians, Schubert varieties, flag manifolds On the geometry of a class of \(N\)-qubit entanglement monotones | 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 suggest a new combinatorial construction for the cohomology ring of the flag manifold. The degree 2 commutation relations satisfied by the divided difference operators corresponding to positive roots define a quadratic associative algebra. In this algebra, the formal analogues of Dunkl operators generate a commutative subring, which is shown to be canonically isomorphic to the cohomology of the flag manifold. This leads to yet another combinatorial version of the corresponding Schubert calculus. The paper contains numerous conjectures and open problems. We also discuss a generalization of the main construction to quantum cohomology. representation of the symmetric group; Pieri rule; Gromov-Witten invariants; Schubert polynomials; cohomology ring of the flag manifold; divided difference operators; quadratic associative algebra; Dunkl operators; Schubert calculus; quantum cohomology Fomin, Sergey; Kirillov, Anatol N., Quadratic algebras, Dunkl elements, and Schubert calculus. Advances in geometry, Progr. Math. 172, 147-182, (1999), Birkhäuser Boston, Boston, MA Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds Quadratic algebras, Dunkl elements, and Schubert calculus | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We study the conformal geometry of surfaces immersed in the four-dimensional conformal sphere \(Q_4\), viewed as a homogeneous space under the action of the Möbius group. We introduce the classes of isotropic surfaces and characterize them as those whose conformal Gauss map is antiholomorphic or holomorphic. We then relate these surfaces to Willmore surfaces and prove some interesting vanishing results and some bounds on the Euler characteristic of the surfaces. Finally, we characterize isotropic surfaces through an Enneper-Weierstrass-type parametrization. Differential geometry of submanifolds of Möbius space, Local submanifolds, Global submanifolds, Homogeneous spaces and generalizations, Grassmannians, Schubert varieties, flag manifolds, Holomorphic bundles and generalizations Remarks on the geometry of surfaces in the four-dimensional Möbius sphere | 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 Harish-Chandra's volume formula shows that the volume of a flag manifold \(G / T\), where the measure is induced by an invariant inner product on the Lie algebra of \(G\), is determined up to a scalar by the algebraic properties of \(G\). This article explains how to deduce Harish-Chandra's formula from Weyl's law by utilizing the Euler-Maclaurin formula. This approach suggests that there may be an elementary explanation available for the appearance of the power series \(x /(1 - e^{- x})\) in the Atiyah-Singer index theorem. compact Lie groups; Laplace-Beltrami operator; heat trace expansion; Weyl's law; Harish-Chandra's volume formula Heat and other parabolic equation methods for PDEs on manifolds, Compact groups, Grassmannians, Schubert varieties, flag manifolds, \(G\)-structures, Perturbations of PDEs on manifolds; asymptotics, Spectral problems; spectral geometry; scattering theory on manifolds, Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.) Harish-Chandra's volume formula via Weyl's law and Euler-Maclaurin formula | 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 simply connected semisimple Lie group, let \(B\) be a Borel subgroup and let \(X=G/B\) be the associated flag manifold. Inside \(X\) is a certain affine subvariety \(Q\), determined by the choice of a principal nilpotent element \(f\) of \(\text{Lie}(G)\), whose closure is denoted \(P_f\) and whose affine coordinate ring, denoted \(A(Q)\), is a polynomial ring. The intersection of \(P_f\) and the Schubert cell defined by \(B\) is denoted \(R\). Furthermore, there is an isomorphism of affine varieties which identifies \(R\) and a certain subset \(Y_0\) of \(\text{Lie}(G)\); under this isomorphism \(Q\cap R\) gets identified with the Toda leaf \(Y^*_0\). The author describes the affine algebras \(A(Q)\), \(A(Y_0)\), \(A(R)\), and \(A(Q\cap R)\) in terms of polynomial generators and relations.
As the author explains in the first sections of the paper, this work was inspired by some conjectures in the case \(G=SL_n(\mathbb{C})\) where the quantum cohomology algebra \(CH(X,\mathbb{C})\) was conjectured and then proven to have a certain description in terms of generators and relations. The author shows this algebra is isomorphic to \(A(Y_0)\) in the \(SL_n\) case.
