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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G_{k+1,n-k}\) denote the Grassmann manifold of complex \((k+1)\)-planes in \(\mathbb C^{n+1}\) and \(H_{l+1}\) the standard non-degenerate Hermitian form on \(\mathbb C^{n+1}\) with \(l+1\) eigenvalues equal to \(1\) and \(n-l\) eigenvalues equal to \(-1\). The domain \(\mathbb D^l_{k,n}:=\{Z\in G_{k+1,n-k}\,|\,\,H_{l+1}|Z>0\}\) is an open orbit of the natural \(SU(l+1,n-l)\)-action on \(G_{k+1,n-k}\) as a group of biholomorphic transformations. Assume that \(k+1\leq l\leq n-k-2\). Then the closure \(\{Z\in G_{k+1,n-k}\,|\,H_{l+1}|\,\,Z \geq 0\}\) of \(\mathbb D^l_{k,n}\) admits a stratification into \(k+2\) orbits
\[
\partial_r\mathbb D^l_{k,n}:=\Big\{Z\in\overline{\mathbb D^l_{k,n}}\;\Big|\; \text{ rang }H_{l+1}|Z=k+1-r\Big\},\quad 0\leq r\leq k+1.
\]
The paper shows the strong relationship between the structure of \(\overline{\mathbb D^l_{k,n}}\) and the rigidity of holomorphic embeddings \(G_{k+1,n-k}\rightarrow G_{k+1,m-k}\): Let \(U\subset G_{k+1,n-k}\) be a connected open set, \(U\cap\partial_{k+1}\mathbb D^l_{k,n}\not=\emptyset\), and \(f:U\rightarrow G_{k+1,m-k}\) a holomorphic embedding with \(f(U\cap\partial_{k+1}\mathbb D^l_{k,n})\subset\partial_{k+1}\mathbb D^l_{k,m}\) if \(l<\frac{n}{2}\) and \(f(U\cap\partial_{k+1}\mathbb D^l_{k,n})\subset\partial_{k+1}\mathbb D^{l+m-n}_{k,m}\) if \(\frac{n}{2}\leq l\). Then \(f\) is the restriction of a holomorphic embedding \(F: G_{k+1,n-k}\rightarrow G_{k+1,m-k}\) induced by a linear injection \(\mathbb C^n\rightarrow\mathbb C^m\). If, in addition, \(f(U\cap\mathbb D^l_{k,n})\subset \mathbb D^l_{k,m}\) and \(l<\frac{n}{2}\) or \(f(U\cap\mathbb D^l_{k,n})\subset\mathbb D^{l+m-n}_{k,m}\) and \(\frac{n}{2}\leq l\), then \(F|\mathbb D^l_{k,n}\) is a proper holomorphic embedding into \(\mathbb D^l_{k,m}\) resp. \(\mathbb D^{l+m-n}_{k,m}\). For \(k=0\) (\(G_{1,n}=\mathbb P^n\)) similar results were obtained by \textit{M. S. Baouendi} and \textit{X. Huang} [J. Differ. Geom. 69, No. 2, 379--398 (2005; Zbl 1088.32003)] about the rigidity of Cauchy-Riemann mappings between real hyperquadrics with positive signature in \(\mathbb P^n\). The proof of the above result in the case \(k=0\) is based on the determination of the cycle space of the maximal linear subvarieties of \(\mathbb D^l_{0,n}\) and the fact that these subvarieties are mapped under \(f\) into maximal linear subvarieties of the target domain. For \(k\geq 1\) the author considers the structures of certain geodesic sub-Grassmannians in \(\mathbb D^l_{k,n}\). He shows that these structures are preserved by the map \(f\) and finally uses a local characterization of standard embeddings of Grassmann manifolds obtained by \textit{N. Mok} [Sci. China, Ser. A 51, No. 4, 660--684 (2008; Zbl 1165.32008)]. extension of local holomorphic embeddings of Grassmann manifolds; maximal linear subvarieties of flag manifolds Ng, S-C, Cycle spaces of flag domains on Grassmannians and rigidity of holomorphic mappings, Math. Res. Lett., 19, 1219-1236, (2012) Homogeneous complex manifolds, Grassmannians, Schubert varieties, flag manifolds Cycle spaces of flag domains on Grassmannians and rigidity of holomorphic mappings | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Grassmann codes were introduced in [\textit{R. Pellikaan} (ed.) et al., Arithmetic, geometry, and coding theory. Proceedings of the international conference held at CIRM, Luminy, France, June 28-July 2, 1993. Berlin: Walter de Gruyter (1996; Zbl 0859.00020)], [\textit{C. Ryan}, Congr. Numerantium 57, 257--271 (1987; Zbl 0638.94021); Congr. Numerantium 57, 273--279 (1987; Zbl 0608.00016)] using the Grassmannian variety. A variation was introduced in [\textit{P. Beelen} et al., IEEE Trans. Inf. Theory 56, No. 7, 3166--3176 (2010; Zbl 1366.94576)], the affine Grassmann code.
This article counts and classifies the minimum weigth codewords of the dual of Grassmann codes and the dual of affine Grasmann codes, by considering lines contained in a Grassmannian variety. Moreover, it proves that the increase of value of successive generalized Hamming weights is at most 2 and that the dual of a Grassmann code is generated by its minimum weight codewords. This has applications for the decoding of Grassmann codes. dual Grassmann code; Hamming weights; Tanner code Beelen, P.; Piñero, F., The structure of dual Grassmann codes, Des. Codes Cryptogr., 79, 451-470, (2016) Geometric methods (including applications of algebraic geometry) applied to coding theory, Applications to coding theory and cryptography of arithmetic geometry, Grassmannians, Schubert varieties, flag manifolds The structure of dual Grassmann codes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author of the paper under review extends to the positive (actually, arbitrary) characteristic case a number of results, previously known only in characteristic 0 (or under some additional assumptions), such as the linearity of general fibers of separable Gauss maps (proven by \textit{P. Griffiths} and \textit{J. Harris} [Ann. Sci. Éc. Norm. Supér. (4) 12, 355--452 (1979; Zbl 0426.14019)]) in characteristic 0 and by \textit{S. Kleiman} and \textit{R. Piene} [Contemp. Math. 123, 107--129 (1991; Zbl 0758.14032)] under the reflexivity assumption) or the characterization of the images of separable Gauss maps due, in characteristic 0, to J.M. Landsberg and, independently, to J. Piontkowski (cf. \textit{T. A. Ivey} and \textit{J. M. Landsberg} [Cartan for beginners: Differential geometry via moving frames and exterior differential systems. Providence, RI: American Mathematical Society (AMS) (2003; Zbl 1105.53001)] and [\textit{G. Fischer} and \textit{J. Piontkowski}, Ruled varieties. An introduction to algebraic differential geometry. Braunschweig: Vieweg (2001; Zbl 0976.14025)]).
Before stating the results, we need to recall the definition of expanding maps (due to the author of the paper under review) and of shrinking maps (studied by Landsberg and Piontkowski) and the related formalism. Let \({\mathbb P}^N\) be the projective \(N\)-space over an algebraically closed field \(K\) of arbitrary characteristic, let \(\mathcal X\) be a subvariety of the Grassmannian \({\mathbb G}(m,\, {\mathbb P}^N)\) of \(m\)-dimensional linear subvarieties of \({\mathbb P}^N\), and let:
\[
\begin{tikzcd} 0 \rar & S \rar["u"] & \mathcal O_{\mathcal X}\otimes_K W \rar["p"] & Q \rar & 0 \end{tikzcd}
\]
be the restriction to \(\mathcal X\) of the tautological exact sequence on \({\mathbb G}(m,\, {\mathbb P}^N)\), where \(W = \text{H}^0({\mathbb P}^N, {\mathcal O}_{{\mathbb P}^N}(1))\) and \(\text{rk}\, Q = m + 1\). The composite map:
\[
\begin{tikzcd} S \rar["u"] & \mathcal O_{\mathcal X}\otimes_K W \rar["d_{\mathcal X}\otimes \, \mathrm{id}"] & \Omega_{\mathcal X}\otimes_K W \simeq \Omega_{\mathcal X}\otimes_{{\mathcal O}_{\mathcal X}} ({\mathcal O}_{\mathcal X}\otimes_K W) \rar["\mathrm{id} \otimes p"] & \Omega_{\mathcal X}\otimes_{{\mathcal O}_{\mathcal X}} Q \end{tikzcd}
\]
turns out to be a morphism of \({\mathcal O}_{\mathcal X}\)-modules \(\varphi : S \rightarrow \Omega_{\mathcal X}\otimes Q\). Let \({\mathcal X}^+\) be the open subset of the nonsingular locus \({\mathcal X}^{\text{sm}}\) of \(\mathcal X\) over which \(\varphi\) has maximum rank, let \(S^+ := \text{Ker}\, \varphi \, |_{{\mathcal X}^+}\), \(u^+ : S^+ \rightarrow {\mathcal O}_{{\mathcal X}^+}\otimes_K W\) the restriction of \(u\) and \(Q^+ := \text{Coker}\, u^+\). If \(\text{rk}\, Q^+ = m^+ + 1\) then the exact sequence:
\[
\begin{tikzcd} 0 \rar & S^+ \rar["u^+"] &\mathcal O_{\mathcal X^+}\otimes_K W \rar & Q^+\rar & 0 \end{tikzcd}
\]
defines a morphism \(\gamma : {\mathcal X}^+ \rightarrow {\mathbb G}(m^+,\, {\mathbb P}^N)\) called the \textit{expanding map} of \(\mathcal X\).
Applying the same kind of construction to the dual exact sequence:
\[
\begin{tikzcd} 0 \rar & Q^\vee \rar & \mathcal O_{\mathcal X}\otimes_K W^\vee \rar & S^\vee \rar & 0 \end{tikzcd}
\]
one gets a morphism of \({\mathcal O}_{\mathcal X}\)-modules \(\psi : Q^\vee \rightarrow \Omega_{\mathcal X}\otimes S^\vee\). Let \({\mathcal X}^-\) be the open subset of \({\mathcal X}^{\text{sm}}\) over which \(\psi\) has maximum rank, \(Q^- := \text{Coker}\, \psi^\vee \, |_{{\mathcal X}^-}\), \(p^-\) the composite morphism \({\mathcal O}_{{\mathcal X}^-}\otimes_K W \overset{p}\longrightarrow Q \rightarrow Q^-\), and \(S^- := \text{Ker}\, p^-\). If \(\text{rk}\, Q^- = m^- + 1\) then the exact sequence:
\[
\begin{tikzcd} 0 \rar & S^- \rar & \mathcal O_{{\mathcal X}^-}\otimes_K W \rar["p^-"] & Q^- \rar & 0 \end{tikzcd}
\]
defines a morphism \(\sigma : {\mathcal X}^- \rightarrow {\mathbb G}(m^-,\, {\mathbb P}^N)\) called the \textit{shrinking map} of \(\mathcal X\).
These constructions are compatible with dominant, \textit{separable} (that is, generically smooth) morphisms \(f : {\mathcal X} \rightarrow {\mathcal Y}\) because for such morphisms the map \(f^\ast \Omega_{\mathcal Y} \rightarrow \Omega_{\mathcal X}\) is, generically, a monomorphism. They can be studied locally using the following observation: if one has a \(K\)-vector space decomposition \(W = W^\prime \oplus W^{\prime \prime}\) then the morphism \(\varphi : {\mathcal O}_{\mathcal X}\otimes_K W^\prime \rightarrow \Omega_{\mathcal X}\otimes_K W^{\prime \prime}\) associated to a short exact sequence of the form:
\[
\begin{tikzcd} 0 \rar & \mathcal O_{\mathcal X}\otimes_K W^\prime \rar["\binom{-\text{id}}{\alpha}"] & \mathcal O_{\mathcal X}\otimes_K W \rar["{(\alpha, \mathrm{id})}"] & \mathcal O_{\mathcal X}\otimes_K W^{\prime \prime} \rar & 0 \end{tikzcd}
\]
is defined by the matrix obtained by applying \(d_{\mathcal X}\) to the entries of the matrix defining \(\alpha : {\mathcal O}_{\mathcal X}\otimes_K W^\prime \rightarrow {\mathcal O}_{\mathcal X}\otimes_K W^{\prime \prime}\).
One deduces, in the global case, that if \(\varphi^\prime\) is the composite map:
\[
\begin{tikzcd} T_{\mathcal X} \otimes S \rar["\text{id} \otimes \varphi"] & T_{\mathcal X} \otimes \Omega_{\mathcal X} \otimes Q \rar & Q \end{tikzcd}
\]
(recall that \(T_{\mathcal X} = {\mathcal H}om_{{\mathcal O}_{\mathcal X}}(\Omega_{\mathcal X}, {\mathcal O}_{\mathcal X})\)) then \(\psi^\vee = - \varphi^\prime\).
One also deduces that if \(m = 0\), i.e., if \({\mathcal X}\subset {\mathbb P}^N\), then the expanding map \(\gamma\) coincides with the \textit{Gauss map} \({\mathcal X}^{\text{sm}} \rightarrow {\mathbb G}(\dim {\mathcal X},\, {\mathbb P}^N)\) associating to \(x \in {\mathcal X}^{\text{sm}}\) the embedded tangent space \({\mathbb T}_x{\mathcal X} \subset {\mathbb P}^N\).
Now, the main results of the paper under review are the following ones:
\begin{itemize}
\item[(1)] Let \(\gamma : X^{\text{sm}} \rightarrow {\mathbb G}(\dim X,\, {\mathbb P}^N)\) be the Gauss map of a projective variety \(X \subset {\mathbb P}^N\) and \({\mathcal Y} \subset {\mathbb G}(\dim X,\, {\mathbb P}^N)\) the closure of the image of \(\gamma\). If \(\gamma : X^{\text{sm}} \rightarrow {\mathcal Y}\) is separable then the closure in \({\mathcal Y} \times {\mathbb P}^N\) of the graph of \(\gamma\) coincides with the closure of \({\mathbb P}((Q_{\mathcal Y}^-)^\vee) \subset {\mathcal Y}^- \times {\mathbb P}^N\). (The author's convention for projective bundles is \({\mathbb P}(E) := \text{Proj}(\mathrm{Sym} E^\vee)\) so that \({\mathbb P}^N = {\mathbb P}(W^\vee)\) and \({\mathbb P}^{N\vee} = {\mathbb P}(W)\)).
\item[(2)] Assume that \({\mathcal X} \subset {\mathbb G}(m,\, {\mathbb P}^N)\) and let \(Y \subset {\mathbb P}(W) = {\mathbb P}^{N\vee}\) be the closure of the image of the projection \({\mathbb P}(S^+) \rightarrow {\mathbb P}(W)\). Since \({\mathbb P}^{N\vee} = {\mathbb G}(N-1,\, {\mathbb P}^N)\) the shrinking map \(\sigma_Y\) associates to \(y \in Y^{\text{sm}}\) the point \(\sigma_Y(y)\) of \({\mathbb G}(N - 1 - \dim Y, \, {\mathbb P}^N)\) corresponding to the intersection of the hyperplanes in \({\mathbb P}^N\) corresponding to the points of the embedded tangent space \({\mathbb T}_yY \subset {\mathbb P}^{N\vee}\). If \(Y\) has dimension \(N - m - 1\) and if \({\mathbb P}(S^+) \rightarrow Y\) is separable then \(\mathcal X\) is the closure of \(\sigma_Y(Y^{\text{sm}})\) and \(\sigma_Y : Y^{\text{sm}} \rightarrow {\mathcal X}\) is separable.
\textit{Dually}, assume that \({\mathcal Y} \subset {\mathbb G}(m^+,\, {\mathbb P}^N)\) and let \(X \subset {\mathbb P}(W^\vee) = {\mathbb P}^N\) be the closure of the image of the projection \({\mathbb P}((Q_{\mathcal Y}^-)^\vee) \rightarrow {\mathbb P}(W^\vee)\). If \(\dim X = m^+\) and if \({\mathbb P}((Q_{\mathcal Y}^-)^\vee) \rightarrow X\) is separable then \(\mathcal Y\) is the closure of the image of the Gauss map \(\gamma : X^{\text{sm}} \rightarrow {\mathbb G}(\dim X,\, {\mathbb P}^N)\) and \(\gamma : X^{\text{sm}} \rightarrow {\mathcal Y}\) is separable.
\item[(3)] In the final section of the paper under review, the author uses the above results to establish a duality on 1-dimensional developable paramater spaces via expanding and shrinking maps. More precisely, consider \({\mathcal X} \subset {\mathbb G}(m,\, {\mathbb P}^N)\), let \(\pi : {\mathbb P}(Q^\vee) \rightarrow {\mathbb P}^N\) be the projection and put \(X = \pi({\mathbb P}(Q^\vee))\). \(\mathcal X\) is \textit{developable} if, for any general point \(x\) of \(\mathcal X\), the Gauss map \(\gamma_X\) is constant on \({\mathbb P}(Q^\vee(x)) \cap X^{\text{sm}}\). \(\gamma^i_{\mathcal X}\) is defined inductively by \(\gamma^1_{\mathcal X} = \gamma_{\mathcal X} =\) the expanding map of \(\mathcal X\), \(\gamma^i_{\mathcal X} := \gamma_{\gamma^{i-1}{\mathcal X}} \circ \gamma^{i-1}_{\mathcal X}\), and \(\sigma^i_{\mathcal X}\) is defined similarly. Moreover, if \(TX\) denotes the closure in \({\mathbb P}^N\) of \(\bigcup_{x\in X^{\text{sm}}}{\mathbb T}_xX\) then \(T^iX\) is defined inductively by \(T^1X = TX\), \(T^iX = T(T^{i-1}X)\). (If \(C \subset {\mathbb P}^N\) is a curve then, in characteristic 0, \(T^iC\) coincides with the \textit{oscullating scroll} of order \(i\) of \(C\)).
\end{itemize}
The main result of the final section of the paper under review asserts that if one considers 1-dimensional subvarieties \({\mathcal X} \subset {\mathbb G}(m,\, {\mathbb P}^N)\), \({\mathcal X}^\prime \subset {\mathbb G}(m^\prime ,\, {\mathbb P}^N)\) and a nonnegative integer \(\varepsilon\) then the following are equivalent:
\begin{itemize}
\item[(a)] \({\mathcal X}^\prime\) is developable, \(X^\prime\) is nondegenerate and is not a cone, \(\gamma^\varepsilon_{{\mathcal X}^\prime}\) is separable and \(\gamma^\varepsilon_{{\mathcal X}^\prime}{\mathcal X}^\prime = {\mathcal X}\);
\item[(b)] \({\mathcal X}\) is developable, \(X\) is nondegenerate and is not a cone, \(\sigma^\varepsilon_{\mathcal X}\) is separable and \(\sigma^\varepsilon_{\mathcal X}{\mathcal X} = {\mathcal X}^\prime\).
\end{itemize}
Moreover, in this case, \(m = m^\prime + \varepsilon\) and \(X = T^\varepsilon X^\prime\). Grassmann variety; Gauss map; separable map; developable variety Furukawa, K., Duality with expanding maps and shrinking maps, and its applications to Gauss maps, \textit{Math. Ann.}, 358, 403-432, (2014) Projective techniques in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Positive characteristic ground fields in algebraic geometry Duality with expanding maps and shrinking maps, and its applications to Gauss 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 The authors follow the spirit of the main researchers in cluster tilting theory, relating a geometric object and its intrinsic combinatorics with a family of algebraic/categorical objetcs. The author study labels of the cyclic polytope with the corresponding notion of mutation, and define a usefull tool to track compatibility: non-\(l\)-intertwining sets of labels.
The authors study \(d\)-precluster tilting subcategories for higher preprojective algebras of tensor algebras, establishing results that apply to higher preprojective algebras of tensor products of algebras of type \(A\).
As the main result, the authors show that mutation on the cyclic polytope is related to higher APR-tilting mutation, as defined by \textit{O. Iyama} and \textit{S. Oppermann} [Trans. Am. Math. Soc. 363, No. 12, 6575--6614 (2011; Zbl 1264.16015)], on the setting of higher preprojective algebras of tensor products of algebras of type \(A\). cyclic polytops; tensor algebras; pre-cluster tilting Cluster algebras, Grassmannians, Schubert varieties, flag manifolds, Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers, Projectives and injectives (category-theoretic aspects), Combinatorial aspects of representation theory The combinatorics of tensor products of higher Auslander algebras of type \(A\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper we establish effective lower bounds on the degrees of the Debarre and Kobayashi conjectures. Then we study a more general conjecture proposed by Diverio-Trapani on the ampleness of jet bundles of general complete intersections in complex projective spaces. Kobayashi hyperbolicity; ample cotangent bundle; Debarre conjecture; Kobayashi conjecture; Diverio-Trapani conjecture Hyperbolic and Kobayashi hyperbolic manifolds, Complete intersections, Grassmannians, Schubert varieties, flag manifolds, Divisors, linear systems, invertible sheaves On the Diverio-Trapani conjecture | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper the tangent bundle of a flag variety F is given as a quotient of a certain universal bundle on F together with a description of the kernel of this quotient involving exact sequences of all the universal bundles on F. The proof uses a reduction to the case of complete flags and an induction on the dimension of the flag. higher direct image; quotient of universal bundle; tangent bundle of a flag variety Grassmannians, Schubert varieties, flag manifolds, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Embeddings in algebraic geometry On the tangent bundle of a 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 This paper provides an interesting approach to the topological study of Schubert varieties, as well as a self-contained exposition on Schubert geometry.
Let \(G_ k(\mathbb{R}^ n)\) denote the Grassmannian of \(k\)-planes in \(\mathbb{R}^ n\). For a sequence \(\underline n=(n_ 0,n_ 1,\dots,n_ s)\) of natural numbers with \(n_ 0=1<n_ 1<\cdots<n_ s=n\), we define Schubert strata \(S(d)\), \(d=(d_ 1,\dots,d_ s)\), by the sets of \(V\in G_ k(\mathbb{R}^ n)\) satisfying \(\dim V\cap\mathbb{R}^{n_ i}=d_ i\), \(i=1,\dots,s\), with respect to the flag \(\mathbb{R}^{n_ 1}\subset\cdots\subset\mathbb{R}^{n_ s}\). Let \(\sigma=(\sigma_ 1,\dots,\sigma_ k)\) be a Schubert symbol and \(\overline{e(\sigma)}\subset G_ k(\mathbb{R}^ n)\) the corresponding Schubert variety. -- Then the main theorem 2.12 in this paper says that there exists an \(\underline n\), such that the stratification of the Schubert variety \(\overline{e(\sigma)}\) given by \(\{S(d)\}\) is coarsest among all topological stratifications of \(\overline{e(\sigma)}\). topological study of Schubert varieties; Schubert geometry; Grassmannian; Schubert symbol; stratification Buoncristiano, S.; Veit, A. B.: The intrinsic stratification of a Schubert variety. Adv. math. 91, No. 1, 1-26 (1992) Grassmannians, Schubert varieties, flag manifolds, Stratifications in topological manifolds, Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations) The intrinsic stratification of a Schubert variety | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors prove that Lusztig's Frobenius map (for quantum groups at roots of unity) can be, after dualizing, viewed as a characteristic zero lift of the geometric Frobenius splitting of \(G/B\) (in char \(p>0\)) introduced by \textit{V. B. Mehta} and \textit{A. Ramanathan} [Ann. Math. (2) 122, 27-40 (1985; Zbl 0601.14043)]. Frobenius map; Schubert variety; quantum group; geometric Frobenius splitting Kumar, S., Littelmann, P.: Frobenius splitting in characteristic zero and the quantum Frobenius map. J. Pure Appl. Algebra 152, 201--216 (2000) Quantum groups (quantized enveloping algebras) and related deformations, Grassmannians, Schubert varieties, flag manifolds, Quantum groups (quantized function algebras) and their representations, Cohomology theory for linear algebraic groups Frobenius splitting in characteristic zero and the quantum Frobenius map | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors study the class of admissible linear embeddings of flag varieties in the context of algebraic geometry. They prove that an admissible linear embedding of flag varieties has a certain explicit form in terms of linear algebra, showing that any direct limit of admissible embeddings of flag varieties is isomorphic to an ind-variety of generalized flags. That is, generalized flags in a countable-dimensional vector space are in a natural one-to-one correspondence with splitting parabolic subgroups \(\mathsf{P}\) of the ind-group \(\mathsf{GL}(\infty)\), and hence the points of homogeneous ind-spaces of the form \(\mathsf{GL}(\infty)/\mathsf{P}\) can be thought of as generalized flags. Ind-varieties are discussed as the ind-group \(\mathsf{SL}(\infty)\), respectively, \(\mathsf{O}(\infty)\) or \(\mathsf{Sp}(\infty)\) for isotropic generalized flags, where the authors construct them in purely algebro-geometric terms. flag variety; homogeneous ind-variety; generalized flag; linear embedding of flag varieties Grassmannians, Schubert varieties, flag manifolds, Group actions on varieties or schemes (quotients) An algebraic-geometric construction of ind-varieties of generalized 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 In this tutorial, we provide an overview of many of the established combinatorial and algebraic tools of Schubert calculus, the modern area of enumerative geometry that encapsulates a wide variety of topics involving intersections of linear spaces. It is intended as a guide for readers with a combinatorial bent to understand and appreciate the geometric and topological aspects of Schubert calculus, and conversely for geometric-minded readers to gain familiarity with the relevant combinatorial tools in this area. We lead the reader through a tour of three variations on a theme: Grassmannians, flag varieties, and orthogonal Grassmannians. The orthogonal Grassmannian, unlike the ordinary Grassmannian and the flag variety, has not yet been addressed very often in textbooks, so this presentation may be helpful as an introduction to type B Schubert calculus. This work is adapted from the author's lecture notes for a graduate workshop during the Equivariant Combinatorics Workshop at the Center for Mathematics Research, Montreal, June 12--16, 2017. Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds Variations on a theme of 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 present a general theory of Schubert polynomials, which are explicit representatives for Schubert classes in the cohomology ring of a flag variety with certain combinatorial properties. The starting point for this theory is a construction of Schubert classes in the cohomology ring of the flag variety of any semi-simple complex Lie group by Bernstein- Gelfand-Gelfand and Demazure. For the groups \(\text{SL}(n, \mathbb{C})\), Lascoux and Schützenberger made the crucial observation that one particular choice of representative of the top cohomology class yields Schubert polynomials simultaneously for all \(n\). In the present work we replicate the theory of \(\text{SL}(n, \mathbb{C})\) Schubert polynomials for the other infinite families of classical Lie groups and their flag varieties---the orthogonal groups \(\text{SO}(2n, \mathbb{C})\) and \(\text{SO}(2n+ 1,\mathbb{C})\) and the symplectic groups \(\text{Sp}(2n, \mathbb{C})\). We define Schubert polynomials to be elements in an inverse limit, which can be calculated as the unique solution of an infinite system of divided difference equations. The solution is derived using two equivalent formulas; one is an analog of the Billey-Jockusch-Stanley formula, while the other expresses our polynomials in terms of \(\text{SL}(n)\) Schubert polynomials and Schur \(Q\)- or \(P\)-functions. Our second formula involves the `shifted Edelman-Greene correspondences' and analogs of the Stanley symmetric functions. The Schubert polynomials form a \(\mathbb{Z}\)-basis for the ring in which they are defined. The non-negative integer coefficients that appear when they are multiplied give intersection multiplicities for Schubert varieties directly, without the need to reduce the product modulo an ideal. Schur functions; Schubert polynomials; Schubert classes; cohomology ring; flag variety; Lie group; orthogonal groups; symplectic groups; divided difference equations; Billey-Jockusch-Stanley formula; Stanley symmetric functions; Schubert varieties Billey, S.; Haiman, M., \textit{Schubert polynomials for the classical groups}, J. Amer. Math. Soc., 8, 443-482, (1995) Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds Schubert polynomials for the classical 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 paper introduces a compactification of the space of proper \(p\times m\) transfer functions with a fixed McMillan degree \(n\). Algebraically, this compactification has the structure of a projective variety and each point of this variety can be given an interpretation as a certain autoregressive system in the sense of Willems. It is shown that the pole placement map with dynamic compensators turns out to be central projection from this compactification to the space of closed-loop polynomials. Using this geometric point of view, necessary and sufficient conditions are given when a strictly proper or proper system can be generically pole assigned by a complex dynamic compensator of McMillan degree \(q\). central projection; autoregressive system; complex dynamic compensator Rosenthal, J., On Dynamic Feedback Compensation and Compactification of Systems, SIAM Journal on Control and Optimization, Vol. 32, pp. 279--300, 1994. Pole and zero placement problems, Multivariable systems, multidimensional control systems, Algebraic methods, Extensions of spaces (compactifications, supercompactifications, completions, etc.), Grassmannians, Schubert varieties, flag manifolds On dynamic feedback compensation and compactification of 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 under review provides a characterization of symplectic Grassmannians among Fano manifolds of Picard number one in terms of their varieties of minimal rational tangents (VMRT).
It is shown that symplectic Grassmannians can be characterized if at every point the VMRT is the expected one. Namely, if \(X\) is a Fano manifold of Picard number one endowed with a beautiful family of rational curves (i.e, a donimating unsplit family with a smooth evaluation morphism), assume that its tangent map \(\tau\) is a morphism and \(\tau_x\) is projectively equivalent to the embedding of \(\mathcal{C}\) into \(\mathbb{P}(H^0(\mathcal{C}, \mathcal{O}_\mathcal{C}(1)))\) for every \(x\in X\), then \(X\) is isomorphisc to a symplectic Grassmannian. Here \(\mathcal{C}\) is defined to be \(\mathbb{P}(\mathcal{O}_{\mathbb{P}^{r-1}}(2)\oplus\mathcal{O}_{\mathbb{P}^{r-1}}(1)^{2n-2r})\) for \(1< r< n\). Fano manifolds; symplectic Grassmannians; VMRT G. Occhetta, L. E. Solá Conde and K. Watanabe. A characterization of symplectic Grassmannians. Math. Z. 286(3-4) (2017), 1421-1433. Fano varieties, Minimal model program (Mori theory, extremal rays), Grassmannians, Schubert varieties, flag manifolds, Homogeneous spaces and generalizations A characterization of symplectic 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 symmetric group and a maximal parabolic subgroup, minimal length coset representatives can be indexed by Young diagrams fitting inside a rectangle. It is shown that parabolic Kazhdan-Lusztig basis elements in the corresponding Hecke algebra can be written as a product of factors which are differences between a standard generator and a rational function in \(v\). The factors depend in a combinatorial way on the Young diagram corresponding to the index of the basis element. A factorization for the dual Kazhdan-Lusztig basis is also obtained. Kazhdan-Lusztig polynomials; Young diagrams; Hecke algebras; Kazdan-Lusztig bases Kirillov, A., Jr., Lascoux, A.: Factorization of Kazhdan-Lusztig elements for Grassmanians. In: Koike, K. et al. (eds.) Combinatorial Methods in Representation Theory, pp. 143--154. Kinokuniya, Tokyo (2000) Hecke algebras and their representations, Grassmannians, Schubert varieties, flag manifolds, Representations of finite symmetric groups, Combinatorial aspects of representation theory Factorization of Kazhdan-Lusztig elements for 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 purpose of this article is to determine the Hilbert function of the tangent cone to points in a Schubert subvariety of a Grassmannian.
\textit{S. S. Abhyankar} [``Enumerative combinatorics of Young tableaux'', Pure and Applied Mathematics 115 (1988; Zbl 0643.05001)] and \textit{J. Herzog} and \textit{N. V. Trung} [Adv. Math. 96, No. 1, 1--37 (1992; Zbl 0778.13022)] gave formulas for the multiplicity and the Hilbert function of the quotient ring of the determinantal ideal of the tangent cone at the identity coset, in the ring of the tangent space of the Grassmann variety. \textit{V. Lakshmibai} and \textit{J. Weyman} [Adv. Math. 84, No. 2, 179--208 (1990; Zbl 0729.14037)] gave a recursive formula for the multiplicity at any point, and from this formula \textit{J. Rosenthal} and \textit{A. Zelevinsky} [J. Algebr. Comb. 13, No. 2, 213--218 (2001; Zbl 1015.14025)] obtained a closed form for the multiplicity at any point.
\textit{V. Kreiman} and \textit{V. Lakshmibai} [in: Algebra and algebraic geometry with applications, 553--563 (2002; Zbl 1092.14060)] obtained an expression for the Hilbert function at the identity coset in terms of the combinatorics of the Weyl group, and they recovered the interpretation of the multiplicity due to Herzog and Trung. They also reformulated their main result in terms of generic determinantal minors. In addition they conjectured an expression for the Hilbert function and the multiplicity when \(x\) is any point.
In the present article the approach of Kreiman and Lakshmibai is clarified, and their expression for the Hilbert function and the multiplicity is extended to all points, as is their combinatorial interpretation of the multiplicity.
\textit{C. Krattenthaler} [Sémin. Lothar. Comb. 45, B45c, 11 p. (2000; Zbl 0965.14023)] has given an interpretation of the Rosenthal-Zelevinsky formula using combinatorial techniques, and he proves the multiplicity formula of Kreiman and Lakshmibai and shows that their conjecture about the Hilbert function is equivalent to a certain finite problem. The approach of Krattenthaler is completely different from that of the present article. While Krattenthaler explores the combinatorial interpretations of the formula, the present article, in the spirit of Lakshmibai-Weyman and Kreiman-Lakshmibai, uses standard monomial theory to translate the problems from geometry to combinatorics. tangent cone; determinantal ideals; standard monomial theory; multiplicity V. Kodiyalam, K. Raghavan, Hilbert functions of points on Schubert varieties in the Grassmannian, J. Algebra 270 (2003), 28--54. Grassmannians, Schubert varieties, flag manifolds, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series Hilbert functions of points on Schubert varieties in 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 \(\Pi\) be a polar space of rank \(n\) and let \({\mathcal G}_k(\Pi)\), \(k\in\{0,\dots, n-1\}\) be the polar Grassmannian formed by \(k\)-dimensional singular subspaces of \(\Pi\). The corresponding Grassmann graph will be denoted by \(\Gamma_k(\Pi)\). We consider the polar Grassmannian \({\mathcal{G}}_{n-1}(\Pi)\) formed by maximal singular subspaces of \(\Pi\) and show that the image of every isometric embedding of the \(n\)-dimensional hypercube graph \(H_n\) in \(\Gamma_{n-1}(\Pi)\) is an apartment of \({\mathcal{G}}_{n-1}(\Pi)\). This follows from a more general result concerning isometric embeddings of \(H_m\), \(m\leq n\) in \(\Gamma_{n-1}(\Pi)\). As an application, we classify all isometric embeddings of \(\Gamma_{n-1}(\Pi)\) in \(\Gamma_{n'-1}(\Pi')\), where \(\Pi'\) is a polar space of rank \(n' \geq n\). apartment; dual polar space; hypercube graph; isometric embedding Pankov, M., Metric characterization of apartments in dual polar spaces, J. Comb. Theory, Ser. A, 118, 1313-1321, (2011) Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations), Polar geometry, symplectic spaces, orthogonal spaces, Buildings and the geometry of diagrams, Grassmannians, Schubert varieties, flag manifolds Metric characterization of apartments in dual polar spaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a connected simply connected semisimple complex algebraic group, and \(G^{\vee}\) be its Langlands dual group. Mirković and Vilonen discovered a family of closed irreducible algebraic subvarieties, called MV cycles, of the affine Grassmannian \(\mathcal{G}\) associated to \(G\), which provide a basis for each finite-dimensional irreducible representation of \(G^{\vee}\). In order to obtain an explicit combinatorial description of MV cycles, \textit{J. E. Anderson} [Duke Math. J. 116, 567--588 (2003; Zbl 1064.20047)] defined MV polytopes for the Lie algebra of the group \(G\) to be the moment map image of these cycles.
In the present paper, an explicit description of the (lowering) Kashiwara operators on MV polytopes in types B and C is given. This provides a simple method for generating MV polytopes inductively. The description can be thought of as a modification of the one in the original Anderson-Mirković conjecture, which Kamnitzer proved in the case of type A, and presented a counterexample in the case of type \(\text{C}_3\). MV polytopes; canonical bases; Kashiwara operators; Lusztig parametrizations; Plücker relations Naito, S.; Sagaki, D., \textit{A modification of the Anderson-mirković conjecture for mirković-vilonen polytopes in types B and C}, J. Algebra, 320, 387-416, (2008) Semisimple Lie groups and their representations, Algebraic cycles, Grassmannians, Schubert varieties, flag manifolds A modification of the Anderson-Mirković conjecture for Mirković-Vilonen polytopes in types \(B\) and \(C\) | 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 A theorem of the first author states that the cotangent bundle of the type \(\mathbf{A}\) Grassmannian variety can be canonically embedded as an open subset of a smooth Schubert variety in a two-step affine partial flag variety (see [\textit{V. Lakshmibai}, Transform. Groups 21, No. 2, 519--530 (2016; Zbl 1390.14148)]).
That theorem was inspired by works of Lusztig and Strickland. In particular, \textit{G. Lusztig} [J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008)] related certain orbit closures arising from the type \(\mathbf{A}\) cyclic quiver with \(h\) vertices to affine Schubert varieties. In the case \(h = 2\), Strickland [\textit{E. Strickland}, J. Algebra 75, 523--537 (1982; Zbl 0493.14030)] relates such orbit closures to conormal varieties of determinantal varieties; furthermore, any determinantal variety can be canonically realized as an open subset of a Schubert variety in the Grassmannian (see [\textit{V. Lakshmibai} and \textit{C. S. Seshadri}, Proc. Indian Acad. Sci., Sect. A 87, No. 2, 1--54 (1978; Zbl 0447.14011)]).