There is much more in the paper than this brief review can summarize. Although it relies extensively on previous work of the author and others, the paper features examples and exposition making it largely self-contained. flag manifold; affine algebras Kostant, Bertram, Flag manifold quantum cohomology, the Toda lattice, and the representation with highest weight \textit{\({\rho}\)}, Selecta Math. (N.S.), 2, 1, 43-91, (1996) Grassmannians, Schubert varieties, flag manifolds, Applications of linear algebraic groups to the sciences Flag manifold quantum cohomology, the Toda lattice, and the representation with highest weight \(\rho\) | 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 semisimple algebraic group. In [Invent. Math. 166, No. 1, 185--228 (2006; Zbl 1106.14037)], \textit{P. Belkale} and \textit{S. Kumar} defined a new product \(\odot_0\) on the cohomology group \(\mathrm{H}^*(G/P,\mathbb{C})\) of any projective \(G\)-homogeneous space \(G/P\). Their definition uses the notion of Levi-movability for triples of Schubert varieties in \(G/P\).
In this article, we introduce a family of \(G\)-equivariant subbundles of the tangent bundle of \(G/P\) and the associated filtration of the de Rham complex of \(G/P\) viewed as a manifold. As a consequence one gets a filtration of the ring \(\mathrm{H}^*(G/P,\mathbb{C})\) and proves that \(\odot_0\) is the associated graded product. One of the aims of this more intrinsic construction of \(\odot_0\) is that there is a natural notion of a fundamental class \([Y]_{\odot_0}\in(\mathrm{H}^*(G/P,\mathbb{C}),\odot_0)\) for any irreducible subvariety \(Y\) of \(G/P\).
Given two Schubert classes \(\sigma_u\) and \(\sigma_v\) in \(\mathrm{H}^*(G/P,\mathbb{C})\), we define a subvariety \( \sum _u^v \) of \(G/P\). This variety should play the role of the Richardson variety; more precisely, we conjecture that \([\sum_u^v]_{\odot_0}=\sigma_u\odot_0\sigma_v\). We give some evidence for this conjecture, and prove special cases.
Finally, we use the subbundles of \(TG/P\) to give a geometric characterization of the \(G\)-homogeneous locus of any Schubert subvariety of \(G/P\). Belkale-Kumar Schubert calculus; Kostant's harmonic forms Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds Distributions on homogeneous spaces 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 A collection \(\mathcal{C}\) of \(k\)-element subsets of \(\lbrace 1,2,\ldots ,m\rbrace\) is weakly separated if for each \(I, J \in \mathcal{C} \), when the integers \(1,2,\ldots ,m\) are arranged around a circle, there is a chord separating \(I\backslash J\) from \(J \backslash I\). \textit{S. Oh} et al. [Proc. Lond. Math. Soc. (3) 110, No. 3, 721--754 (2015; Zbl 1309.05182)] constructed a correspondence between weakly separated collections which are maximal by inclusion and reduced plabic graphs, a class of networks defined by \textit{A. Postnikov} [``Total positivity, Grassmannians and networks'', Preprint, \url{arXiv:math/0609764}] which give coordinate charts on the Grassmannian of \(k\)-planes in \(m\)-space. As a corollary, they proved Scott's Purity Conjecture, which states that a weakly separated collection is maximal by inclusion if and only if it is maximal by size. In this note, we describe maximal weakly separated collections corresponding to symmetric plabic graphs, which give coordinate charts on the Lagrangian Grassmannian, and prove a symmetric version of the Purity Conjecture. plabic graphs; weakly separated collections; plabic tilings; symmetric plabic graphs; total positivity; Lagrangian Grassmannian Combinatorial aspects of algebraic geometry, Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.), Structural characterization of families of graphs, Grassmannians, Schubert varieties, flag manifolds The purity conjecture in type \(C\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials There are, in positive characteristic, non-reduced parabolic subgroup schemes of semisimple algebraic groups. They have been classified and described by \textit{C. Wenzel} [cf. e.g. Trans. Am. Math. Soc. 337, No. 1, 211-218 (1993; Zbl 0785.20024)]. The homogeneous spaces with non-reduced parabolic stabilizers are called \textit{unseparated} flag varieties. The most basic result about these spaces is their rationality. This has been established by Wenzel who has given a \(T\)-stable parametrization of a `big cell' in these varieties. (Here \(T\) is a maximal torus in the corresponding semisimple group.) After this, there is a host of questions about these spaces many of which remain unanswered. One of the authors of the paper under review has shown that the canonical line bundles on these spaces are in general neither ample nor negative ample and that they in general violate the Kodaira vanishing theorem [see \textit{N. Lauritzen}, C. R. Acad. Sci., Paris, Sér. I 315, 715-718 (1992; Zbl 0774.14014)]. All these facts are discussed in the paper under review.