In this paper, the authors extend the result of [\textit{V. Lakshmibai}, Transform. Groups 21, No. 2, 519--530 (2016; Zbl 1390.14148)] to cominuscule generalized Grassmannians of arbitrary finite type (such Grassmannians occur in types \(\mathbf{A}-\mathbf{E}\)). Schubert varieties; Grassmannian; affine flag varieties Grassmannians, Schubert varieties, flag manifolds The cotangent bundle of a cominuscule Grassmanian | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 universal family \(\pi:\mathcal{X}\rightarrow|\mathcal{O}_{\mathbb{P}}^n(d)|\) of hypersurfaces of degree \(d\) in the complex projective plane \(\mathbb{P}^n\) comes equipped with a polarization \(\mathcal{L}\). Furthermore, the sheaf \(\pi_*\mathcal{L}^{\otimes^k}\), with \(k\geq 1\), is locally free and is called the \(k\)-th Verlinde bundle of the family \(\pi\), denoted by \(V_k\). In the present paper, the author studies the splitting behavior of such Verlinde bundles. Let \(Z\) be the set of jumping lines of \(V_{d+1}\) in the Grassmannian of lines in \(|\mathcal{O}(d)|\), \(\mathrm{Gr}(1,|\mathcal{O}(d)|)\). In the main theorem of this paper, the author calculates for \(n\leq 3\) the dimension of \(Z\) as well as the class of \(Z\) in the Chow ring \(CH(\mathrm{Gr}(1),|\mathcal{O}(d)|)\). [\textit{J. N. Iyer}, ``Bundles of verlinde spaces and group actions'', Preprint \url{arXiv:1309.7562}] Verlinde bundles; jumping lines; cohomology class Vector bundles on surfaces and higher-dimensional varieties, and their moduli, (Equivariant) Chow groups and rings; motives, Grassmannians, Schubert varieties, flag manifolds Verlinde bundles of families of hypersurfaces and their jumping lines | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper is a review of results on the structure of homogeneous ind-varieties \(G/P\) of the ind-groups \(G = \operatorname{GL}_\infty (\mathbb{C})\), \(\operatorname{SL}_\infty (\mathbb{C})\), \(\operatorname{SO}_\infty (\mathbb{C} )\), and \(\operatorname{Sp}_\infty (\mathbb{C} )\), subject to the condition that \(G/P\) is the inductive limit of compact homogeneous spaces \(G_n /P_n \). In this case, the subgroup \(P \subset G\) is a splitting parabolic subgroup of \(G\) and the ind-variety \(G/P\) admits a ``flag realization''. Instead of ordinary flags, one considers generalized flags that are, in general, infinite chains \(\mathcal{C}\) of subspaces in the natural representation \(V\) of \(G\) satisfying a certain condition; roughly speaking, for each nonzero vector \(\upsilon\) of \(V\), there exist the largest space in \(\mathcal{C} \), which does not contain \(\upsilon \), and the smallest space in \(\mathcal{C} \), which contains \(v\). We start with a review of the construction of ind-varieties of generalized flags and then show that these ind-varieties are homogeneous ind-spaces of the form \(G/P\) for splitting parabolic ind-subgroups \(P \subset G\). Also, we briefly review the characterization of more general, i.e., nonsplitting, parabolic ind-subgroups in terms of generalized flags. In the special case of the ind-grassmannian \(X\), we give a purely algebraic-geometric construction of \(X\). Further topics discussed are the Bott-Borel-Weil theorem for ind-varieties of generalized flags, finite-rank vector bundles on ind-varieties of generalized flags, the theory of Schubert decomposition of \(G/P\) for arbitrary splitting parabolic ind-subgroups \(P \subset G\), as well as the orbits of real forms on \(G/P\) for \(G = \operatorname{SL}_\infty (\mathbb{C} )\). ind-variety; ind-group; generalized flag; Schubert decomposition; real form Infinite-dimensional Lie groups and their Lie algebras: general properties, Infinite-dimensional Lie (super)algebras, Grassmannians, Schubert varieties, flag manifolds Ind-varieties of generalized flags: a survey | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This is a detailed version of the author's report intended for the 25th Arbeitstagung Bonn 1984, Proc. Meet. Max-Placnk-Inst. Math., Lect. Notes 1111, 59-101 (1985). Let \(x_ i\) be commutative and \(\xi_ k\) skew- commutative variables. The scheme Spec \({\mathbb{Z}}[x_ 1,...,x_ m;\xi_ 1,...,\xi_ n]\) may be considered as a geometric object of dimension (1,m\(| n)\) where 1, m and n correspond to the ''arithmetic'' direction \({\mathbb{Z}}\), ''even'' directions \(x_ 1,...,x_ m\) and ''odd'' directions \(\xi_ 1,...,\xi_ n\), respectively. According to the author, this paper aims at giving meaning to the following metaphysical principle: ''all three types of dimensions have equal rights'' - thus developing A. Weil's ideas. The first part of the paper deals with the analogy between the ''arithmetic'' and the ''even'' dimensions. Here, an attemps is made to lay the foundations of ''arithmetic geometry'' (''A-geometry'') which would contain analogues of the main notions and theorems of algebraic and analytic geometries. In the second part of the paper, the relation between ''even'' and ''odd'' dimensions is discussed on the basis of the principle: ''even geometry is the collective effect of infinite- dimensional odd geometry''. The technical part of the paper (the material involved) is clear from the contents: 1. A-manifolds and A-divisors; 2. Riemann-Roch theorems; 3. Problems and perspectives of A-geometry; 4. superspaces; 5. Schubert supercells, 6. Geometry of supergravity.
Throughout the text, the interplay of mathematical and physical notions is stressed, the latter pertaining mostly to elementary particles, field theory and supergravity. supermanifold; grand unification; arithmetic geometry; A-geometry; even geometry; odd geometry; Riemann-Roch theorems; Schubert supercells; supergravity DOI: 10.1070/RM1984v039n06ABEH003181 Generalizations (algebraic spaces, stacks), Supergravity, Grassmannians, Schubert varieties, flag manifolds, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Riemann-Roch theorems New directions in geometry | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The survey is devoted to the concept of ``standard monomials'', introduced in the 1940's by Hodge to study the Schubert varieties in the Grassmannian/flag varieties, and developed in the 1970's into a ``theory'' by Seshadri, in collaboration with Musili, Lakshmibai, Littelmann, etc., to study the Schubert varieties in the generalized flag varieties \(G/Q\), where \(G\) is a semisimple group and \(Q\) is a parabolic subgroup of \(G\).
The author begins with the classical results on the Grassmannians and the Schubert varieties, such as the Plücker coordinates, the quadratic relations and the standard monomials in the Plücker coordinates that form a basis in the algebra of regular functions on the affine cones over the Grassmannians. This nice basis, parametrized by the standard Young tableaux, allows to prove many important geometric properties of the Grassmannians and the Schubert varieties -- the projective normality, the projective factoriality (for the Grassmannians), the projective Cohen-Macaulay property.
In the next sections possible generalizations to the cases of the flag variaties corresponding to irreducible \(G\)-modules with 1) a minuscule highest weight; 2) a quasi-minuscule highest weight; 3) a classical type highest weight are given. Some applications of the theory to the study of determinantal varieties, ladder determinantal varieties, varieties of idempotents, varieties of complexes, quiver varieties and singular loci of the Schubert varieties are discussed. For the later developments of the standard monomial theory see the preceding review [\textit{V.Lakshmibai}, in: A tribute to C. S. Seshadri. Birkhäuser, Trends in Mathematics, 283--309 (2003; Zbl 1056.14065)], which may be found in the same volume.
This report may be recommended both to beginners who are looking for main definitions, problems and results in the area, and to specialists who would like to have a systematic historical review of the subject. Grassmannians; semisimple algebraic groups; flag varieties; Schubert varieties; minuscule weights Musili, C.: The development of standard monomial theory. I. In: A tribute to C. S. Seshadri, Chennai, 2002. Trends Math., pp. 385--420. Birkhäuser, Basel (2003) Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations), History of mathematics in the 20th century The development of standard monomial theory. I | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author studies the manifold \(N^n_m\) of nondegenerate \(m\)-pairs of the real projective space \(\mathbb{R} P(n)\), i.e., the manifold whose points are pairs \((A,B)\) of an \(m\)-plane \(A\) and an \(n-m-1\) plane \(B\) not intersecting in \(\mathbb{R} P(n)\). In particular, the author constructs a hyperbolic Kählerian metric \(g\) on \(N^n_m\) which is semi-Riemannian of signature \(((m+ 1)(n-m),(m+ 1)(n-m))\) and a Kähler complex structure \(J\) on \(N^n_m\) such that \(g(JX, JY)= -g(X,Y)\) for all \(X,Y\in TM\) and the form \(\Omega(X, Y)= g(X, JY)\) is symplectic i.e., \(d\Omega= 0\). It is proved that \((N^n_m,g)\) is an Einstein manifold and that \(N^n_0\) has constant holomorphic sectional curvature. At the end, the author proves that \((N^n_m,\Omega)\) is symplectomorphic to the cotangent bundle of the Grassman manifold \(G_{m,n}\) of \(m\)-dimensional subspaces of \(\mathbb{R} P(n)\) with the standard symplectic structure of the cotangent bundle. hyperbolic Kähler manifold; symplectic manifold Symplectic manifolds (general theory), Grassmannians, Schubert varieties, flag manifolds Symplectic geometry on the manifold of nondegenerate \(m\)-pairs | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 matrix with real entries is called totally positive if all its minors are positive. G. Lusztig extended this classical subject to the totally positive variety \(G_{>0}\) in an arbitrary reductive group \(G\) and the totally positive varieties \(\left( P \setminus G \right) _{>0}\) for any parabolic subgroup \(P\) of \(G\).
In 2001, in order to study total positivity in algebraic groups and canonical bases in quantum groups, \textit{S. Fomin} and \textit{A. Zelevinsky} introduced the class of cluster algebras [J. Am. Math. Soc. 15, No. 2, 497--529 (2002; Zbl 1021.16017)]. \textit{C. Geiß}, \textit{B. Leclerc} and \textit{J. Schröer} have studied cluster algebras associated with Lie groups of type \(\mathbb{A}\), \(\mathbb{D}\), \(\mathbb{E}\), and have modelled them by categories of modules over the Gelfand-Ponomarev preprojective algebras \(\Lambda\) of the same type. They have shown in [Invent. Math. 165, No. 3, 589--632 (2006; Zbl 1167.16009)] that each reachable maximal rigid \(\Lambda\)-module can be thought of as a seed of a cluster algebra structure on \(\mathbb{C}[N]\), the coordinate ring of a maximal unipotent subgroup of \(G\). They also attached to each standard parabolic subgroup \(P\) of \(G\) a certain subcategory \(\mathcal{C}_P\) of \(\text{mod} \Lambda\) and showed that each reachable maximal rigid \(\Lambda\)-module in \(\mathcal{C}_P\) gives a seed for a cluster algebra structure on \(\mathbb{C}[N_P]\), the coordinate ring of the unipotent radical of \(P\).
In [{\textit{C. Geiss}}, {\textit{B. Leclerc}} and {\textit{J. Schröer}}, EMS Series of Congress Reports, 253--283 (2008; Zbl 1203.16014)], they conjectured that each basic maximal rigid \(\Lambda\)-module in \(\mathcal{C}_P\) gives rise to a total positivity criterium for the partial flag variety \(P \setminus G\). In the paper under review, the author partially answers this conjecture and showed that every reachable basic maximal rigid module in \(\mathcal{C}_P\) gives rise to a total positivity criterium which leads to a (generally infinite) number of criteria. total positivity; partial flag varieties; preprojective algebras N. Chevalier, Total positivity criteria for partial flag varieties, J. Algebra 348 (2011), 402--415. Grassmannians, Schubert varieties, flag manifolds, Cluster algebras Total positivity criteria for partial 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 The main result of this work is the characterization of the covering relations of the Bruhat order of the maximal parabolic quotients of type B. Our approach is mainly combinatorial and is based in the pattern of the corresponding permutations also called signed \(k\)-Grassmannians permutations. We obtain that a covering relation can be classified in four different pairs of permutations. This answers a question raised by \textit{T. Ikeda} and \textit{T. Matsumura} [Math. Z. 280, No. 1--2, 269--306 (2015; Zbl 1361.14029)] providing a nice combinatorial model for maximal parabolic quotients of type B. signed \(k\)-Grassmannians permutations Permutations, words, matrices, Grassmannians, Schubert varieties, flag manifolds, Reflection and Coxeter groups (group-theoretic aspects) Covering relations of \(k\)-Grassmannian permutations of type B | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper we calculate Reidemeister torsion of flag manifold \(K/T\) of compact semi-simple Lie group \(K = SU_{n+1}\) using Reidemeister torsion formula and Schubert calculus, where \(T\) is maximal torus of \(K\). We find that this number is 1. Also we explicitly calculate ring structure of integral cohomology algebra of flag manifold \(K/T\) of compact semi-simple Lie group \(K = SU_{n+1 }\) using root data, and Groebner basis techniques. Reidemeister torsion; flag manifolds; Weyl groups; Schubert calculus; Groebner-Shirshov bases; graded inverse lexicographic order Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc., Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Gröbner-Shirshov bases, Semisimple Lie groups and their representations, Loop groups and related constructions, group-theoretic treatment On Reidemeister torsion of flag manifolds of compact 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 Let \(k\) be an algebraically closed field and let \(G\) be a reductive \(k\)-group. Assume that the characteristic of \(k\) is either zero or greater then \(2h\), where \(h\) denotes the Coxeter number of \(G\). Put \(F := k((\varpi))\) and form the loop group \(LG\) accordingly. For any \(\gamma \in \mathfrak{g}(F)\), the affine Springer fiber \(\mathrm{Spr}_\gamma\) classifies the Iwahori subgroups whose Lie algebras contain \(\gamma\). It follows from Lusztig's works that there is an action of \(\tilde{W} \times \pi_0(LG_\gamma)\) on \(H_\bullet(\mathrm{Spr}_\gamma)\) or on \(H^\bullet_c(\mathrm{Spr}_\gamma)\); here \(LG_\gamma\) stands for the centralizer and \(\tilde{W} = X_*(T) \rtimes W\) stands for the extended affine Weyl group. Therefore, the spherical part \(\mathbb{Q}_\ell[X_*(T)]^W\) of \(\mathbb{Q}_\ell[\tilde{W}]\) acts as well.
The following conjecture is due to Goresky, Kottwitz and MacPherson (and independently by Bezrukavnikov and Varshavsky): for any regular semisimple \(\gamma\), the actions of \(\mathbb{Q}_\ell[\tilde{W}]\) factor through a canonical algebra homomorphism \(\sigma_\gamma: \mathbb{Q}_\ell[X_*(T)]^W \to \mathbb{Q}_\ell[\pi_0(LG_\gamma)]\). The main local results of the paper is that (i) the conjecture holds for \(H^\bullet_c(\mathrm{Spr}_\gamma)\), and (ii) it also holds for \(H_\bullet(\mathrm{Spr}_\gamma)\) upon passing to graded pieces with respect to a certain filtration \(\mathrm{Fil}^p H_\bullet(\mathrm{Spr}_\gamma)\). More generally, for any parahoric subgroup \(P \subset LG\), the corresponding assertions for \(\mathbb{Q}_\ell[X_*(T)]^{W_P}\) and \(\mathrm{Spr}_{P, \gamma}\) will follow from the conjecture above, which is just the case \(P = I\) (Iwahori subgroup).
The argument is global. It makes use of the parabolic Hitchin fibration \(f^{\mathrm{par}}: \mathcal{M}^{\mathrm{par}} \to \mathcal{A}^{\mathrm{Hit}} \times X\) over a curve \(X\) and considering an action of \(\mathbb{Q}_\ell[X_*(T)]^W\) on \(Rf^{\mathrm{par}}_! \mathbb{Q}_\ell|_{(\mathcal{A}^\heartsuit \times X)'}\). The theories in [\textit{B. C. Ngô}, Publ. Math., Inst. Hautes Etud. Sci. 111, 1--271 (2010; Zbl 1200.22011)] and [\textit{Z. Yun}, Adv. Math. 228, No. 1, 266--328 (2011; Zbl 1230.14048)] play a crucial role here. It also gives an impressive application of Yun's global Springer theory to local problems. Springer fiber; Hitchin fiber; global Springer theory Yun, Z.: The spherical part of the local and global Springer actions. Math. ann. 359, No. 3-4, 557-594 (2014) Vector bundles on curves and their moduli, Grassmannians, Schubert varieties, flag manifolds, Reflection and Coxeter groups (group-theoretic aspects), Linear algebraic groups over local fields and their integers The spherical part of the local and global Springer actions | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials For a split reductive group \(\mathbf{G}\) over \(k = \mathbb{F}_q\), we denote by \(X\) the variety of Borel subgroups of \(\mathbf{G}\). Deligne and Lusztig introduced the locally closed subsets of \(X\) given by
\[
X(w) = \left\{ x \in X: \mathrm{inv}(x,F(x)) = w \right\},
\]
where \(F: X \to X\) is the Frobenius morphism over \(k\), \(\mathrm{inv}\) measures the relative position in terms of Bruhat decomposition, and \(w\) ranges over the elements of the Weyl group. These are quasi-projective varieties defined over \(k\) endowed with natural \(G = \mathbf{G}(k)\)-actions, and \(\dim X(w) = \ell(w)\).
The virtual \(G\)-representations \(\sum_i (-1)^i \mathrm{H}^i_c(X(w), \overline{\mathbb{Q}}_\ell)\) are known to detect all \(\ell\)-adic irreducible characters of \(G\). Nevertheless, much less is known about the individual cohomologies \(\mathrm{H}^i_c(X(w), \overline{\mathbb{Q}}_\ell)\) as \(G \times \mathrm{Gal}_k\)-modules. The goal of the article under review is to give two inductive procedure to unravel their structures when \(\mathbf{G} = \mathbf{GL}_n\).
The algorithm makes use of squares, which are quadruples \(\{w', sw', w's, w\} \subset W\) appearing originally in the BGG resolutions of finite-dimensional Lie algebra representations. More generally, one considers hyper-squares in \(W\) or even the monoid \(F^+\) freely generated by the simple reflections in \(W\). To each element \(w\) in \(F^+\) one can attach the Demazure compactification \(\overline{X}(w)\) of Deligne-Lusztig varieties. This is connected to our original question by a spectral sequence
\[
E_1^{p, q} = \bigoplus_{v \leq w, \; \ell(v) = \ell(w) - p} \mathrm{H}^q(\overline{X}(v)) \Rightarrow \mathrm{H}^{p+q}_c(X(w)).
\]
For the precise inductive recipe, we refer the readers to the last few results recorded in Section 1. Deligne-Lusztig variety Group actions on varieties or schemes (quotients), Discrete subgroups of Lie groups, Finite ground fields in algebraic geometry, Representations of finite groups of Lie type, Grassmannians, Schubert varieties, flag manifolds, Representation theory for linear algebraic groups, Linear algebraic groups over finite fields The cohomology of Deligne-Lusztig varieties for the general linear group | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This book is a new monograph on classification of prehomogeneous vector spaces, which were introduced by M. Sato. T. Shintani's note of Sato's lecture is the first one on this topic. [\textit{M. Sato} and \textit{T. Shintani}, Theory of prehomogeneous vector spaces, Sugaku no Ayumi 15, No. 1, 85--157 (1970), partly translated into English in Nagoya Math. J. 120, 1--34 (1990; Zbl 0703.22011)]. \textit{M. Sato} and \textit{T. Kimura} [Nagoya Math. J. 65, 1--155 (1977; Zbl 0321.14030)] completed a classification of prehomogeneous vector spaces based on castling transformation. On the other hand, \textit{H. Rubenthaler}'s thesis [Publication d'I.R.M.A. Strasbourg (1982; Zbl 0546.22019)] studied prehomogeneous vector spaces from another point of view based on Vinberg's theorem on the graduation of semisimple Lie algebras. He called it prehomogeneous vector space of commutative parabolic type.
The author of this article gives a complete classification of prehomogeneous vector spaces of this type. However, the detailed study of the structure of prehomogeneous vector spaces and zeta functions associated with them has been done only in some special cases. The future study will be developed on the classification by these authors. classification of prehomogeneous vector spaces; castling transformation; semisimple Lie algebras A. Mortajine: Classification des espaces préhomogènes de type parabolique réguliers et de leurs invariants relatifs, Hermann, Paris, 1991. Semisimple Lie groups and their representations, Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.), Research exposition (monographs, survey articles) pertaining to topological groups, Prehomogeneous vector spaces, Homogeneous spaces and generalizations, Linear algebraic groups over the reals, the complexes, the quaternions, Simple, semisimple, reductive (super)algebras, Grassmannians, Schubert varieties, flag manifolds, Harmonic analysis on homogeneous spaces Classification of prehomogeneous vector spaces of regular parabolic type and their relative invariants | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(\Lambda \) be an artin algebra. In his seminal [in: Represent. Theory of Algebras, Proc. Phila. Conf., Lect. Notes pure appl. Math. 37, 1--244 (1978; Zbl 0383.16015)], \textit{M. Auslander} introduced the concept of morphisms being determined by modules. Auslander was very passionate about these investigations (they also form part of the final chapter of the Auslander-Reiten-Smalø book and could and should be seen as its culmination). The theory presented by Auslander has to be considered as an exciting frame for working with the category of \(\Lambda \)-modules, incorporating all what is known about irreducible maps (the usual Auslander-Reiten theory), but the frame is much wider and allows for example to take into account families of modules--an important feature of module categories. What Auslander has achieved is a clear description of the poset structure of the category of \(\Lambda \)-modules as well as a blueprint for interrelating individual modules and families of modules. Auslander has subsumed his considerations under the heading of ``morphisms being determined by modules''. Unfortunately, the wording in itself seems to be somewhat misleading, and the basic definition may look quite technical and unattractive, at least at first sight. This could be the reason that for over 30 years, Auslander's powerful results did not gain the attention they deserve. The aim of this survey is to outline the general setting for Auslander's ideas and to show the wealth of these ideas by exhibiting many examples. Auslander bijections; Auslander-Reiten theory; right factorization lattice; morphisms determined by modules; finite length categories: global directedness; local symmetries; representation type; Brauer-Thrall conjectures; Riedtmann-Zwara degenerations; hammocks; Kronecker quiver; quiver Grassmannians; Auslander varieties; modular lattices; meet semi-lattices \beginbarticle \bauthor\binitsC. M. \bsnmRingel, \batitleThe Auslander bijections: How morphisms are determined by modules, \bjtitleBull. Math. Sci. \bvolume3 (\byear2013), page 409-\blpage484. \endbarticle \endbibitem Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers, Abelian categories, Grothendieck categories, Representation type (finite, tame, wild, etc.) of associative algebras, Representations of quivers and partially ordered sets, Grassmannians, Schubert varieties, flag manifolds, \(K_0\) of other rings The Auslander bijections: how morphisms are determined by modules | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials For a connected reductive group \(G\) and a Borel subgroup \(B\), we study the closures of double classes \(BgB\) in a \((G\times G)\)-equivariant ``regular'' compactification of \(G\). We show that these closures \(\overline{BgB}\) intersect properly all \((G\times G)\)-orbits, with multiplicity one, and we describe the intersections. Moreover, we show that almost all \(\overline {BgB}\) are singular in codimension two exactly. We deduce this from more general results on \(B\)-orbits in a spherical homogeneous space \(G/H\); they lead to formulas for homology classes of \(H\)-orbit closures in \(G/B\), in terms of Schubert cycles. Bruhat decomposition; equivariant compactification; regular embedding; spherical homogeneous space; Schubert cycles M. Brion. ''The behaviour at infinity of the Bruhat decomposition''. Comment. Math. Helv. 73(1998), pp. 137--174.DOI. Group actions on varieties or schemes (quotients), Homogeneous spaces and generalizations, Grassmannians, Schubert varieties, flag manifolds The behavior at infinity of the Bruhat decomposition | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 projected Gromov-Witten variety is the union of all rational curves of fixed degree that meet two opposite Schubert varieties in a homogeneous space \( X = G/P\). When \( X\) is cominuscule we prove that the map from a related Gromov-Witten variety is cohomologically trivial. This implies that all (3-point, genus zero) \(K\)-theoretic Gromov-Witten invariants of \(X\) are determined by projected Gromov-Witten varieties, which extends an earlier result of \textit{A. Knutson} et al. [J. Reine Angew. Math. 687, 133--157 (2014; Zbl 1345.14047)], and provides an alternative version of the `quantum equals classical' theorem. Our proof uses that any projected Gromov-Witten variety in a cominuscule space is also a projected Richardson variety. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), \(K\)-theory of schemes, Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Rational and unirational varieties, Rationally connected varieties Projected Gromov-Witten varieties in cominuscule spaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G_{\mathbb{R}}\) be a real semisimple Lie group with finite center. Let \(K_ \mathbb{R}\) be a maximal compact subgroup of \(G_ \mathbb{R}\). Let \(G\) (respectively \(K\)) be the complexification of \(G_ \mathbb{R}\) (respectively \(K_{\mathbb{R}}\)). In this paper, the representation theory of \(G_ \mathbb{R}\) is studied using the geometry of the flag manifold of \(G\). Using Beilinson-Bernstein's results one has a correspondence between \{representations of \(G_ \mathbb{R}\)\} and \{Harish-Chandra modules\}. On the other hand, using Mirkovic-Uzawa-Vilonen's results, one has a correspondence between \{\(G_ \mathbb{R}\)-equivariant sheaves\} and \{\(K\)- equivariant sheaves\} (on the flag manifold of \(G\)). The author establishes a correspondence between \{representations of \(G_ \mathbb{R}\)\} and \{\(G_ \mathbb{R}\)-equivariant sheaves\} on the one hand, and \{Harish- Chandra modules\} and \{\(K\)-equivariant sheaves\} on the other, thus making the epicture complete.
This paper makes an important contribution to the representation theory of real semi-simple Lie groups. representations; Harish-Chandra modules; \(G_ \mathbb{R}\)-equivariant sheaves; \(K\)-equivariant sheaves; flag manifolds; representations of real semisimple Lie groups Kashiwara, M.: \(D\)-modules and representation theory of Lie groups. Ann. Inst. Fourier (Grenoble) \textbf{43}(5), 1597-1618 (1993) Semisimple Lie groups and their representations, Grassmannians, Schubert varieties, flag manifolds, Representations of Lie and linear algebraic groups over real fields: analytic methods, Classical groups (algebro-geometric aspects) \(D\)-modules and representation theory of 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 A flag domain \(D=G/V\) for \(G\) a simple real non-compact group \(G\) with compact Cartan subgroup is non-classical if it does not fiber holomorphically or anti-holomorphically over a Hermitian symmetric space. We prove that for \(\Gamma\) an infinite, finitely generated discrete subgroup of \(G\), the analytic space \(\Gamma\backslash D\) does not have an algebraic structure. We also give another proof of the theorem of Huckleberry that any two points in a non-classical domain \(D\) can be joined by a finite chain of compact subvarieties of \(D\). Hodge theory; flag domain; Hermitian symmetric space Griffiths, P.; Robles, C.; Toledo, D., Quotients of non-classical flag domains are not algebraic, Algebraic Geom., 1, 1-13, (2014) Differential geometry of homogeneous manifolds, Grassmannians, Schubert varieties, flag manifolds, Complex manifolds Quotients of non-classical flag domains are not algebraic | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Relying on recent advances in the theory of motives we develop a general formalism for derived categories of motives with \(\mathbf{Q}\)-coefficients on perfect \(\infty\)-prestacks. We construct Grothendieck's six functors for motives over perfect (ind-)schemes perfectly of finite presentation. Following ideas of Soergel-Wendt, this is used to study basic properties of stratified Tate motives on Witt vector partial affine flag varieties. As an application we give a motivic refinement of Zhu's geometric Satake equivalence for Witt vector affine Grassmannians in this set-up. motives; perfect schemes; Witt vector affine flag variety; Satake equivalence Motivic cohomology; motivic homotopy theory, Grassmannians, Schubert varieties, flag manifolds, Representation theory for linear algebraic groups Tate motives on Witt vector affine 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 The main result of the article is: Let \(X\) be a Schubert variety in the Grassmann manifold \(G(m,n)\) of \(m\)-dimensional subspaces of an \(n\)- dimensional vector space over the real or complex numbers. Then there exists a Riccati flow on \(G(m,n)\), and a stable manifold \(W\) of this flow, such that \(W\) is exactly the smooth locus of \(X\). The approach to the study of the singularities of Schubert varieties via the Riccati flow gives a new point of view on the topic, with interesting consequences. It is, for example, possible to prove, without using representation theory, that Schubert varieties over the complex numbers are smooth if and only if their singular cohomology satisfies Poincaré duality. stable manifold of Riccati flow; singularities of Schubert varieties Wolper, J.S.: The Riccati flow and singularities of Schubert varieties. Proceedings of the AMS 123 (1995) 703--709 Grassmannians, Schubert varieties, flag manifolds, Singularities in algebraic geometry, Dynamics induced by flows and semiflows, Topology of real algebraic varieties The Riccati flow and singularities of Schubert varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We show that the monotone Lagrangian torus fiber of the Gelfand-Cetlin integrable system on the complex Grassmannian \(\operatorname{Gr}(k, n)\) supports generators for all maximum modulus summands in the spectral decomposition of the Fukaya category over \(\mathbb{C}\), generalizing the example of the Clifford torus in projective space. We introduce an action of the dihedral group \(D_n\) on the Landau-Ginzburg mirror proposed by \textit{R. J. Marsh} and \textit{K. Rietsch} [Adv. Math. 366, Article ID 107027, 131 p. (2020; Zbl 1453.14104)] that makes it equivariant and use it to show that, given a lower modulus, the torus supports nonzero objects in none or many summands of the Fukaya category with that modulus. The alternative is controlled by the vanishing of rectangular Schur polynomials at the \(n\)-th roots of unity, and for \(n = p\) prime this suffices to give a complete set of generators and prove homological mirror symmetry for \(\operatorname{Gr}(k, p)\). Fukaya category; homological mirror symmetry; Grassmannian Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category, Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Mirror symmetry (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Kähler manifolds Fukaya category of Grassmannians: rectangles | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 two Schubert classes \(\sigma_{\lambda}\) and \(\sigma_{\mu}\) in the quantum cohomology of a Grassmannian, we construct a partition \(\nu \), depending on \(\lambda\) and \(\mu \), such that \(\sigma_{\nu}\) appears with coefficient 1 in the lowest (or highest) degree part of the quantum product \(\sigma_{\lambda}\bigstar \sigma_{\mu}\). To do this, we show that for any two partitions \(\lambda\) and \(\mu\), contained in a \(k \times (n - k)\) rectangle and such that the \(180^{\circ}\)-rotation of one does not overlap the other, there is a third partition \(\nu\), also contained in the rectangle, such that the Littlewood-Richardson number \(c_{\lambda \mu}^{\nu}\) is 1. quantum cohomology; toric tableau; Littlewood-Richardson number Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of representation theory A note on quantum products of Schubert classes in a 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 These notes form an enlarged version of a series of four lectures, given in the Curves Seminar at Queen's in the winter of 2000. This is a purely expository account and except possibly in the commission of errors, no originality is in evidence. Since the general theorems of the subject are very well-documented, there are few complete proofs to be found here. Instead, I have consistently preferred to work out a specific low dimensional example to illustrate a theorem, and to let the reader do the general case. My intent has been to stay away from some of the notational clutter which must be suffered by anyone studying the Grassmann varieties. Research exposition (monographs, survey articles) pertaining to algebraic geometry, Grassmannians, Schubert varieties, flag manifolds Notes on Grassmannians and Schubert varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In the present article, two birational isomorphisms of the space \({\mathbb{P}}^ 6 \)and the Grassmannian G(1,4) are considered, which are determined by means of a normcurve in \({\mathbb{P}}^ 6 \)and of Segre's variety in \({\mathbb{P}}^ 9.\) Their decomposition into a product of monoidal transformations is described. decomposition into monoidal transformations; Grassmannian Grassmannians, Schubert varieties, flag manifolds, Projective techniques in algebraic geometry, Rational and birational maps Two geometric constructions of birational isomorphisms of the space \(P_ 6\) and the Grassmannian 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 \(k\) connected semisimple complex algebraic group. Fix a maximal torus \(T\) and a Borel subgroup \(B\) such that \(T\subset B\subset G\). Let \(W\) be the Weyl group of \(G\) relative to \(T\). For any \(w\) in \(W\), let \(X_w=\overline{BwB/B}\) denote the Schubert variety corresponding to \(w\). This article is concerned with the problem of whether there is a flat family over \(\text{Spec}\,\mathbb{C}[t]\) such that the general fiber is \(X_w\) and the special fiber is a toric variety.
The existence of such a degeneration was obtained by \textit{N. Gonciulea} and \textit{V. Lakshmibai} for the flag variety \(G/B\) when \(G= \text{SL}_n\) [Transform. Groups 1, 215--248 (1996; Zbl 0909.14028)]. Their proof is based on the theory of standard monomials. The cornerstone of their proof is the following: fundamental weights are minuscule weights, hence, a basis of every fundamental representation is endowed with a structure of a distributive lattice.
A complete study of the degeneration problem was made in the case when \(G\) has rank two [\textit{R. Dehy}, J. Algebra 228, 60--90 (2000; Zbl 0973.17033)]. Note also that the \(A_n\) case was studied by \textit{R. Dehy} and \textit{R. Yu} for a class of elements \(w\) in the Weyl group [J. Algebr. Comb. 10, 149--172 (1999; Zbl 0966.17004); and Ann. Inst. Fourier 51, 1525--1538 (2001; Zbl 1017.14019)]. The proofs rely on the theory of standard monomials as well. A natural question would be: is there a (flat) toric degeneration of the flag variety \(G/B\) which restricts to a toric degeneration of the Schubert varieties \(X_w\) for any \(w\) in the Weyl group? The author proves that every Schubert variety of \(G\) has a flat degeneration into a toric variety, generalizing the results cited above. The basic tool used is Lusztig's canonical basis and the string parametrization of this basis. Caldero, P.\!, Toric degenerations of Schubert varieties, Transform. Groups, 7, 51-60, (2002) Grassmannians, Schubert varieties, flag manifolds, Toric varieties, Newton polyhedra, Okounkov bodies, Fibrations, degenerations in algebraic geometry Toric degenerations of Schubert varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials For a scheme \(X\), let \(D(X)\) denote the bounded derived category of coherent sheaves on \(X\). One approach to study \(D(X)\) is to `break' it up into simpler subcategories. This notion of \textit{semiorthogonal decomposition} was developed by \textit{A. I. Bondal} and \textit{M. M. Kapranov} [Math. USSR, Izv. 35, No. 3, 519--541 (1990; Zbl 0703.14011); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 53, No. 6, 1183--1205 (1989)]. Using this notion, \textit{D. O. Orlov} [Russ. Acad. Sci., Izv., Math. 41, No. 1, 133--141 (1993; Zbl 0798.14007); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 56, No. 4, 852--862 (1992)] gave the semiorthogonal decompositions for projective, Grassmann and flag bundles. This description generalizes the full exceptional collections on such varieties by \textit{A. A. Beilinson} [Funkts. Anal. Prilozh. 12, No. 3, 68--69 (1978; Zbl 0402.14006)] and \textit{M. M. Kapranov} [Invent. Math. 92, No. 3, 479--508 (1988; Zbl 0651.18008)]. In the paper under review, the author carries out a similar analysis for twisted Grassmanninans. derived category; Azumaya algebra; twisted form of Grassmannian; semiorthogonal decomposition Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Derived categories, triangulated categories, Grassmannians, Schubert varieties, flag manifolds Semiorthogonal decompositions for 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 In this paper, due to their relevance to the fixed point set of the Peterson variety, a familiar formula for counting special involutions of the form \(a_1\), \(a_1-1\), \(a_2-2\), \(\ldots\), \(1\), \(a_2\), \(a_2-1\), \(\ldots\), \(a_1+1\), \(\ldots\), \(n\), \(n-1\), \(\ldots\), \(a_r+1\) in the symmetric group \(S_n\) is given and their correspondence to the composition set of a positive integer \(n\) is established. Lastly, the filling of composition diagrams with the set of these special involutions is also described. Peterson variety; symmetric group; compositions; involutions Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Symmetric groups On the special involutions of the symmetric group \(S_n\) | 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 The author studies the variety \(H(r,m)\) of Hankel \(r\)-planes in \(\mathbb{P}^{m}\) called a Hankel variety. In the case \(r<m-2\) \(H(r,m)\) is a proper subvariety of the Grassmannian \(G(r.m)\subseteq \mathbb{P}^{N}\), where \(N=\left( \begin{matrix} m+1 \\ r+1 \end{matrix} \right) -1 \). This subvariety is invariant under the action of special projectivities \(\omega\) of \(\mathbb{P}^{N}\) strictly linked to the standard rational normal curve in \(\mathbb{P}^{N}\). The locus of Hankel \(l+1\)-planes containing a non Hankel \(l\)-plane is described. It is shown that the singular locus of \(H(r,m)\) is closely related to invariant subvarieties of \(H(r,m)\) under the action of certain projectivities \(\overline{\omega}\) of \(\mathbb{P}^{N}\) induced by the projectivities \(\omega\). Grassmannian; singularities; Hankel variety; projective space; standard rational normal curve G. Failla, \textit{On certain Loci of Hankel r-planes of }P\textit{m}, Mathematical Notes 92 (2012) 4, 544--553. Projective techniques in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds On certain loci of Hankel \(r\)-planes of \(\mathbb P^m\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We discuss the construction of higher-dimensional surfaces based on the harmonic maps of \(S^2\) into \(\mathbb{C}\mathbb{P}^{N-1}\) and other Grassmannians. We show that there are two ways of implementing this procedure -- both based on the use of the relevant projectors. We study various properties of such projectors and show that the Gaussian curvature of these surfaces, in general, is not constant. We look in detail at the surfaces corresponding to the Veronese sequence of such maps and show that for all of them this curvature is constant but its value depends on which mapping is used in the construction of the surface.{
\copyright 2010 American Institute of Physics} Hussin, V; Yurduşen, I; Zakrzewski, WJ, Canonical surfaces associated with projectors in Grassmannian sigma models, J. Math. Phys., 51, 103509, (2010) Harmonic maps, etc., Grassmannians, Schubert varieties, flag manifolds, Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations), Differential geometric aspects of harmonic maps Canonical surfaces associated with projectors in Grassmannian sigma models | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The main object of the paper under review is a generalized Severi-Brauer \(K\)-variety \(\mathrm{SB}(r,A)\). This is the \(K\)-form of the Grassmann variety \(\mathbb G(r,n)\) which represents the right \(rn\)-dimensional ideals of a central simple \(K\)-algebra \(A\) of degree \(n\). The author shows that for every \(0<r<n\), the variety \(\mathrm{SB}(r,A)\) is \(K\)-birationally equivalent to the direct product of \(\mathrm{SB}(g.c.d.(n,r),A)\) and some projective \(K\)-space. Another result establishes, for central simple \(K\)-algebras \(A\), \(B\) of coprime degrees, a \(K\)-birational equivalence of the classical Severi--Brauer variety \(\mathrm{SB}(A\otimes_KB)\) and the direct product of \(\mathrm{SB}(A)\times _K\mathrm{SB}(B)\) by some projective \(K\)-space. (Note that these statements are stronger than just stable \(K\)-birational equivalence of varieties in question.)
These results are deduced, by twisting, from the corresponding birational isomorphisms of Grassmann varieties equivariant with respect to the natural action of the group \(\mathrm{PGL}_1(A)\).