Moreover the authors compute the characters of parabolic subgroup schemes. It is proven that a line bundle has global sections if and only if it is dominant. Then the authors describe the canonical line bundle and give a character formula for spaces of sections in homogeneous line bundles which coincides with the Weyl character formula for usual flag varieties. -- An unfortunate fact is that varieties of unseparated flags only are Frobenius split in the trivial cases, when they are isomorphic to ordinary flag varieties. The reason for this is that the canonical line bundle in general fails to be negative.
Then the authors give a systematic method for computing the weight defining the canonical line bundle and prove a vanishing theorem for sufficiently dominant line bundles. Finally they consider the simplest nontrivial case of non-reduced parabolic stabilizers and also give examples showing that the vanishing theorem for ample line bundles and Kodaira's vanishing theorem break down for small dominant weights for varieties of unseparated flags. unseparated flag varieties; characters of parabolic subgroup schemes; weight; vanishing theorem; dominant line bundles; parabolic stabilizers W. Haboush, N. Lauritzen, \textit{Varieties of unseparated flags}, in: Linear Algebraic Groups and Their Representations (Los Angeles, CA, 1992), Contemp. Math., Vol. 153, Amer. Math. Soc., Providence, RI, 1993, pp. 35-57. Grassmannians, Schubert varieties, flag manifolds, Representation theory for linear algebraic groups, Vanishing theorems in algebraic geometry, Infinitesimal methods in algebraic geometry Varieties of unseparated flags | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(C\) be a smooth curve of genus \(g\). Here we construct (under geometric restrictions, like \(C\) hyperelliptic or a complete intersection) spanned rank \(n\) vector bundles \(E\) on \(C\) with canonical determinant and with a \((2n+1)\)-dimensional linear subspace \(W\subseteq H^0(E)\) such that the natural wedge map \(\bigwedge^n(W)\to H^0(\det(E))\) is injective. The motivation came from a paper by Pirola and Rizzi, who used \((E,W)\) to get certain non-trivial higher cycle maps on the relative jacobian of an \(n\)-dimensional family of curves \({\mathcal C}\to S\) with \(C\) as a fiber. spanned vector bundle; canonical determinant; higher cycle map; Jacobian; Griffiths group Vector bundles on curves and their moduli, Jacobians, Prym varieties, Algebraic cycles, Grassmannians, Schubert varieties, flag manifolds Spanned vector bundles with canonical determinant on special 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 \(G\) be a connected semisimple Lie group and let \(X=G/B\), where \(B\) is a Borel subgroup of \(G\). Let \(H\) be a Cartan subgroup of \(B\) and let \(T\) be the maximal compact torus of \(H\). Then \(T\) acts on the Bott--Samelson variety \(\Gamma\) and the natural map \(g:\Gamma\rightarrow X\) is \(T\)-equivariant. The author calculates the restriction to fixed points of two bases of the equivariant cohomology of Bott towers and, by restriction, obtains similar results for \(\Gamma\). The morphism \(g^*\) and the multiplicative structure of \(H^*_T(\Gamma)\) are described: a method to calculate structure constants of \(H^*_T(\Gamma)\) is given. equivariant cohomology; flag manifolds Willems, Matthieu: Cohomologie équivariante des tours de Bott et calcul de Schubert équivariant. J. inst. Math. jussieu 5, No. 1, 125-159 (2006) Equivariant homology and cohomology in algebraic topology, Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Toric varieties, Newton polyhedra, Okounkov bodies, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras Equivariant cohomology of Bott towers and equivariant Schubert calculus. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The \(K\)-theory ring of a homogeneous space \(X\) has a natural basis parameterized by the Schubert varieties of \(X\). An important question in enumerative geometry is the determination of the structure constants of the ring with respect to this basis. The paper under review studies the special case when \(X\) is a cominuscule Grassmannian, and one of the multiplied classes corresponds to a special Schubert variety (Pieri rule). The authors describe the structure constants both as integers determined by positive recursive identities and as the number of certain combinatorial objects called tableaux. The result reproves a formula of Lenard in type \(A\), and is new for orthogonal and Lagrangian Grassmannians. The proof is based on calculating sheaf Euler characteristics of special triple intersections of Schubert varieties. Pieri rule; cominuscule Grassmannian; \(K\)-theory Buch, Ander Skovsted; Ravikumar, Vijay, Pieri rules for the \(K\)-theory of cominuscule Grassmannians, J. Reine