As an application, the author obtains some statements in the spirit of Amitsur's conjecture, such as the following one: if \(A\) and \(A'\) are central simple \(K\)-algebras of degree \(n\) which generate the same subgroup of the Brauer group \(\mathrm{Br}(K)\), then for any \(r\) prime to \(n\) such that \(3\leq r\leq n-3\) the varieties \(\mathrm{SB}(r,A)\) and \(\mathrm{SB}(r,A')\) are \(K\)-birationally equivalent. Grassmann variety; Severi-Brauer variety; Amitsur's conjecture; torsor M. Florence, Géométrie birationnelle équivariante des grassmanniennes, J. reine angew. Math., 674, 81-98, (2013) Grassmannians, Schubert varieties, flag manifolds, Other nonalgebraically closed ground fields in algebraic geometry Equivariant birational geometry 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 \textit{A. Björner} and \textit{T. Ekedahl} [Ann. Math. (2) 170, No. 2, 799--817 (2009; Zbl 1226.05268)] prove that general intervals \([e, w]\) in Bruhat order are ``top-heavy'', with at least as many elements in the \(i\)-th corank as the \(i\)-th rank. Well-known results of \textit{J. B. Carrell} [Proc. Symp. Pure Math. 56, 53--61 (1994; Zbl 0818.14020)] and of \textit{V. Lakshmibai} and \textit{B. Sandhya} [Proc. Indian Acad. Sci., Math. Sci. 100, No. 1, 45--52 (1990; Zbl 0714.14033)] give the equality case: \([e, w]\) is rank-symmetric if and only if the permutation \(w\) avoids the patterns 3412 and 4231 and these are exactly those \(w\) such that the Schubert variety \(X_w\) is smooth. In this paper we study the finer structure of rank-symmetric intervals \([e, w]\), beyond their rank functions. In particular, we show that these intervals are still ``top-heavy'' if one counts cover relations between different ranks. The equality case in this setting occurs when \([e, w]\) is self-dual as a poset; we characterize these \(w\) by pattern avoidance and in several other ways. Weyl group; Bruhat order; Schubert variety; intersection cohomology; Kazhdan-Lusztig polynomial Combinatorial aspects of algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Classical problems, Schubert calculus Self-dual intervals in the Bruhat order | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Deligne-Illusie-Raynaud (see [\textit{P. Deligne} and \textit{L. Illusie}, Invent. Math. 89, 247--270 (1987; Zbl 0632.14017)]) have shown that if a smooth projective variety \(X\) over an algebraically closed field \(K\) of characteristic \(p>0\) is liftable to \(W_2(K)\) (the ring of the second Witt vectors) and \(\dim X\leq p\), then the Kodaira vanishing theorem holds on \(X\). Although every \(K3\) surface in positive characteristic can be lifted to characteristic zero, there do exist some Calabi-Yau threefolds in characteristic 2 or 3 which cannot be lifted to characteristic zero. For instance, \textit{T. Ekedahl} [``On non-liftable Calabi-Yau threefolds'', preprint, \url{arXiv:math/0306435}] proved that the Hirokado variety is not liftable to \(W_2(K)\) or characteristic zero. The paper under review proved a Kodaira-type vanishing theorem on the Hirokado variety \(X\), explicitly, \(H^1(X,L^{-1})=0\) holds for any ample line bundle \(L\) with \(H^0(X,L^3)\neq 0\). The proof is a consequence of Ekedahl's interpretation of the Hirokado variety as a Deligne-Lusztig type variety associated to the Grassmannian \(\mathrm{Gr}(2,4)\) together with the theory of pre-Tango structure. Rudakov, A.N., Shafarevich, I.R.: Inseparable morphisms of algebraic surfaces. (Russ.) Izv. Akad. Nauk SSSR Ser. Mat. \textbf{40}(6), 1269-1307 (1976). 1439 Vanishing theorems in algebraic geometry, Calabi-Yau manifolds (algebro-geometric aspects), Positive characteristic ground fields in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds Kodaira type vanishing theorem for the Hirokado 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 A flag domain \(D\) is an open orbit of a real form \(G_{0}\) in a flag manifold \(Z = G/P\) of its complexification. If \(D\) is holomorphically convex, then, since it is a product of a Hermitian symmetric space of bounded type and a compact flag manifold, \(\mathrm{Aut}(D)\) is easily described. If \(D\) is not holomorphically convex, then in previous work it was shown that \(\mathrm{Aut}(D)\) is a Lie group whose connected component at the identity agrees with \(G_{0}\), except possibly in situations which arise in Onishchik's list of flag manifolds where \(\mathrm{Aut}(Z)^{0} =\hat{ G}\) is larger than \(G\). In the present work the group \(\mathrm{Aut}(D)^{0} =\hat{ G}_{0}\) is described as a real form of \(\hat{G}\). Using an observation of Kollar, new and much simpler proofs of much of our previous work in the case where \(D\) is not holomorphically convex are given. flag domains; automorphism groups; finiteness theorem Grassmannians, Schubert varieties, flag manifolds, Complex Lie groups, group actions on complex spaces, Noncompact Lie groups of transformations, Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects) Cycle connectivity and automorphism groups of flag domains | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 investigate properties of modules introduced by \textit{W. Kraśkiewicz} and \textit{P. Pragacz} [C. R. Acad. Sci., Paris, Sér. I 304, 209--211 (1987; Zbl 0642.13011); Eur. J. Comb. 25, No. 8, 1327--1344 (2004; Zbl 1062.14065)] which realize Schubert polynomials as their characters. In particular, we give some characterizations of modules having filtrations by Kraśkiewicz-Pragacz modules. In finding criteria for such filtrations, we calculate generating sets for the annihilator ideals of the lowest vectors in Kraśkiewicz-Pragacz modules and derive a projectivity result concerning Kraśkiewicz-Pragacz modules. Schubert polynomials; Kraśkiewicz-Pragacz modules Watanabe, M.: An approach toward Schubert positivities of polynomials using kraśkiewicz-pragacz modules. European J. Combin. 58, 17-33 (2016) Combinatorial aspects of representation theory, Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds An approach towards Schubert positivities of polynomials using Kraśkiewicz-Pragacz modules | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Bernstein, Frenkel, and Khovanov have constructed a categorification of tensor products of the standard representation of \(\mathfrak{sl}_2\), where they use singular blocks of category \(\mathcal{O}\) for \(\mathfrak{sl}_n\) and translation functors. Here we construct a positive characteristic analogue using blocks of representations of \(\mathfrak{s}\mathfrak{l}_n\) over a field \(\mathbf{k}\) of characteristic \(p\) with zero Frobenius character, and singular Harish-Chandra character. We show that the aforementioned categorification admits a Koszul graded lift, which is equivalent to a geometric categorification constructed by Cautis, Kamnitzer, and Licata using coherent sheaves on cotangent bundles to Grassmanians. In particular, the latter admits an abelian refinement. With respect to this abelian refinement, the stratified Mukai flop induces a perverse equivalence on the derived categories for complementary Grassmanians. This is part of a larger project to give a combinatorial approach to Lusztig's conjectures for representations of Lie algebras in positive characteristic. categorification; modular representation; Fourier-Mukai transform; localization Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.), Grassmannians, Schubert varieties, flag manifolds, Classical groups (algebro-geometric aspects) Categorification via blocks of modular representations for \(\mathfrak{sl}_n\) | 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 Let \(X\) be a partial flag variety, stratified by orbits of the Borel. We give a criterion for the category of modular perverse sheaves to be equivalent to modules over a Koszul ring. This implies that modular category \(\mathcal O\) is governed by a Koszul-algebra in small examples. Weidner, Jan, Grassmannians and Koszul duality, Math. Z., 0025-5874, 278, 3-4, 1033-1064, (2014) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Grassmannians, Schubert varieties, flag manifolds, Quadratic and Koszul algebras Grassmannians and Koszul duality | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials From the abstract: Given a homogeneous ideal \(I \) in a polynomial ring over a field, one may record, for each degree \(d\) and for each polynomial \(f\in I_d\), the set of monomials in \(f\) with nonzero coefficients. These data collectively form the \(tropicalization\) of \(I\) . Tropicalizing ideals induces a ``matroid stratification'' on any (multigraded) Hilbert scheme. Very little is known about the structure of these stratifications. In this paper, we explore many examples of matroid strata, including some with interesting combinatorial structure, and give a convenient way of visualizing them. We show that the matroid stratification in the Hilbert scheme of points (\(\mathbb{P}^1)^{[k]}\) is generated by all Schur polynomials in \(k\) variables. We end with an application to the \(T\) -graph problem of (\(\mathbb{A}^2)^{[n]}\); classifying this graph is a longstanding open problem, and we establish the existence of an infinite class of edges. Schur polynomials; tropical ideals; multigraded Hilbert schemes Parametrization (Chow and Hilbert schemes), Algebraic combinatorics The matroid stratification of the Hilbert scheme of points on \(\mathbb{P}^1\) | 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 Let \(G\) be the group scheme \(\mathrm{SL}_{d+1}\) over \(\mathbb{Z}\) and let \(Q\) be the parabolic subgroup scheme corresponding to the simple roots \(\alpha_2,\cdots,\alpha_{d-1}\). Then \(G/Q\) is the \(\mathbb{Z}\)-scheme of partial flags \(\{D_1\subset H_d\subset V\}\). We will calculate the cohomology modules of line bundles over this flag scheme. We will prove that the only non-trivial ones are isomorphic to the kernel or the cokernel of certain matrices with multinomial coefficients. cohomology; line bundles; flag schemes; Weyl modules; multinomial coefficients Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of algebraic geometry, Representation theory for linear algebraic groups On the cohomology of line bundles over certain flag schemes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We derive a family of high-order, structure-preserving approximations of the Riemannian exponential map on several matrix manifolds, including the group of unitary matrices, the Grassmannian manifold, and the Stiefel manifold. Our derivation is inspired by the observation that if \(\Omega\) is a skew-Hermitian matrix and \(t\) is a sufficiently small scalar, then there exists a polynomial of degree \(n\) in \(t\Omega\) (namely, a Bessel polynomial) whose polar decomposition delivers an approximation of \(e^{t\Omega}\) with error \(O(t^{2n+1})\). We prove this fact and then leverage it to derive high-order approximations of the Riemannian exponential map on the Grassmannian and Stiefel manifolds. Along the way, we derive related results concerning the supercloseness of the geometric and arithmetic means of unitary matrices. matrix manifold; retraction; geodesic; Grassmannian; Stiefel manifold; Riemannian exponential; matrix exponential; polar decomposition; unitary group; geometric mean; Karcher mean E. S. Gawlik and M. Leok, \textit{High-Order Retractions on Matrix Manifolds Using Projected Polynomials}, preprint, , 2017. Other matrix algorithms, Local Riemannian geometry, Grassmannians, Schubert varieties, flag manifolds, Factorization of matrices High-order retractions on matrix manifolds using projected 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 Consider the ring \(\mathcal{S}\) of symmetric polynomials in \(k\) variables over an arbitrary base ring \(\mathfrak{k}\). Fix \(k\) scalars \(a_1, a_2, \ldots, a_k\) in \(\mathfrak{k}\). Let \(I\) be the ideal of \(\mathcal{S}\) generated by \(h_{n-k+1}-a_1, h_{n-k+1}-a_2,\ldots, h_{n-k+1}-a_k\), where \(h_i\) is the \(i\)-th complete homogeneous symmetric polynomial.
The quotient ring \(\mathbf{S}/I\) generalizes both the usual and the quantum cohomology of the Grassmannian.
We show that \(\mathbf{S}/I\) has a \(\mathfrak{k}\)-module basis consisting of (residue classes of) Schur polynomials fitting into a \(k \times (n-k)\)-rectangle; and that its multiplicative structure constants satisfy the same \(S_3\)-symmetry as those of the Grassmannian cohomology. We conjecture the existence of a Pieri rule (proven in two particular cases) and a positivity property generalizing that of Gromov-Witten invariants. symmetric functions; partitions; Schur functions; Gröbner bases; Grassmannian; cohomology Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Classical problems, Schubert calculus A quotient of the ring of symmetric functions generalizing quantum 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 \(\mathbb{F}_q\) denote a finite field having \(q\) elements, where \(q\) is a prime power. For a vector space \(E\) of finite dimension \(m\) over \(\mathbb{F}_q\) and \(\ell\le m\), let \(G(\ell, m)\) denote the Grassmannian variety of vector subspaces of dimension \(\ell\) of \(E\). A projective variety \(X\) is called a linear section of the of the Grassmannian \(G(\ell,m)\) if \(X=G(\ell, m)\cap Z(g_1,\ldots, g_N)\), where \(g_1, g_2, \ldots, g_N\) are linearly independent functionals in the ideal that they generate and \(X(\mathbb{F}_q)=\{P_1, \ldots, P_M\}\) is a non-empty set of \(\mathbb{F}_q\)-rational points of \(X\). In this paper, authors study parity-check codes by showing that for every linear section of a Grassmannian, there exists a parity check code with good properties depending on the linear sections. For the Lagrangian-Grassmannian variety, they reveal that these parity-check codes are the low density parity check (LDPC) codes. They also obtained some properties of parity check codes associated to linear sections of Grassmannians. algebraic geometry codes; Grassmann codes; Lagrangian-Grassmannian codes; Schubert codes; parity check codes; LDPC codes Grassmannians, Schubert varieties, flag manifolds, Applications to coding theory and cryptography of arithmetic geometry, Geometric methods (including applications of algebraic geometry) applied to coding theory, Linear codes (general theory) Codes on linear sections of the Grassmannian | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author investigates the algebro-geometric structure of Drinfeld-Anderson motives (generalizing Anderson's \(t\)-motives for the case of arbitrary global function fields). In the first part of the paper, he studies shtukas related to Drinfeld-Anderson motives. The main result of the second part is a theorem of uniformization of such motives using Sato-Grassmannians. \(A\)-motives; shtukas; Sato Grassmannians; Drinfeld-Anderson motives Potemine, I. Yu.: Drinfeld -- Anderson shtukas and uniformization of A-motives via Sato grassmannians. Contrib. algebra geom. 41 (2001) Drinfel'd modules; higher-dimensional motives, etc., Grassmannians, Schubert varieties, flag manifolds Drinfeld-Anderson shtukas and uniformization of \(A\)-motives via Sato 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 This paper is concerned with the study of rational points on certain projective varieties over number fields. These varieties are fiber bundles over generalized flag varieties and the fibers are equivariant compactifications of algebraic tori. Before giving a detailed description we will explain the basic problem in general and elementary terms.
We have restricted ourself to the case of split tori and split groups because
this simplifies some technical details. The general case can be treated similarly.
We consider these results as an important step towards an understanding of the arithmetic of spherical varieties. For example, choosing \(P = B\) a Borel subgroup, \(T = U\backslash B\) where \(U\) is the unipotent radical of \(B\) and \(\eta: B\to T\) the natural projection, we obtain an equivariant compactification of \(U\backslash G\), a horospherical variety. Twisted products over \(P\backslash G\) have also been studied in [M. Strauch, Height zeta functions of fibre bundles over generalized flag varieties (German). Bonn. Math. Schrift. 309 (1998; Zbl 0922.14011)], where the fiber is a flag variety (of a different type). In this situation the asymptotic behavior turns out to be the predicted one. We give a brief description of the remaining sections.
Section 2 recalls the relevant facts we need concerning generalized flag varieties, i.e., a description of line bundles on \(W = P\backslash G\), the cone of effective divisors in \(\text{Pic}(W)_R\), metrization of line bundles, height zeta functions. The exposition is based entirely on the paper [\textit{J. Franke}, \textit{Yu. I. Manin} and \textit{Y. Tschinkel}, Invent. Math. 95, 421--435 (1989; Zbl 0674.14012)].
The next section contains the corresponding facts for toric varieties. It is a
summary of a part of [\textit{V. V. Batyrev} and \textit{Y. Tschinkel}, Int. Math. Res. Not. 1995, No. 12, 591--635 (1995; Zbl 0890.14008)]. We give the explicit calculation of the Fourier transform \(\hat H_\Sigma(\cdot ; \varphi)\) and show that Poisson's summation formula can be used to give an expression of the height zeta function \(Z_T (\mathcal L_\phi; s)\).
In Section 4 we introduce twisted products, discuss line bundles on them, the
Picard group, metrizations of line bundles, etc. It ends with a formula for the height zeta function
\[
Z_{Y^o} (\mathcal L^Y_\varphi\otimes\pi^*\mathcal L_\lambda, s)
\]
in the domain of absolute convergence.
The first part of Section 5 explains the method for the proof that the height
zeta function can be continued meromorphically to a halfspace beyond the abscissa of absolute convergence. Moreover, we state a theorem which gives a description of the coefficient of the Laurent series at the pole in question. This coefficient will be the leading one, provided that it does not vanish. One can relate the coefficient to arithmetic and geometric invariants of the pair \((U;\mathcal L)\) but we decided not to pursue this, since there are detailed expositions of all the necessary arguments in [\textit{E.~Peyre}, Duke Math. J. 79, 101--218 (1995; Zbl 0901.14025), \textit{V. V. Batyrev} and \textit{Y. Tschinkel} [Int. Math. Res. Not. 1995, No. 12, 591-635 (1995; Zbl 0890.14008), and Astérisque 251, 299--340 (1998; Zbl 0926.11045)].
These two theorems (meromorphic continuation of certain integrals and the
description of the coefficient) will be proved in a more general context in Section
7. The second part of Section 5 contains the proof that the hypothesis of
these theorems are fulfilled in our case. It ends with the main theorem on the
asymptotic behavior of the counting function \(N_{Y^o}(\mathcal L;H)\), assuming that the coefficient of the Laurent series mentioned above does not vanish. Section 6 is devoted to the proof of this fact. In Section 8 we prove some statements on Eisenstein series (well-known to the experts) which are used in Section 5. And finally, in Section 9 we explain in detail some special cases of our main theorem. Asymptotics of rational points; height zeta functions Strauch M., Tschinkel Y.: Height zeta functions of toric bundles over flag varieties. Selecta Math. 5, 352--396 (1999) Arithmetic varieties and schemes; Arakelov theory; heights, Varieties over global fields, Heights, Rational points, Grassmannians, Schubert varieties, flag manifolds, Toric varieties, Newton polyhedra, Okounkov bodies Height zeta functions of toric bundles over 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 The Schur polynomials \(s_{\lambda }\) are essential in understanding the representation theory of the general linear group. They also describe the cohomology ring of the Grassmannians. For \(\rho = (n, n-1, \dots , 1)\) a staircase shape and \(\mu \subseteq \rho\) a subpartition, the Stembridge equality states that \(s_{\rho /\mu } = s_{\rho /\mu^T}\). This equality provides information about the symmetry of the cohomology ring. The stable Grothendieck polynomials \(G_{\lambda }\), and the dual stable Grothendieck polynomials \(g_{\lambda }\), developed by \textit{A. S. Buch} [Acta Math. 189, No. 1, 37--78 (2002; Zbl 1090.14015)], \textit{T. Lam} and \textit{P. Pylyavskyy} [Int. Math. Res. Not. 2007, No. 24, Article ID rnm125, 48 p. (2007; Zbl 1134.16017)], are variants of the Schur polynomials and describe the \(K\)-theory of the Grassmannians. Using the Hopf algebra structure of the ring of symmetric functions and a generalized Littlewood-Richardson rule, we prove that \(G_{\rho /\mu } = G_{\rho /\mu^T}\) and \(g_{\rho /\mu } = g_{\rho /\mu^T}\), the analogues of the Stembridge equality for the skew stable and skew dual stable Grothendieck polynomials. Stembridge equality; Grothendieck polynomial; Young tableau; Hopf algebra Symmetric functions and generalizations, Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Connections of Hopf algebras with combinatorics The Stembridge equality for skew stable Grothendieck polynomials and skew dual stable Grothendieck polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The Schubert calculus on the space of \(d\)-dimensional linear subspaces of a smooth \(n\)-dimensional quadric lying in the complex projective space is the object of study in this article. Following Hodge and Pedoe the author develops the intersection theory of this space in a purely combinatorial manner. It is proved in particular that if a triple intersection of Schubert cells on this space is nonempty then a certain combinatorial relation holds among the Schubert symbols involved, similar to the classical one. The sufficiency of these necessary conditions is also studied. Several examples are examined to illustrate the necessity and sufficiency of these conditions. subspaces of a quadric; Schubert calculus; intersection theory; intersection of Schubert cells Sertöz, S.: A triple intersection theorem for the varieties \(SO(n)/pd\), Fund. math. 142, No. 3, 201-220 (1993) Grassmannians, Schubert varieties, flag manifolds, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry A triple intersection theorem for the varieties \(SO(n)/P_ d\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 a formula of \textit{S. C. Billey}, \textit{W. Jockusch} and \textit{R. P. Stanley} [Some combinatorial properties of Schubert polynomials, J. Algebr. Comb. 2, No. 4, 345-374 (1993; Zbl 0790.05093)], \textit{S. Fomin} and \textit{A. N. Kirillov} [Yang-Baxter equation, symmetric functions, and Schubert polynomials, Proceedings of the conference on power series and algebraic combinatorics, Firenze (1993)] have introduced a new set of diagrams that encode the Schubert polynomials. In this paper, these objects are called rc-graphs. Here, two variants of an algorithm for constructing the set of all rc-graphs for a given permutation are defined and proved. This construction makes many of the identities known for Schubert polynomials more apparent, and yields new ones. In particular, we find a new proof of Monk's rule using an insertion algorithm on rc- graphs. This insertion rule is a generalization of the Schensted insertion for tableaux. We find two conjectures of analogs of Pieri's rule for multiplying Schubert polynomials. The authors also extend the algorithm to generate the double Schubert polynomials. Schubert polynomials; rc-graphs; Monk's rule; Pieri's rule N. Bergeron and S. Billey. ''RC-graphs and Schubert polynomials''. Experiment. Math. 2 (1993), pp. 257--269.DOI. Symmetric functions and generalizations, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Enumerative combinatorics, Combinatorial identities, bijective combinatorics, Permutations, words, matrices, Grassmannians, Schubert varieties, flag manifolds rc-graphs 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 The subject of this paper is the big quantum cohomology rings of symplectic isotropic Grassmannians $\mathrm{IG}(2, 2n)$. We show that these rings are regular. In particular, by ``generic smoothness'', we obtain a conceptual proof of generic semisimplicity of the big quantum cohomology for $\mathrm{IG}(2, 2n)$. Further, by a general result of \textit{C. Hertling} [Frobenius manifolds and moduli spaces for singularities. Cambridge: Cambridge University Press (2002; Zbl 1023.14018)], the regularity of these rings implies that they have a description in terms of isolated hypersurface singularities, which we show in this case to be of type $A_{n-1}$. By the homological mirror symmetry conjecture, these results suggest the existence of a very special full exceptional collection in the derived category of coherent sheaves on $\mathrm{IG}(2, 2n)$. Such a collection is constructed in the appendix by Alexander Kuznetsov. semisimplicity of quantum cohomology; unfoldings of singularities; Lefschetz exceptional collections Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Mirror symmetry (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Gromov-Witten invariants, quantum cohomology, Frobenius manifolds On quantum cohomology of Grassmannians of isotropic lines, unfoldings of $A_n$-singularities, and Lefschetz exceptional collections | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(k\) be a finite field, \(\overline{k}\) an algebraic closure of \(k\), \(L = \overline{k}((t))\) the field of Laurent series, and \(\sigma\) the Frobenius on \(\overline{k}\) and \(L\). Let \(G\) be a split connected reductive group over \(k\), let \(A\) be a split maximal torus, and \(B\) a Borel subgroup containing \(A\).
The affine Deligne-Lusztig variety associated with an element \(b\in G(L)\) and a dominant coweight \(\mu \in X_*(A)\) is by definition
\[
X^G_\mu(b) = X_\mu(b) = \{ g \in G(L)/K;\;g^{-1}b\sigma(g) \in K t^\mu K \}.
\]
Here \(t^\mu\) denotes the image of \(t\) under the homomorphism \(L^\times = \mathbb G_m(L) \rightarrow A(L) \subset G(L)\). This is a locally closed subset of the affine Grassmannian \(G(L)/K\). The closed affine Deligne-Lusztig variety is defined as the union
\[
X_{\leq \mu}(b) = \bigcup_{\lambda \leq \mu} X_\lambda(b).
\]
It is a closed subset of \(G(L)/K\). We consider both of these sets as subschemes (with the reduced scheme structure). They are schemes locally of finite type over \(\overline{k}\). These constructions are obviously analogous to the construction of (usual) Deligne-Lusztig varieties. In the current situation, the root system is replaced by the affine root system, and the (partial) flag variety is replaced by a (partial) affine flag variety, specifically by the affine Grassmannian. We can identify the set of connected components of \(G(L)/K\) with the algebraic fundamental group \(\pi_1(G)\) of \(G\), and denote by \(\kappa_G \colon G(L)/K \rightarrow \pi_1(G)\) the induced map.
In the paper under review, the sets of connected components of these schemes are studied. To simplify the statements, let us assume throughout this review that the group \(G\) is simple. First, there is the Hodge-Newton decomposition (which was essentially already obtained by \textit{R. E. Kottwitz} [Int. Math. Res. Not. 2003, No. 26, 1433--1447 (2003; Zbl 1074.14016)]): Take a standard parabolic subgroup \(P \subseteq G\) with Levi subgroup \(A\subset M \subset P\) which contains the centralizer of the Newton vector of \(b\). We may then assume that \(b\in M\), and in this situation, if \(\kappa_M(b)=\mu\), then the natural inclusion \(X_\mu^M(b) \rightarrow X_\mu^G(b)\) is an isomorphism.
Now assume that \(G\) does not contain a proper standard parabolic subgroup to which we can apply the Hodge-Newton decomposition (the pair \((b,\mu)\) being fixed such that \(X_\mu(b)\neq\emptyset\)). Then either \(b\) is \(\sigma\)-conjugate to the translation element \(t^\mu\) and \(t\mu\) is central, in which case \(X_\mu(b)\cong X_{\leq \mu}(b) \cong G(k((t)))/G(k[[t]])\) are discrete, or \(\kappa_G\) induces a bijection \(\pi_0(X_{\leq \mu}(b)) \cong \pi_1(G)\).
Finally, those cases where \(X_\mu(b)\) has dimension \(0\) are characterized. It is also shown that the set of connected components of \(X_\mu(b)\) and \(X_{\leq\mu}(b)\) are different in general. affine Deligne-Lusztig varieties Viehmann E.: Connected components of affine Deligne--Lusztig varieties. Math. Ann. 340, 315--333 (2008) Grassmannians, Schubert varieties, flag manifolds, Algebraic moduli problems, moduli of vector bundles, Formal groups, \(p\)-divisible groups, Linear algebraic groups over local fields and their integers Connected components of closed 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 Sei \(G\) eine topologische Gruppe, \(d\) eine rechtsinvariante Metrik auf \(G\) und \(H\) eine abgeschlossene Untergruppe von \(G\). Der Verf. zeigt, daß dann der Raum \((G\setminus H)/H\) der Linksnebenklassen von \(H\) in \(G\) durch die Hausdorffmetrik \(h_ d\) metrisierbar ist und daß zwischen \(h_ d\) und d folgende Beziehungen bestehen: (1) \(h_ d(xH,yH)=\inf_{u,v\in H}d(xu,yv)=d(x,yH)\). (2) Eine Folge \(x_ nH\) konvergiert gegen \(xH\) bezüglich \(h_ d\) genau dann, wenn eine Folge \(y_ n\) in \(G\) existiert mit \(y_ n\equiv x_ n (mod H)\) und \(y_ n\to x\) bezüglich \(d\). (3) Ist d eine zweiseitig invariante Metrik auf \(G\), so wirkt \(G\) invariant auf \(G\setminus H\) bezüglich \(h_ d\). Ist \(H\) sogar ein Normalteiler von \(G\), so ist \(h_ d\) eine zweiseitig invariante Metrik auf der Faktorgruppe \(G/H\).
Sei d eine zweiseitig invariante Metrik auf \(G\) und \(K_ H\) der Ineffektivitätskern der Untergruppe \(H\) in \(G\), d.h. die Menge \(\{ u;\) \(u\in G\) mit \(uxH=xH\) für alle \(x\in G\}\). Dann wirkt die mit \(h_ d\) metrisierte Gruppe \(G/K_ H\) effektiv und invariant auf dem mit \(h_ d\) metrisierten Raum \(G/H\). Die Konvergenz in \(G/K_ H\) impliziert die uniforme Konvergenz der entsprechenden Isometrien auf \(G\setminus H.\)
Außerdem zeigt der Verf. folgende Sachverhalte: Für die volle euklidische Bewegungsgruppe B von \({\mathbb{R}}^ n\) gibt es keine zweiseitig invariante Metrik, die die kompakt-offene Topologie auf B induzierte. Die orthogonale Gruppe \(O_ n({\mathbb{R}})\) wirkt auf der Stiefelmannigfaltigkeit \(\text{Stief}(k,{\mathbb{R}}^ n)\) der orthonormierten k-Beine des Vektorraums \({\mathbb{R}}^ n\) sowohl bezüglich der Hausdorffmetrik (auf \(\text{Stief}(k,{\mathbb{R}}^ n))\) als auch bezüglich der punktweisen Metrik (für zwei k-Beine \((b_ i)\) und \((b'_ i)\) ist der Abstand durch \(\max_{i=1,...,k}\| b_ i-b'_ i\|\) gegeben) invariant; diese beiden Metriken sind verschieden, induzieren aber dieselbe Topologie.
Bekanntlich gibt es eine kanonische Abbildung \(\phi\) von \(\text{Stief}(k,{\mathbb{R}}^ n)\) in die Graßmannmannigfaltigkeit \(\text{Grass}(k,{\mathbb{R}}^ n)\) aller k-dimensionalen Teilräume von \({\mathbb{R}}^ n:\) Für \(B\in \text{Stief}(k,{\mathbb{R}}^ n)\) ist \(\phi(B)\) der Aufspann von \(B\). In \(\text{Grass}(k,{\mathbb{R}}^ n)\) gibt es eine ''Projektormetrik'' pr, die durch die orthogonalen Projektionen \(P_ M\) von \({\mathbb{R}}^ n\) auf k-dimensionale Teilräume M bestimmt ist; für k-dimensionale Teilräume M, N definiert man \(pr(M,N)=\| P_ M-P_ N\|\). Die Gruppe \(O_ n({\mathbb{R}})\) wirkt auf \(\text{Grass}(k,{\mathbb{R}}^ n)\) invariant (ineffektiv) sowohl bezüglich der Hausdorff- als auch bezüglich der Projektormetrik; diese beiden Metriken sind zwar verschieden, induzieren auf \(\text{Grass}(k,{\mathbb{R}}^ n)\) aber dieselbe Topologie, und die kanonische Abbildung \(\phi\) ist offen. Stiefel manifold; Grassmannian; homogeneous spaces; metric groups; Hausdorff distance; topological group; invariant metric; Hausdorff; metric; quotient space Structure of general topological groups, Metric spaces, metrizability, General properties and structure of locally compact groups, General properties and structure of real Lie groups, Grassmannians, Schubert varieties, flag manifolds On homogeneous spaces of metric 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 Consider the affine space consisting of pairs of matrices \((A,B)\) of fixed size, and its closed subvariety given by the rank conditions \(\operatorname{rank} A\leq a\), \(\operatorname{rank} B\leq b\), and \(\operatorname{rank}(A\cdot B)\leq c\), for three non-negative integers \(a,b,c\). These varieties are precisely the orbit closures of representations for the equioriented \(\mathbb{A}_3\) quiver. In this paper we construct the (equivariant) minimal free resolutions of the defining ideals of such varieties. We show how this problem is equivalent to determining the cohomology groups of the tensor product of two Schur functors of tautological bundles on a 2-step flag variety. We provide several techniques for the determination of these groups, which is of independent interest. Representations of quivers and partially ordered sets, Grassmannians, Schubert varieties, flag manifolds, Group actions on varieties or schemes (quotients), Determinantal varieties, Syzygies, resolutions, complexes and commutative rings, Sheaves in algebraic geometry Minimal free resolutions of ideals of minors associated to pairs of matrices | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a semisimple algebraic group and \(B \subset G\) be a Borel subgroup. For \(P_1,P_2,\dots,P_r\) parabolic subgroups of \(G\) containing \(B\) one can define a Bott-Samelson variety as
\[
(P_1\times ^B P_2\times ^B \dots \times ^B P_r)/B.
\]
These varieties are also described as desingularizations of Schubert varieties in the complete flag variety \(G/B\) and can be constructed recursively as towers of \({\mathbb P}^1\)-bundles (for details, see Section 3 of the paper under review). In this last interpretation, Bott-Samelson varieties have been useful to characterize some varieties whose all elementary contractions are \({\mathbb P}^1\)-bundles, as complete flag varieties (see [\textit{G. Occhetta} et al., Annali della Scuola normale superiore di Pisa, Classe di scienze XVII(2): 573--607 (2017; Zbl 1390.14128)]). The idea is to use the intersection matrix of the relative anticanonical bundles of the contractions dot their different fibers and show it is the Cartan matrix of a particular semisimple Lie algebra. Then prove, via a recursive construction of Bott-Samelson varieties starting form a point, that the variety is isomorphic to the complete flag variety determined by this Lie algebra. This process needs to control the construction of the \({\mathbb P}^1\)-bundle structure at any step which has to be compatible with that of the corresponding flag bundle. For groups \(G\) whose Dynkin diagram is simply laced, this has been done in the reference above, because the corresponding \(H^1\)'s are one dimensional, so that the construction is unique; and also for those whose Dynkin diagram is not simply laced, except \(F_4\) and \(G_2\), by means of the choice of what the authors call a \textit{good word}, avoiding possible choices of \(H^1\) groups of dimension bigger than one. This paper completes the picture dealing with the cases \(F_4\) and \(G_2\), previously considered by different techniques. In this cases a good word cannot be chosen. In the paper under review the authors interpretate the excess of parameters for these two groups as the existence of a geometrical structure (orthogonal or skew-symmetric) on a partial flag for which the complete flag is defined using the notion of isotropy with respect to this structure. Moreover, they show that, also for \(F_4\) and \(G_2\), one can choose a \textit{flag-compatible} word, which satisfies that two successors can be different as \({\mathbb P}^1\)-bundles but isomorphic as varieties, unifying the characterization of complete flag varieties. Fano manifolds; rational homogeneous manifolds; Bott-Samelson varieties Fano varieties, Minimal model program (Mori theory, extremal rays), Grassmannians, Schubert varieties, flag manifolds, Homogeneous spaces and generalizations Deformation of Bott-Samelson varieties and variations of isotropy structures | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors study the lifting of the Schubert stratification of the homogeneous space of complete real flags of \(\mathbb R^{n+1}\) to its universal covering group \(\mathrm{Spin}_{n+1}\). They call the lifted strata the Bruhat cells of \(\mathrm{Spin}_{n+1}\), in keeping with the homonymous classical decomposition of reductive algebraic groups. They present explicit parameterizations for these Bruhat cells in terms of minimal-length expressions \(\sigma = a_{i_1} \dots a_{i_k}\) for permutations \(\sigma \in S_{n+1}\) in terms of the n generators \(a_i = (i, i + 1)\). These parameterizations are compatible with the Bruhat orders in the Coxeter-Weyl group \(S{n+1}\). This stratification is an important tool in the study of locally convex curves; they present a few such applications. Schubert stratifications; signed Bruhat cells; Bruhat stratifications; locally convex curves; Coxeter group; symmetry group Grassmannians, Schubert varieties, flag manifolds, Stratifications in topological manifolds, Covering spaces and low-dimensional topology Locally convex curves and the Bruhat stratification of the spin group | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The sheaf of principal parts \(J^k(E)\) has been studied by many authors (DiRocco, Grothendieck, Laksov, Maakestad, Perkinson, Piene, Sommese etc). The sheaf \(J^k(E)\) of an \(O_X\)-module \(E\) where \(X\) is a scheme has a left and right structure as \(O_X\)-module and the left structure of \(J^k(E)\) has been studied by several authors in the case where \(X\) is projective \(n\)-space over an algebraically closed field of characteristic zero. The aim of the paper under review is to complete this study and to relate it to the theory of representations of quivers.
The projective space \(\mathbb{P}(V^*)\) may be realized as a quotient \(\mathrm{SL}(V)/P\) where \(P\) is a parabolic subgroup of \(\mathrm{SL}(V)\) and there is an equivalence of categories between the category of \(P\)-modules and the category of \(\mathrm{SL}(V)\)-linearized vector bundles on \(\mathbb{P}=\mathbb{P}(V^*)\). The category \(C\) of vector bundles on \(P\) with an \(\mathrm{SL}(V)\)-linearization is an abelian category, hence by Freyd's full embedding theorem it follows the category \(C\) is equivalent to a full subcategory of the category \(mod(A)\) of left modules on an associative ring \(A\). One aim of the paper is to describe the associative ring \(A\) associated to projective space \(P\) and to construct the \(A\)-module corresponding to the sheaf of principal parts \(J^k(E)\).
In section one of the paper the author gives the motivation for the writing of the paper and the main results of the paper.
In section two of the paper the author gives the definition of the sheaf of principal parts \(J^k(E)\) of any \(O_X\)-module \(E\) on any scheme \(X\) following the standard construction using the infinitesimal neighborhood of the diagonal and mentions some properties of this construction: He gives a description of the fiber of the principal parts, the fundamental exact sequences and the relationship with sheaves of differential operators.
In section three the author introduce the concepts of algebraic groups, homogeneous spaces and homogeneous vector bundles and mention the fact that if \(E\) is a \(G\)-linearized homogeneous vector bundle on a homogeneous space \(G/H\) it follows \(J^*(E)\) has a canonical \(G\)-linearization. The author mentions the notions of a Cartan decomposition of a parabolic Lie algebra, the notion of a maximal weight vector of a \(p\)-module where \(p\) is a parabolic Lie algebra and the notion of an irreducible homogeneous vector bundle. The author ends section three with an introduction to the notion of quiver representations. He also constructs the quiver \(Q_V\) associated to projective space \(\mathbb{P}=\mathbb{P}(V^*)\). He moreover gives an explicit construction of the equivalence between the category of representations of \(Q_V\) and the category of homogeneous vector bundles on \(P\).
In section four the author mentions known results on \(J^k(O(d))\) on \(P\) and introduces some notions defined in [\textit{D. Perkinson}, Compos. Math. 104, 27--39 (1996; Zbl 0895.14016)]. He uses these notions and some explicit formulas to prove the existence of a decomposition \(J^k(O(d))\cong Q_{k,d}\oplus J^d(O(d))\) where \(Q_{k,d}\) is an explicitly defined vector bundles on \(P\). The author defines a map
\[
n^{k-d}:S^k(V)\otimes O(d-k) \rightarrow S^k(V) \otimes O_P
\]
and an isomorphism \(Q_{k,d}\cong \ker(n^{k-d})\). He proves that the map \(n^{k-d}\) is an \(\mathrm{SL}(V)\)-invariant differential operator. The author moreover proves some properties of the bundles \(Q_{k,d}\) and use these properties to give an explicit construction of the \(Q_V\)-representation of \(J^k(O(d))\). The paper ends with a proof of the fact that the Taylor truncation map has maximal rank in the cases where \(h \leq k\).
Note: In Proposition 4.6 the author states the existence of an \(\mathrm{SL}(V)\)-equivariant isomorphism
\[
J^k(O(d)) \cong S^k(V) \otimes O(d-k)
\]
In other papers [the reviewer, Proc. Am. Math. Soc. 133, No. 2, 349--355 (2005; Zbl 1061.14040)] it was proved that \(J^k(O(d))\cong S^k(V^*)\otimes O(d-k)\). One may suspect there is an error in the paper since \(S^k(V)\) and \(S(V^*)\) are different as \(\mathrm{SL}(V)\)-modules. The difference between the author's paper and the reviewer's paper is that Re is considering \(\mathbb{P}(V)\) -- projective space parametrizing lines in \(V^* \) where Maakestad is considering \(\mathbb{P}(V^*)\) -- projective space parametrizing lines in \(V\). principal parts; quiver representation; stable; vector bundle; projective space Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Grassmannians, Schubert varieties, flag manifolds, Sheaves and cohomology of sections of holomorphic vector bundles, general results Principal parts bundles on projective spaces and quiver 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 Kohnert's algorithm for the generation of Schubert polynomials is derived from Monk's rule for the multiplication of Schubert polynomials. Schubert polynomials R. Winkel. ''A derivation of Kohnert's algorithm from Monk's rule''. Sém. Lothar. Combin. 48 (2002), Art. B48f.URL. Symmetric functions and generalizations, Combinatorial aspects of representation theory, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds A derivation of Kohnert's algorithm from Monk's rule | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials As a result of the solution of Horn's conjecture and the saturation conjecture by \textit{A. Klyachko} [Sel. Math., New Ser. 4, No. 3, 419--445 (1998; Zbl 0915.14010)], and \textit{A. Knutson} and \textit{T. Tao} [J. Am. Math. Soc. 12, No. 4, 1055--1090 (1999; Zbl 0944.05097)], one can decide when an intersection of Schubert classes in a Grassmannian is non-zero by solving a series of inequalities coming from the answer to similar questions for smaller Grassmannians. This indicates that there should be geometric proofs of the results of Klyachko and Knutson/Tao [see \textit{W. Fulton}, Astérisque 252, 255--269, Exp. No. 845 (1998; Zbl 0929.15006)]. In this article the author provides such proofs and obtains results that generalize the original conjecture of Horn, and also give a stronger form of the saturation conjecture.