Angew. Math., 668, 109-132, (2012) Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Applications of methods of algebraic \(K\)-theory in algebraic geometry Pieri rules for the \(K\)-theory of cominuscule 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 Based on Thomas and Yong's \(K\)-theoretic jeu de taquin algorithm, we prove a uniform Littlewood-Richardson rule for the \(K\)-theoretic Schubert structure constants of all minuscule homogeneous spaces. Our formula is new in all types. For the main examples of Grassmannians of type A and maximal orthogonal Grassmannians it has the advantage that the tableaux to be counted can be recognized without reference to the jeu de taquin algorithm. \(K\)-theory in geometry, Grassmannians, Schubert varieties, flag manifolds \(K\)-theory of minuscule 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 book under review is concerned with the study of equivariant embeddings of homogeneous spaces: if \(G/H\) is a homogeneous \(G\)-space, an equivariant embedding of \(G/H\) is a \(G\)-variety \(X\) containing \(G/H\) as a dense open orbit. Given a homogeneous space \(G/H\), one would (to the extent possible) like to classify all such embeddings, as well as study their geometric properties. Common examples of such equivariant embeddings are flag varieties, toric varieties, and spherical varieties. This monograph provides a compact survey of these issues, including many results of Brion, Knop, Luna, and Vust, among others, as well as the author's own significant contributions to the field.
At the heart of the text is the Luna-Vust theory, which classifies equivariant embeddings of \(G/H\) using `combinatorial' language involving \(B\)-invariant divisors of \(G/H\) (here \(B\) denotes a Borel subgroup of \(G\)) and \(G\)-invariant valuations of the function field of \(G/H\). The author actually presents his own generalization of this theory in which he replaces \(G/H\) by any \(G\)-variety. In general, the difficulty of describing the \(B\)-invariant divisors and \(G\)-invariant valuations makes such classification impossible in practice, but for special cases, this does become tractable. Indeed, if the complexity of the \(G\)-action (defined to be the codimension of a general \(B\)-orbit) is 0 or 1, the above classification becomes genuinely combinatorial. The author devotes considerable time to these cases, in which one may also describe many geometric aspects of a given equivariant embedding.
The author wisely assumes that the reader has a background in algebraic geometry and representation theory; this keeps the length of the monograph manageable. The book begins with a basic discussion of algebraic homogeneous spaces in Chapter 1. Chapter 2 deals with two invariants of a \(G\)-variety \(X\): its complexity (defined above) and its rank (the rank of the lattice of weights of \(B\)-eigenfunctions). The main results are general formulae for computing these invariants.
Chapter 3 contains the author's presentation of the Luna-Vust theory mentioned above. Particular attention is paid to the complexity 0 and complexity 1 cases. A general description of \(B\)-stable divisors is given, which is again refined for the complexity 0 and 1 cases. In these cases, the author also presents a number of results on intersection theory.
Chapter 4 contains a discussion of \(G\)-invariant valuations, important due to their role in the Luna-Vust theory. Finally, Chapter 5 returns to the study of complexity 0 \(G\)-varieties, also known as spherical \(G\)-varieties. A number of characterizations of such varieties are presented, as well as descriptions of some special subclasses, and results on Frobenius splittings. The monograph also includes several appendices which cover a number of topics in algebraic geometry and representation theory which are used elsewhere in the text.
The author admirably fills a gap in the current literature on algebraic groups with this book. It should make a welcome addition to any mathematician's library. Homogeneous spaces; spherical varieties; Luna-Vust theory D. Timashev: \textit{Homogeneous Spaces and Equivariant Embeddings}, Enc. of Math. Sc. 138, Springer, Heidelberg et al. (2011). Group actions on varieties or schemes (quotients), Research exposition (monographs, survey articles) pertaining to algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Homogeneous spaces and generalizations Homogeneous spaces and equivariant embeddings | 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 Many results involving Schur functions have analogues involving \(k\)-Schur functions. Standard strong marked tableaux play a role for \(k\)-Schur functions similar to the role standard Young tableaux play for Schur functions. We discuss results and conjectures toward an analogue of the hook-length formula.