For relations to the characterization of eigenvalues of a sum of hermitian matrices in terms of the summands, and the connections to representation theory, combinatorics and geometric invariant theory see the beautiful article [\textit{W. Fulton}, Bull. Am. Math. Soc., New Ser. 37, No. 3, 209--249 (2000; Zbl 0994.15021)]. Schubert classes; Grassmannians; hermitian matrices; eigenvalues; transversality Geometric proofs of Horn and saturation conjectures. \textit{Journal of Alge-} \textit{braic Geometry }15(2006), 133--173.arXiv:math/0208107.Zbl 1090.14014 MR 2177198 Grassmannians, Schubert varieties, flag manifolds, Determinantal varieties, Representation theory for linear algebraic groups, Classical problems, Schubert calculus Geometric proofs of Horn and saturation conjectures | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 give combinatorial formulas for the Hilbert coefficients, \(h\)-polynomial and the Cohen-Macaulay type of Schubert varieties in Grassmannians in terms of the posets associated with them. As a consequence, necessary conditions for a Schubert variety to be a complete intersection and combinatorial criteria are given for a Schubert variety to be Gorenstein and almost Gorenstein, respectively. Grassmannian; Schubert variety; Hilbert coefficients; Gorenstein ring; almost Gorenstein ring Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Dimension theory, depth, related commutative rings (catenary, etc.), Grassmannians, Schubert varieties, flag manifolds Hilbert coefficients of Schubert varieties in 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 This paper is concerned with the drapeau theorem for differential systems. By a differential system \((R,D)\), we mean a distribution \(D\) on a manifold \(R\). The derived system \(\partial D\) is defined, in terms of sections, by \(\partial \mathcal{D}=\mathcal{D}+[\mathcal{D},\mathcal{D}]\). Moreover higher derived systems \(\partial^i D\) are defined by \(\partial^i D =\partial(\partial^{i-1} D)\). The differential system \((R,D)\) is called regular if \(\partial^i D\) are subbundles of \(TR\) for every \(i\geq 1\). We say that \((R,D)\) is an \(m\)-flag of length \(k\), if \((R,D)\) is regular and has a derived length \(k\), i.e., \(\partial^k D =TR\), such that \(\operatorname{rank}D=m+1\) and \(\operatorname{rank}\partial^{i} D=\partial^{i-1}D+m\) for \(i=1,\dots k\). Especially \((R,D)\) is called a Goursat flag (un drapeau Goursat) of length \(k\) when \(m=1\).
The main purpose of this paper is to clarify the procedure of ``rank 1 prolongation'' of an arbitrary differential system \((R,D)\) of rank \(m+1\), and to give good criteria for an \(m\)-flag of length \(k\) to be special. A generalisation of the drapeau theorem for an \(m\)-flag of length \(k\) for \(m\geq 2\) is proved. differential system; Goursat flag; \(m\)-flag of length \(k\); rank 1 prolongation K. Shibuya and K. Yamaguchi, Drapeau theorem for differential systems, Differential Geometry and its Applications 27 (2009), 793--808. Vector distributions (subbundles of the tangent bundles), General geometric structures on manifolds (almost complex, almost product structures, etc.), Exterior differential systems (Cartan theory), Jets in global analysis, Grassmannians, Schubert varieties, flag manifolds Drapeau theorem for differential 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 Grassmannian \(\mathrm{Gr}(d,n)\) parametrizing the \(d\)-planes inside \(\Bbbk^n\) can be seen as a subvariety of some projective space via the Plücker embedding. Its intersection \(\mathrm{Gr}_0(d,n)\) with the main strata of the projective space consists of \(d\)-planes with non-zero Plücker coordinates. The tropicalization \(T\mathrm{Gr}_0(d,n)\) of \(\mathrm{Gr}_0(d,n)\) is a fan inside \(\Lambda^d\mathbb{R}^n\) that can be seen from several points of view:
\begin{itemize}
\item First, it is the image by the valuation of the \(\mathbb{K}\)-points of \(G_0(d,n)\) for \(\mathbb{K}\) some algebraically closed valued field with surjective valuation,
\item Then, it is the moduli space of linear tropical space that are realizable over valued extensions of \(\Bbbk\),
\item Finally, it consists of the vectors \(w\in\Lambda^d\mathbb{R}^n\) such that the initial degeneration \(\mathrm{in}_w \mathrm{Gr}_0(d,n)\) is non-empty. The goal of the paper is to study the properties of some of the initial degenerations of the Grassmannian.
\end{itemize}
The cones of \(T \mathrm{Gr}(d,n)\) are in bijection with some subdivisions \(\Delta_w\) of the hypersimplex \(\Delta(d,n)\) into matroid polytopes. Here, \(w\) is some point that belongs to the relative interior of the considered cone. As the ideals giving the initial degenerations \(\mathrm{in}\mathrm{Gr}_0(d,n)\) can be quite difficult to handle, even with a computer, the paper studies them using a closed embedding inside some varieties constructed from \(\Delta_w\) which are easier to manipulate. The description of this variety is as follows.
To each \(d\)-plane \(L\) inside \(\Bbbk^n\) is associated a matroid obtained via the hyperplane arrangement that the coordinate hyperplanes of \(\Bbbk^n\) realize inside the subspace \(L\). Such a matroid is called realizable. Conversely, to each realizable matroid \(M\), we can consider the \textit{thin Schubert cell} \(\mathrm{Gr}_M\subset\mathrm{Gr}(d,n)\) consisting of subspaces inducing the matroid \(M\).
Consider the subdivision \(\Delta_w\) of \(\Delta(d,n)\) into matroid polytopes. Let \(Q\) be one of the matroid polytopes and let \(M_Q\) be the corresponding matroid. A face \(Q'\) of \(Q\) corresponds to a matroid \(M_{Q'}\) which is the direct sum of a quotient and a contraction of \(M_Q\). Using this description, the paper gives a way to lift the inclusion \(Q'\subset Q\) to an application on the level of thin Schubert cells corresponding to \(M_Q\) and \(M_{Q'}\): \(\mathrm{Gr}_{M_Q}\rightarrow \mathrm{Gr}_{M_{Q'}}\). Therefore, it is possible to consider the inverse limit \[\varprojlim_{Q\in\Delta_w}\mathrm{Gr}_{M_Q},\] associated to the subdivision obtained from a cone of \(T\mathrm{Gr}_0(d,n)\).
For an element \(w\) in the fan \(T \mathrm{Gr}_0(d,n)\), the paper then constructs a closed immersion \[\mathrm{in}_w\mathrm{Gr}_0(d,n)\rightarrow\varprojlim_{Q\in\Delta_w}\mathrm{Gr}_{M_Q}.\] It then uses it to deduce results such as the smoothness and irreducibleness of the initial degenerations of \(\mathrm{Gr}_0(3,7)\), the fact that \(\mathrm{Gr}_0(3,9)\) has a non-connected initial degeneration.
They also use the setting to prove the \(n=7\) case of the following conjecture of Keel and Tevelev [26]. Let \(X(3,n)\) be the normalization of the quotient of \(\mathrm{Gr}(3,n)\) by a maximal torus \(H\subset \mathrm{PGL}(d)\), and \(X_0(3,n)\) the same quotient with \(\mathrm{Gr}_0(3,n)\) instead. As it is already known that \(X(3,n)\) is not log-canonical for \(n\geqslant 9\), the conjecture states that \(X(3,n)\) is a schön and log canonical compactification for \(X_0(3,n)\) when \(n=6,7\) and \(8\). The conjecture was already proven for \(n=6\) by
\textit{M. Luxton} [The log canonical compactification of the moduli space of six lines in \({{\mathbb{P}}}^2\). Texas: University of Texas at Austin (PhD Thesis) (2008)]. Grassmannian; initial degeneration; tropical; matroid Combinatorial aspects of tropical varieties, Grassmannians, Schubert varieties, flag manifolds, Embeddings in algebraic geometry Initial degenerations 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 Tropical algebraic geometry is a branch of algebraic geometry where an algebraic variety is studied by transforming it into a ``polyhedral object'' called its \textit{tropicalization}. The past decade has seen an upsurge in characterizing properties of the algebraic variety that are captured by tropicalization and its applications. Grassmanians are an important class of algebraic varieties and the study of the tropical geometry associated with them goes back to [\textit{D. Speyer} and \textit{B. Sturmfels}, Adv. Geom. 4, No. 3, 389--411 (2004; Zbl 1065.14071)]. Real Grassmanians have a subset associated to them that has lately received a lot of attention, namely its positive part also known as the \textit{positive Grassmanian} [\textit{L. Williams}, ``The positive Grassmanian, the amplituhedron, and cluster algebras'', Preprint, \url{arXiv:2110.10856}].
Extending the work of [\textit{D. Speyer} and \textit{L. Williams}, J. Algebr. Comb. 22, No. 2, 189--210 (2005; Zbl 1094.14048)] for the positive part of both \(\mathrm{Gr}(3,6)\) and \(\mathrm{Gr}(3,7)\), the authors provide a combinatorial-geometric description of the tropicalization of the positive part of the Grassmanian \(\mathrm{Gr}(k,n)\) (Theorem 6.4). The main idea is to study configuration spaces of \(n\) principal flags for \(\mathrm{SL}_k\) and describe its tropicalization in terms of objects called \textit{higher laminations} [\textit{I. Le}, Geom. Topol. 20, No. 3, 1673--1735 (2016; Zbl 1348.30023)]. Roughly speaking, a higher lamination is an equivalences class of certain configurations of points in the affine building of \(\mathrm{SL}_k\). The authors then study the tropicalization of a map between the configuration space of \(n\) principal flags for \(\mathrm{SL}_k\) and the (affine cone) over the Grassmanian \(\mathrm{Gr}(k,n)\). The fibers of this tropicalization map are equivalence classes of higher laminations called \textit{horocycle laminations}, leading the authors to the combinatorial-geometric description of the tropicalization of the positive Grassmanian. The paper is largely self-contained with sections on relevant topics such as cluster algebras, configuration spaces and affine buildings. Geometric aspects of tropical varieties, Cluster algebras, Grassmannians, Schubert varieties, flag manifolds Tropicalization of positive 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 describe moduli spaces of invariant generalized complex structures and moduli spaces of invariant generalized Kähler structures on maximal flag manifolds under \(B\)-transformations. We give an alternative description of the moduli space of generalized complex structures using pure spinors, and describe a cell decomposition of these moduli spaces induced by the action of the Weyl group generalized geometry; flag manifolds; homogeneous spaces Moduli problems for differential geometric structures, Generalized geometries (à la Hitchin), Grassmannians, Schubert varieties, flag manifolds Invariant generalized complex geometry on maximal flag manifolds and their moduli | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 a family of spaces called \textit{critical varietieso}. The positive part of each critical variety is a subset of one of Postnikov's positroid cells inside the totally nonnegative Grassmannian. The combinatorics of positroid cells is captured by the dimer model on a planar bipartite graph \(G\), and the critical variety is obtained by restricting to Kenyon's critical dimer model associated to a family of isoradial embeddings of \(G\). This model is invariant under square/spider moves on \(G\), and we give an explicit boundary measurement formula for critical varieties which does not depend on the choice of \(G\). Special cases include critical electrical networks and Baxter's critical \(Z\)-invariant Ising model associated to rhombus tilings of polygons in the plane. In the case of regular polygons, our formula yields new simple expressions for response matrices of electrical networks and for correlation matrices of the Ising model. Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of algebraic geometry Critical 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 authors study the actions of a complex Lie group on a Grassmann manifold. The approach is elementary in that they use a Theorem of \textit{W. L. Chow} [Ann. Math. (2) 50, 32--67 (1949; Zbl 0040.22901)] as presented by \textit{St. Lojasiewicz} in his book [`Introduction to complex analytic geometry', Birkhäuser (1991; Zbl 0747.32001)] and the theory of constructible sets [Lojasiewicz, loc. cit.]. actions of complex Lie groups; Grassmann manifold Complex Lie groups, group actions on complex spaces, Grassmannians, Schubert varieties, flag manifolds, General properties and structure of complex Lie groups On the holomorphic actions of a complex Lie group on the Grassmannian manifold | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The following is a classic open problem in the Schubert calculus of the flag variety:
Given any reasonably nice subvariety \(Y \subset \mathrm{Flags}(\mathbb C^n)\), express the homology class of \(Y\) as an integral linear combination of Schubert classes.
In this paper, the authors consider the case where \(Y\) is the Peterson variety. By analyzing the cellular structure of the Peterson variety and the group action of a one-dimensional torus on this variety, they reduce the computations in the intersection theory of the flag variety to a systematic combinatorial analysis of the elements of the symmetric group.
In the process, they give a partial solution to the first problem introduced above.
their proof counts the points of intersection between certain Schubert varieties in the full flag variety and the Peterson variety, and shows that these intersections are proper and transverse. Schubert calculus; intersection theory; Peterson variety Insko, Erik, Schubert calculus and the homology of the Peterson variety, Electron. J. Combin., 22, 2, (2015) Grassmannians, Schubert varieties, flag manifolds, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Classical problems, Schubert calculus, Classical real and complex (co)homology in algebraic geometry Schubert calculus and the homology of the Peterson 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 present the linearized metrizability problem in the context of parabolic geometries and sub-Riemannian geometry, generalizing the metrizability problem in projective geometry studied by \textit{R. Liouville} [J. de l'Éc. Polyt. cah. LIX. 7--76 (1889; JFM 21.0317.02)].
The paper under review is concerned with bracket-generating distributions arising in parabolic geometries, which are Cartan-Tanaka geometries modelled on homogeneous spaces \(G/P\), where \(G\) is a semisimple Lie group and \(P\) a parabolic subgroup of \(G\). On a manifold \(M\) equipped with such a parabolic geometry, each tangent space is modelled on the \(P\)-module \(g/p\).
First, motivating examples are provided:
1. Parabolic geometries and Weyl structures.
2. Projective parabolic geometries.
3. Parabolic geometries on filtered manifolds.
4. \(BGG\) operators, local metrizability of the homogeneous model, and normal solutions.
Next, a general method for linearizability and a classification of all cases with irreducible defining distribution where this method applies are given. These tools lead to natural sub-Riemannian metrics on generic distributions of interest in geometric control theory.
By using an algebraic linearization condition, the authors establish a linearization principle. The metrizability procedure is illustrated by showing how the well-known example of projective geometry fits into the general method. In addition, the metric tractor bundle is investigated.
Let \(p_0=p/{p^\perp}\) and \(h\) a \(p_0\)-module. The main result is a classification of metric parabolic geometries with irreducible \(h\).
Finally, examples of reducible cases where the linearized metrizability procedure works are given. Cartan geometry; parabolic geometry; Bernstein-Gelfand-Gelfand resolution; projective metrizability; sub-Riemannian metrizability; overdetermined linear PDE; Weyl connections Other connections, Sub-Riemannian geometry, Grassmannians, Schubert varieties, flag manifolds, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Semisimple Lie groups and their representations, General geometric structures on manifolds (almost complex, almost product structures, etc.), Differential geometry of homogeneous manifolds, Natural bundles, Invariance and symmetry properties for PDEs on manifolds, Nonlinear systems in control theory Subriemannian metrics and the metrizability of parabolic geometries | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 article, the authors study the flop of the total space of a certain kind of vector bundle on a Grassmannian to the total space of a bundle on the dual Grassmannian. They show these spaces have many derived equivalences. From these, they show that the derived category on each space has many non-trivial auto-equivalences. These auto-equivalences are algebraically constructed as window-shifts. In this case, the authors give a geometric interpretation of one of the window-shifts as a spherical twist.
In order to achieve all these results, the authors first demonstrate the principles of their arguments using the example of the standard Atiyah flop in 3 dimensions. Following this, they give a clear exposition of the general heuristic argument. Finally, the authors conclude with the necessary detailed arguments. derived categories; Grassmannians; derived equivalences; spherical functors Donovan, W., Segal, E.: Window shifts, flop equivalences and Grassmannian twists. Compos. Math. \textbf{150}(6), 942-978 (2014). arXiv:1206.0219 Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Derived categories, triangulated categories, Grassmannians, Schubert varieties, flag manifolds Window shifts, flop equivalences and Grassmannian twists | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We show a canonical injective morphism from the quantum cohomology ring \(QH^\ast (G/P)\) to the associated graded algebra of \(QH^\ast (G/B)\), which is with respect to a nice filtration on \(QH^\ast (G/B)\) introduced by Leung and the author. This tells us the vanishing of a lot of genus zero, three-pointed Gromov-Witten invariants of flag varieties \(G/P\).
For Part I, see [\textit{N. C. Leung} and \textit{C. Li}, J. Differ. Geom. 86, No. 2, 303--354 (2010; Zbl 1316.14107)]. quantum cohomology; flag varieties; filtered algebras; Gromov-Witten invariants Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds Functorial relationships between \(QH^\ast (G/B)\) and \(QH^\ast (G/P)\). 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 Consider the commutative unipotent group \((\mathbb{G}_a)^n\) over the field of complex numbers \(\mathbb{C}\). It is an important problem to study equivariant compactifications of the group \((\mathbb{G}_a)^n\). In other words, we are interested in actions with an open orbit of the group \((\mathbb{G}_a)^n\) on complete \(n\)-dimensional algebraic varieties \(X\).
In [\textit{B. Hassett} and \textit{Y. Tschinkel}, Int. Math. Res. Not. 1999, No. 22, 1211--1230 (1999; Zbl 0966.14033)], it was shown that equivariant compactifications of \((\mathbb{G}_a)^n\) with \(X\) being a projective space \(\mathbb{P}^n\) are in bijection with commutative associative local algebras over \(\mathbb{C}\) of dimension \(n+1\); see also [\textit{F. Knop} and \textit{H. Lange}, Math. Ann. 267, 555--571 (1984; Zbl 0544.14028)]. In particular, starting from \(n=6\) the number of equivalence classes of such compactifications is infinite.
Hassett and Tschinkel [Zbl 0966.14033] asked the same question for \(X\) being a non-degenerate projective quadric. By [\textit{E. V. Sharoǐko}, Sb. Math. 200, No. 11, 1715--1729 (2009; Zbl 1205.13030); translation from Mat. Sb. 200, No. 11, 145--160 (2009)], in this case an equivariant compactification of \((\mathbb{G}_a)^n\) exists and is unique.
Let \(G\) be a semisimple complex linear algebraic group, \(P\) a parabolic subgroup of \(G\), and \(X=G/P\) the corresponding homogeneous space. Such varieties \(X\) are called generalized flag varieties, they are known to be the only complete homogeneous spaces of linear algebraic groups. In [\textit{I. V. Arzhantsev}, Proc. Am. Math. Soc. 139, No. 3, 783--786 (2011; Zbl 1217.14032)], all homogeneous spaces \(G/P\) that admit an action with open orbit of the group \((\mathbb{G}_a)^n\) are found, and the question on the uniqueness of such an action is raised. In [\textit{B. Fu} and \textit{J.-M. Hwang}, Math. Res. Lett. 21, No. 1, 121--125 (2014; Zbl 1327.32030)], the uniqueness result is proved for a wide class of projective varieties including the Grassmanians \(\text{Gr}(k,m)\) different from projective spaces. The latter are precisely the varieties of the form \(G/P\) with \(G=\text{SL}(m)\) that admit an action of the group \((\mathbb{G}_a)^n\) with an open orbit.
In the paper under review, the uniqueness result is obtained for all generalized flag varieties \(G/P\), which are different from projectvie spaces and admit an action of the group \((\mathbb{G}_a)^n\) with an open orbit. The author establishes a correspondence between such actions and nilpotent multiplications on the nilpotent radical of the corresponding parabolic Lie subalgebra considered as an \(L\)-module with respect to the adjoint action of the Levi subgroup \(L\) of the parabolic subgroup \(P\).
Let \(V\) be a finite-dimensional module of a reductive algebraic group \(L\). One says that a bilinear map \(V\times V\to V, (v,w)\mapsto v\cdot w\), is an \(L\)-compatible nilpotent multiplication if this map is commutative, associative, the operator of multiplication \(V\to V, w\mapsto v\cdot w\) by any element \(v\in V\) is nilpotent and coincides with the operator \(V\to V, w\mapsto xw\), for some \(x\) in the Lie algebra of the group \(L\).
In Theorem 21, a classification of \(L\)-compatible nilpotent multiplications on simple modules \(V\) for a simple algebraic group \(L\) is obtained. This classification leads to the uniqueness result (Theorem 25).