New tools, residue and quotient tables, are presented, which allow for efficient computation of strong covers and have the potential to describe many other phenomena in \(k\)-function theory. \(k\)-Schur functions; strong marked tableaux; enumeration Fishel, S.; Konvalinka, M.: Results and conjectures on the number of standard strong marked tableaux. J. combin. Theory ser. A 131, 153-186 (2015) Combinatorial aspects of representation theory, Symmetric functions and generalizations, Combinatorial aspects of simplicial complexes, , Grassmannians, Schubert varieties, flag manifolds Results and conjectures on the number of standard strong marked tableaux | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In the paper an explicit formula is proven for the multiplication of an arbitrary Schubert cycle by a special Schubert cycle in the Chow (or cohomology) ring of the homogeneous spaces \(Sp (2m)/P\) and \(SO (2m + 1)/P\), where \(P\) is a maximal parabolic subgroup. These homogeneous spaces are interpreted as Grassmannians of isotropic subspaces of a fixed dimension in \(2m\)-dimensional (resp. \((2m + 1)\)-dimensional) vector space endowed with a non-degenerate symplectic (resp. orthogonal) form. The method follows an earlier paper by the author [Manuscr. Math. 79, No. 2, 127-151 (1993; Zbl 0789.14041)] and uses the divided difference description of Borel's characteristic map in the basis of Schubert cycles given by \textit{I. N. Bernstein}, \textit{I. M. Gel'fand} and \textit{S. I. Gel'fand} [Russ. Math. Surv. 28, No. 3, 1-26 (1973); translation from Usp. Mat. Nauk 28, No. 3(171), 3-26 (1973; Zbl 0286.57025)] and \textit{M. Demazure} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 7, 53-88 (1974; Zbl 0312.14009)]. This allows one to reformulate the original intersection theory problem into some questions of purely algebro-combinatorial nature. As a by-product one obtains some Giambelli-type formulas for these isotropic Grassmannians. Schubert cycle; isotropic Grassmannians Pragacz, P., Ratajski, J.: A Pieri-type theorem for Lagrangian and odd orthogonal Grassmannians. J. Reine Angew. Math. 476, 143--189 (1996) Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Homogeneous spaces and generalizations, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Enumerative problems (combinatorial problems) in algebraic geometry A Pieri-type theorem for Lagrangian and odd orthogonal 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 Starting with Grassmann's work, a short review is given of the development of geometric algebra, and the reasons why it is a useful system for describing much of physics. Applications are then discussed in cosmology, including a novel boundary condition for the universe, and efficient ways to encode Bianchi cosmology. Predictions for the cosmic microwave background in such models, and in another area owing much to Grassmann (string theory), are also discussed. Cosmology; Grassmann; geometric algebra; cosmic microwave background; Bianchi models; closed universes; string cosmology History of relativity and gravitational theory, Relativistic cosmology, Grassmannians, Schubert varieties, flag manifolds, History of mathematics in the 20th century, Biographies, obituaries, personalia, bibliographies Grassmann, geometric algebra and cosmology | 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 Tensors, tensor spaces, tensor algebras, and tensor calculus have become a ubiquitous, powerful and universal toolkit in mathematics, statistics, physics, communication theory, and other natural sciences. Many topics involving the study of tensors are of classical nature, but just as many of them are currently very active areas of research both in mathematics and in its various applications.
As the author of the book under review points out, this text has three intended uses: as a classroom textbook, a reference work for researchers in the sciences, and a versatile presentation of classical and modern results in geometry based on methods of tensor algebra. Also, as there is barely a reasonable reference for all the remarkable developments of modern tensor calculus and its important applications, the author's goal has been to fill this gap in the literature.
As for the contents, the book is divided into four parts, each of which contains several chapters and their respective sections.
The first part is titled ``Motivation from applications, multilinear algebra, and elementary results'' and comprises the first three chapters. The introductory Chapter 1 starts with motivating problems, including some basic definitions from multilinear algebra, the complexity of matrix multiplication, the significance of tensor decomposition in physics and algebraic statistics, a brief discussion of the complexity problem ``P vs. NP'', and a first glimpse of tensor network states in quantum mechanics. Chapter 2 introduces the language of tensors in the framework of multilinear algebra. In this context, the fundamental notions of rank and border rank of tensors are discussed, first steps towards the decomposition of tensor spaces are provided, and three appendices are added. Apart from some basic facts from linear algebra, wiring diagrams are explained in one of these appendices. Chapter 3 presents various results on rank and border rank of tensors that can be stated without the use of algebraic geometry and representation theory. In general, this chapter gives a comprehensive report on the present state of art, and the main purpose of this chapter is to provide, in elementary terms, a reference for researchers in the sciences.
Part 2 contains the following seven chapters and comes with the heading ``Geometry and representation theory''. Chapter 4 is devoted to the necessary algebraic geometry in higher tensor calculus, thereby introducing the essentials of projective algebraic geometry as far as needed. Algebraic varieties, Veronese embeddings, Grassmannians, tangent and cotangent spaces, group actions, and homogeneous varieties are among the basic concepts touched upon here. Chapter 5 deals with secant varieties of projective varieties, as many results on border rank of tensors are more easily proved in this geometric context. In this chapter, the reader also meets Terracini's Lemma, the polynomial Waring problem, secant varieties of Segre varieties, and a geometric formulation of some conjectures from complexity theory and signal processing. Chapter 6 turns to the representation theory for tensor spaces. Apart from the basic material on representations of finite groups, the main topics of this chapter are: the Littlewood-Richardson rule, Pieri's formula, weights and weight spaces, homogeneous varieties, and symmetric functions.
Chapter 7 discusses tests for the border rank of tensors by using equations for secant varieties in general. Strassen's equations, Young flattenings, and Friedland's equations complete the topical material of this chapter. Chapter 8 investigates further classes of algebraic varieties related to tensor spaces. This includes tangential varieties, dual varieties, Fano varieties of lines, Chow varieties of zero cycles, Brill's equations, and other topics. Chapter 9 describes the rank of tensors in a more general geometric context, where the main discussion regards the ranks of symmetric tensors and their bounds.
Chapter 10 focuses on spaces of tensors admitting suitable normal forms. Normal forms for points in small secant varieties as well as normal forms, ranks, and border ranks of special (``small'') tensors play a distinguished role in this last chapter of Part 2.
Concrete applications of tensor algebra and tensor geometry are the main theme of Part 3, which contains the subsequent four chapters. Chapter 11 returns to the complexity of matrix multiplication. Its goal is to bring the reader up to date on what is known regarding this current topic of research, including new proofs of many standard results, partly in the algebro-geometric context of secant varieties. Chapter 12 addresses the problem of tensor decomposition in concrete applications. The author discusses here two instructive examples from signal processing, namely blind source separation and deconvolution of DS-CMDA signals, explains the study of cumulants along the way, and turns then to exact tensor decomposition algorithms, including the celebrated uniqueness theorem by \textit{J. B. Kruskal} (1977). Chapter 13 gives an introduction to several algebraic versions of the famous complexity problem ``P vs. NP'', including holographic algorithms and geometric complexity theory, whereas Chapter 14 describes applications of varieties of tensors in phylogenetics, algebraic statistics, and quantum mechanics.
Part 4 is devoted to advanced topics, which are treated in the remaining three chapters. Chapter 15 gives a brief outline of a theorem by \textit{J. Alexander} and \textit{A. Hirschowitz} [``Polynomial interpolation in several variables'', J. Algebr. Geom. 4, No. 2, 201--222 (1995; Zbl 0829.14002)], which determines the dimensions of the varieties of symmetric tensors of bounded border rank. Chapter 16 includes a brief description of the rudiments of the representation theory of complex simple Lie groups and algebras. This is used to present a proof of Kostant's theorem on the ideals of homogeneous varieties, the statement of the Bott-Borel-Weil theorem on the cohomology of homogeneous vector bundles, and a discussion of the inheritance principle for \(G\)-modules from a general point of view. Finally, Chapter 17 explains the basics of a set of techniques of studying algebraic \(G\)-varieties via desingularizations by means of vector bundles over a homogeneous variety. These ideas were initiated by G. Kempf a few decades ago, but were recently further expanded, developed and utilized by \textit{J. Weyman} [Cohomology of vector bundles and syzygies. Cambridge Tracts in Mathematics 149. Cambridge: Cambridge University Press (2003; Zbl 1075.13007)]. This chapter provides a brief introduction to the Kempf-Weyman method for determining ideals and singularities of \(G\)-varieties, intended primarily to serve as a guide to Weyman's book (cited above). Fundamental techniques such as Koszul sequences, syzygies, and minimal free resolutions for varieties are explained in the course of this chapter, and subspace varieties are discussed, in this context, at the end of this concluding chapter.
Numerous examples and exercises enhance the ample material covered in this modern text on tensors and their applications. Hints and answers to selected exercises are provided in an extra section, and the book comes with a very rich and up-to-date bibliography (of about 340 references), followed by a just as carefully arranged index.
All in all, this is an interesting, useful and versatile text on tensor calculus. It illuminates its modern setting in (algebraic) geometry in very instructive a manner, demonstrates its growing use in various branches of sciences in a highly enlightening way and provides a wealth of information on tensors in table format for working researchers. The style of writing stands out by its lucidity, distinctness, relatively elementary language, and particular user-friendliness, thereby leading the reader to the forefront of current research in some of the topics touched upon in this book. multilinear algebra; tensor products; algebraic varieties; secant varieties; representation theory; complexity theory; matrices; monograph; textbook; matrix multiplication; tensor decomposition; tensor network; border rank; tensor calculus; projective algebraic geometry; Terracini's Lemma; polynomial Waring problem; Segre varieties; signal processing; Littlewood-Richardson rule; Pieri's formula; Strassen's equation; Young flattening; Friedland's equation; Fano varieties of line; Chow varieties of zero cycle; Brill's equation; normal form; Konstant's theorem; Bott-Borel-Weil theorem; Koszul sequences; syzygies J. M. Landsberg, \textit{Tensors: Geometry and Applications}, American Mathematical Society, Providence, RI, 2012. Vector and tensor algebra, theory of invariants, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to linear algebra, Multilinear algebra, tensor calculus, Determinantal varieties, Grassmannians, Schubert varieties, flag manifolds, Exterior algebra, Grassmann algebras, Homogeneous spaces and generalizations, Group actions on varieties or schemes (quotients), Representation theory for linear algebraic groups, Characterization and structure theory of statistical distributions, Signal theory (characterization, reconstruction, filtering, etc.), Differential geometric aspects in vector and tensor analysis Tensors: geometry 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 Let \(w_0\) be the element of maximal length in the symmetric group \(S_n\), and let \(\text{Red}(w_0)\) be the set of all reduced words for \(w_0\). We prove the identity
\[
\sum_{(a_1,a_2,\dots)\in\text{Red}(w_0)}(x+ a_1)(x+ a_2)\cdots= {n\choose 2}! \prod_{1\leq i<j\leq n} {2x+ i+ j-1\over i+j-1},\tag{\(*\)}
\]
which generalizes Stanley's formula for the cardinality of \(\text{Red}(w_0)\), and Macdonald's formula \(\sum a_1a_2\cdots= \left(\begin{smallmatrix} n\\ 2\end{smallmatrix}\right)!\). Our approach uses an observation, based on a result by \textit{M. L. Wachs} [J. Comb. Theory, Ser. A 40, 276-289 (1985; Zbl 0579.05001)], that evaluation of certain specializations of Schubert polynomials is essentially equivalent to enumeration of plane partitions whose parts are bounded from above. Thus, enumerative results for reduced words can be obtained from the corresponding statements about plane partitions, and vice versa. In particular, identity \((*)\) follows from Proctor's formula for the number of plane partitions of a staircase shape, with bounded largest part. Similar results are obtained for other permutations and shapes; \(q\)-analogues are also given. Young tableaux; Ferrers shape; reduced words; identity; Stanley's formula; Macdonald's formula; Schubert polynomials; enumeration of plane partitions; permutations; shapes; \(q\)-analogues S. Fomin and A. N. Kirillov, \textit{Reduced words and plane partitions}, J. Algebraic Combin., 6 (1997), pp. 311--319. Combinatorial aspects of partitions of integers, Exact enumeration problems, generating functions, Symmetric functions and generalizations, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds, Representations of finite symmetric groups, Combinatorial aspects of representation theory Reduced words and plane partitions | 0 |
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