For uniqueness results for non-commutative unipotent group actions with an open orbit on generalized flag varieties, see [\textit{D. Cheong}, Transform. Groups 22, No. 1, 163--186 (2017; Zbl 1454.14126)]. semisimple group; parabolic subgroup; flag variety; commutative unipotent group; nilpotent algebra S. Billy, V. Lakshimibai, \textit{Singular Loci of Schubert Varieties}, Progress in Mathematics, Vol. 182, Birkhäuser, Boston, 2000. Grassmannians, Schubert varieties, flag manifolds, Group actions on varieties or schemes (quotients) Unipotent commutative group actions on flag varieties and nilpotent multiplications | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 for any positive integers \(n_1,n_2,\ldots,n_k\) there exists a real flag manifold \(F(1,\ldots,1,n_1,n_2,\ldots,n_k)\) with cup-length equal to its dimension. Additionally, we give a necessary condition that an arbitrary real flag manifold needs to satisfy in order to have cup-length equal to its dimension. cup-length; flag manifold; Lyusternik-Shnirel'man category Grassmannians, Schubert varieties, flag manifolds, Lyusternik-Shnirel'man category of a space, topological complexity à la Farber, topological robotics (topological aspects), Algebraic topology of manifolds On real flag manifolds with cup-length equal to its dimension. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 reductive algebraic group over an algebraically closed field \(\Bbbk\) of positive characteristic \(p\), \(B\) a Borel subgroup of \(G\), \(P\) a minimal parabolic subgroup of \(G\) containing \(B\), and \(\pi\colon G/B\to G/P\) the natural morphism. Using Orlov's semiorthogonal decomposition of the bounded derived category of coherent sheaves on \(G/B\) to those on \(G/P\), \textit{A. Samokhin} [J. Algebra 324, No. 6, 1435-1446 (2010; Zbl 1221.14055)] derived a short exact sequence relating the Frobenius direct image of the structure sheaf of \(G/B\) to that of \(G/P\), which was a key to his proof of the \(D\)-affinity of \(G/B\) for the symplectic group \(G\) of degree 4 over \(\Bbbk\). In this note we obtain his exact sequence from a short exact sequence of \(G_nB\)-modules, \(G_n\) the \(n\)-th Frobenius kernel of \(G\), using representation theory. reductive algebraic groups; Borel subgroups; parabolic subgroups; short exact sequences; Frobenius kernels; derived categories of coherent sheaves; semiorthogonal decompositions; Frobenius direct images; representations Representation theory for linear algebraic groups, Grassmannians, Schubert varieties, flag manifolds, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Cohomology theory for linear algebraic groups On a lemma of Samokhin. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 irreducible decompositions of the degeneracy loci of matrices which are defined by rank conditions on upper left submatrices. By introducing the concepts of standard and essential rank functions, we give an explicit classification of these degeneracy loci. Based on standard rank functions, we design an algorithm to write a degeneracy locus as a union of its irreducible components. This gives an answer to a problem raised by Sturmfels. degeneracy loci; irreducible decomposition; Ehresmann-Bruhat-Chevalley partial order Determinantal varieties, Algebraic combinatorics, Combinatorics of partially ordered sets Irreducible decompositions of degeneracy loci of matrices | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(B_r\) be the polynomial ring with \(r\) variables over the rational field \(\mathbb Q\). A \(\mathbb Q\)-basis for \(B_r\) can be given in terms of polynomials called Schur determinants \(S_\lambda\), where \(\lambda\) is a partition of length at most \(r\), and \(S_\lambda\) is the quotient of the determinant of a matrix whose rows are given by powers of the variables defined by \(\lambda\), divided by the corresponding Vandermonde determinant on the variables. This basis determines an isomorphism of \(B_r\) with the \(r\)-th exterior algebra of a vector space \(V=\bigoplus_{i\geq 0} \mathbb Q\cdot b_i\) of infinite countable dimension. Thus, one obtains via contraction, for all \(k\), an action on the polynomial ring \(B_r\) of the Lie algebra \(gl(\bigwedge^k V)\) of finite endomorphisms of \(\bigwedge^k V\). The main results of the paper under review provides an expression of the \(gl(\bigwedge^k V)\)-action on \(B_r\) through a compact formula. The authors determine the formula in terms of certain operators \(\Gamma\) on \(\bigwedge V\). These operators \(\Gamma\) are an approximate version of the vertex operators occurring in the bosonic vertex representation of the Lie algebra of all matrices of infinite size. exterior algebras; vertex operators; Hasse-Schmidt derivations; bosonic and fermionic representations; symmetric functions Grassmannians, Schubert varieties, flag manifolds, Exterior algebra, Grassmann algebras, Vertex operators; vertex operator algebras and related structures, Symmetric functions and generalizations Polynomial ring representations of endomorphisms of exterior powers | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 articles of this volume will be reviewed individually. Proceedings, conferences, collections, etc. pertaining to nonassociative rings and algebras, Lie algebras and Lie superalgebras, Quantum groups and related algebraic methods applied to problems in quantum theory, Representation theory for linear algebraic groups, Schur and \(q\)-Schur algebras, Grassmannians, Schubert varieties, flag manifolds, Proceedings of conferences of miscellaneous specific interest Recent advances in representation theory, quantum groups, algebraic geometry, and related topics. AMS special sessions on ``Geometric and algebraic aspects of representation theory and quantum groups'' and ``Noncommutative algebraic geometry'', Tulane University, New Orleans, LA, USA, October 13--14, 2012 | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In the paper under review the author studies algebraic varieties defined by minors of a fixed size of a matrix such that all minors are restricted to lie in a ladder shaped region. The main result are explicit formulas for the Hilbert functions and Hilbert series related to these varieties. In particular, the author shows that, although the formulas in the general case are complicated, they can be used to derive fairly simple estimates for some useful geometric invariants of the varieties such as the degree of the Hilbert polynomial. determinantal ideals; Hilbert functions; determinantal varieties; Schubert varieties; ladder determinantal varieties; Hilbert series DOI: 10.1016/S0012-365X(01)00256-4 Determinantal varieties, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Exact enumeration problems, generating functions, Combinatorial identities, bijective combinatorics, Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds, Linkage, complete intersections and determinantal ideals Hilbert functions of ladder determinantal varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In the paper under review, the authors study intersections of quadrics, components of the hypersurface in the Grassmannian $\mathrm{Gr} (3, \mathbb{C}^{3})$, in the context of classical point-line arrangements. Let $\mathcal{A}$ be a generic arrangement in $\mathbb{C}^{3}$ and $\mathcal{A}_{\infty}$ the arrangement of lines in $H_{\infty} \cong \mathbb{P}^{2}$ directions at infinity of planes in $\mathcal{A}$. The space of generic arrangements of $n$ lines in $(\mathbb{P}^{2})^{n}$ is a Zariski open set $U$ in the space of all arrangements of $n$ lines in $(\mathbb{P}^{2})^{n}$. On the other hand in $\mathrm{Gr}(3, \mathbb{C}^n)$ there is an open set $U'$ consisting of $3$-spaces intersecting each coordinate hyperplane transversally (i.e. having dimension of intersection equal $2$). Observe that generic arrangements in $\mathbb{C}^{3}$ can be regarded as points in $\mathrm{Gr}(3, \mathbb{C}^n)$. Let $\{s_1 < \dots < s_6 \} \subset \{1, \dots , n\}$ be a set of indices of a generic arrangement $A = \{H^{0}_{1}, \dots, H^{0}_{n}\}$ in $\mathbb{C}^{3}$, $\alpha_{i}$ the normal vectors of $H^{0}_{i}$'s and $\beta_{i\,j\,l} = \det(\alpha_{i}, \alpha_{j}, \alpha_{l})$. For any permutation $\sigma \in S_{6}$ denote by $\sigma = \{\{i_{1},i_{2}\},\{i_{3},i_{4}\},\{i_{5},i_{6}\}\}$, $i_{j} = s_{\sigma(j)}$, and $Q_{\sigma}$ the quadric in $\mathrm{Gr}(3,\mathbb{C}^{n})$ given by the equation $\beta_{i_{1}\, i_{3} \, i_{4}} \beta_{i_{2} \, i_{5} \, i_{6}} - \beta_{i_{2} \,i_{3} \, i_{4}} \beta_{i_{1} \, i_{5} \, i_{6}} = 0$.
Main theorem (Pappus's theorem). For any disjoint classes $[\sigma_{1}]$ and $[\sigma_{2}]$, there exists a unique class $[\sigma_{3}]$ disjoint from $[\sigma_{1}]$ and $[\sigma_{2}]$ such that $\{Q_{\sigma_{1}}, Q_{\sigma_{2}}, Q_{\sigma_{3}}\}$ is a Pappus configuration, i.e. \[ Q_{\sigma_{i_{1}}} \cap Q_{\sigma_{i_{2}}} = \bigcap_{i=1}^{3} Q_{\sigma_{i}} \] for any $\{i_1, i_2 \} \subset [3]$.
In the rest of the paper, the authors retrieve the Hesse configuration of lines studying intersections of six quadrics of the form $Q_{\sigma}$ for opportunely chosen $[\sigma]$. It is worth emphasizing that the study is made in a wider context of the so-called discriminantal arrangements. discriminantal arrangements; intersection lattice; Grassmannian; Pappus's theorem Arrangements of points, flats, hyperplanes (aspects of discrete geometry), Combinatorial aspects of matroids and geometric lattices, Grassmannians, Schubert varieties, flag manifolds Pappus's theorem in Grassmannian \(Gr(3, \mathbb{C}^n)\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G'\) be a complex semisimple Lie group, \(Q\) be a parabolic subgroup and \(G\) be a real form of \(G'\).
In this paper the authors concentrate on the CR structure of the minimal orbit \(M\) in \(G'/Q\). Each of these \(M\) is in one to one correspondence associated to a special parabolic CR algebra which is called minimal. The authors achieve the classification of minimal orbits through the classification of parabolic minimal CR algebras. They compute the fundamental group of \(M\) and they show, under a certain condition, that all \(G\)-homogeneous CR manifolds that are locally CR diffeomorphic to \(M\) are simply connected and globally CR diffeomorphic to \(M\). Moreover every \(G\)-homogeneous CR manifold that is locally CR diffeomorhic to \(M\) admits a fundamental reduction. complex flag manifolds; homogeneous CR manifolds; minimal orbits of a real form; parabolic CR algebra A. Altomani - C. Medori - M. Nacinovich, The CR structure of minimal orbits in complex flag manifolds, J. Lie Theory, 16 (2006), pp. 483--530. Zbl1120.32023 MR2248142 CR structures, CR operators, and generalizations, Grassmannians, Schubert varieties, flag manifolds, Simple, semisimple, reductive (super)algebras, General properties and structure of real Lie groups, Homogeneous complex manifolds The CR structure of minimal orbits in complex flag manifolds | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We give an inductive procedure for finding the extremal rays of the equivariant Littlewood-Richardson cone, which is closely related to the solution space to S. Friedland's majorized Hermitian eigenvalue problem. In so doing, we solve the ``rational version'' of a problem posed by \textit{C. Robichaux} et al. [Lond. Math. Soc. Lect. Note Ser. 473, 284--335 (2022; Zbl 1489.14033)]. Our procedure is a natural extension of P. Belkale's algorithm for the classical Littlewood-Richardson cone. The main tools for accommodating the equivariant setting are certain foundational results of \textit{D. Anderson} et al. [Compos. Math. 149, No. 9, 1569--1582 (2013; Zbl 1286.15023)]. We also study two families of special rays of the cone and make observations about the Hilbert basis of the associated lattice semigroup. Horn's conjecture; Hermitian matrices; extremal rays; equivariant Littlewood-Richardson cone Inequalities involving eigenvalues and eigenvectors, Positive matrices and their generalizations; cones of matrices, Hermitian, skew-Hermitian, and related matrices, Eigenvalues, singular values, and eigenvectors, Inequalities and extremum problems involving convexity in convex geometry, Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of algebraic geometry, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Classical problems, Schubert calculus, Equivariant algebraic topology of manifolds Extremal rays of the equivariant Littlewood-Richardson cone | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 count the \(\mathbb F_q\)-rational points of GIT quotients of quiver representations with relations. We focus on two types of algebras -- one is one-point extended from a quiver \(Q\), and the other is the Dynkin \(A_2\) tensored with \(Q\). For both, we obtain explicit formulas. We study when they are polynomial-count. We follow the similar line as in the first paper [J. Algebra 372, 542-559 (2012; Zbl 1283.16014)] but algebraic manipulations in Ringel-Hall algebra will be replaced by corresponding geometric constructions. quiver representations; quivers with relations; Ringel-Hall algebras; GIT quotients; one-point extensions; representation varieties; moduli spaces; quiver Grassmannians; quiver flags; tensor product algebras; polynomial-counts; positivity Representations of quivers and partially ordered sets, Algebraic moduli problems, moduli of vector bundles, Enumerative problems (combinatorial problems) in algebraic geometry, Cluster algebras, Quantum groups (quantized enveloping algebras) and related deformations, Grassmannians, Schubert varieties, flag manifolds Counting using Hall algebras. II: Extensions from quivers. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper, we construct stable Bott-Samelson classes in the projective limit of the algebraic cobordism rings of full flag varieties, upon an initial choice of a reduced word in a given dimension. Each stable Bott-Samelson class is represented by a bounded formal power series modulo symmetric functions in positive degree. We make some explicit computations for those power series in the case of infinitesimal cohomology. We also obtain a formula of the restriction of Bott-Samelson classes to smaller flag varieties. Schubert calculus; cobordism; flag variety; Bott-Samelson resolution Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Grassmannians, Schubert varieties, flag manifolds, Generalized (extraordinary) homology and cohomology theories in algebraic topology, Bordism and cobordism theories and formal group laws in algebraic topology, Symmetric functions and generalizations, Classical problems, Schubert calculus, Combinatorial aspects of algebraic geometry Stability of Bott-Samelson classes in algebraic cobordism | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper provides an example of a simple highest weight module over \(\mathfrak{sl}_{12}(\mathbb{C})\) whose characteristic variety is reducible. The proof of reducibility is rather indirect, it uses the theories of \(p\)-canonical bases, \(W\)-graphs and perverse sheaves. More precisely, the paper gives two permutations \(x\) and \(y\) in \(S_{12}\) which lie in the same right Kazhdan-Lusztig cell and such that a normal slice to the Schubert variety corresponding to \(y\) along the Schubert cell corresponding to \(x\) is isomorphic to the Kashiwara-Saito singularity. This is equivalent to the reducibility of a certain characteristic variety. That characteristic varieties in other types can be reducible was already known. characteristic variety; Schubert variety; Kazhdan-Lusztig cell; highest weight module; Kashiwara-Saito singularity Williamson, Geordie, A reducible characteristic variety in type \(A\).Representations of reductive groups, Prog. Math. Phys. 312, 517-532, (2015), Birkhäuser/Springer, Cham Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.), Intersection homology and cohomology in algebraic topology, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Grassmannians, Schubert varieties, flag manifolds A reducible characteristic variety in type \(A\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In the paper under review, the authors develop the theory of \textit{poset pinball}, a combinatorial game introduced in [\textit{M. Harada} and \textit{J. Tymoczko}, ``Poset pinball, GKM-compatible subspaces, and Hessenberg varieties'', \url{arXiv:1007.2750}] to study the equivariant cohomology ring of a GKM-compatible subspace \(X\) of a GKM space (see Definition 4.5 of [loc. cit.]); Harada and Tymoczko also proved that, in certain circumstances, a \textit{successful outcome of Betti poset pinball} yields a module basis for the equivariant cohomology ring of \(X\). In this paper, the authors first define a \textit{dimension pair algorithm}, which yields a successful outcome of Betti poset pinball for any type \(A\) regular nilpotent Hessenberg and any type \(A\) nilpotent Springer variety, considered as GKM-compatible subspaces of the flag variety. This algorithm is motivated by a correspondence between Hessenberg affine cells and certain Schubert polynomials. Second, in a special case of regular nilpotent Hessenberg varieties, the authors prove that the pinball outcome is \textit{poset-upper-triangular}, and hence the corresponding classes form a \(H^\ast_{S^1}\)(pt)-module basis for the \(S^{1}\)-equivariant cohomology ring of the Hessenberg variety. poset pinball; equivariant cohomology; type \(A\) regular nilpotent Hessenberg varieties Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of representation theory, Equivariant homology and cohomology in algebraic topology, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Poset pinball, the dimension pair algorithm, and type \(A\) regular nilpotent Hessenberg varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The aim of the article is to give resolution of the singularities of a Schubert cycle in the Grassmannian \(G(r,n)\) by blowing-up certain sub- Schubert cycles.
Let \(K\) a field, \(r\) and \(n\) positive integers with \(r<n\), and \(G(r,n)\) the Grassmannian of \(r\)-dimensional subspaces of \(K^ n\). We call \(T\) the set of \(r\)-uples \((t) = (t_ 1, \dots, t_ r)\) of integers satisfying \(1 \leq t_ 1 < \cdots < t_ r \leq n\), and for each \((t)\) in \(T\) we introduce \(r(n - r)\) variables \(X^{(t)}_{i,j}\), \(1 \leq i \leq r\), \(1 \leq j \leq n\) and \(j \neq t_ 1, \dots, t_ r\), and we define an affine scheme \(U^{(t)} = \text{Spec} K[X^{(t)}_{i,j}]\). The scheme structure on the Grassmannian \(G(r,n)\) is obtained by gluing the affine sets \(U^{(t)}\). For any \((l)\) in \(T\), we define the Schubert cycle \(S_{(l)}\). For any \((t)\) and \((l)\) in \(T\), we define a closed subscheme \(S^{(t)}_{(l)}\) of \(U^{(t)}\) whose equations are the \((r - i + 1)\) minors of some matrix. The structure scheme on the Schubert cycle \(S_{(l)}\) is obtained by gluing the sets \(S^{(t)}_{(l)}\). The scheme \(S_{(l)}\) is not necessarily smooth and its dimension is equal to \(\sum^ r_{i = 1} (l_ i - i)\). We want to study the blowing-up of a Schubert cycle \(S_{(l)}\) in \(G(r,n)\), more precisely if we order the sub-Schubert cycles \(\sigma_ 1 \leq \cdots \leq \sigma_ k\) of \(S_{(l)}\) by dimension, we want to determine their strict transforms by this blowing-up. We put an order on \(T\) (inverse lexicographic): \((k) > (l)\) if and only if there exists \(s \geq 0\) so that \(k_ i = l_ i\) for \(i \leq s\) and \(k_{s + 1} < l_{s + 1}\).
Proposition: For any \((l)\) in \(T\), \(S^{(l)}_{(l)}\) is the first nontrivial Schubert cycle in \(S^{(l)}\), i.e. \(S^{(l)}_{(l)} \neq \emptyset\) and \(S^{(l)}_{(t)} = \emptyset\) for \((t) < (l)\).
Blowing-up \(S_{(l)}\), the strict transforms of the Schubert cycles in \(U^{(l)} = \text{Spec} K [X_{i,j}^{(l)}]\) are isomorphic to the Schubert cycles in \(U^{(l')}\), where \((l) > (l')\). More precisely, we can cover the blowing-up \(\widetilde {U^{(l)}}\) of \(U^{(l)}\) by affine chart \(B_{(l)} (s,t)\), with \(1 \leq s \leq r\), \(t \geq l_ s + 1\) and \(t \neq l_ 1, \dots, l_ r\). For any \((s,t)\), we can construct \((l')\) such that \((l) > (l')\) and the chart \(B_{(l)} (s,t)\) is isomorphic to \(U^{(l')}\), then the strict transform of a Schubert cycle in this chart is isomorphic to the Schubert cycle in \(U^{(l')}\). -- From the proposition, the author deduces the principal result:
Corollary: Let \(\sigma\) be a Schubert cycle in \(G(r,n)\), let \(\{\sigma_ i\}\) be the Schubert cycles contained in \(\sigma\) ordered so that \(\dim \sigma_ i \leq \dim \sigma_{i + 1}\). Then \(\sigma\) can be desingularized by blowing up \(\sigma_ 1\), then by blowing up the strict transform of \(\sigma_ 2\), and so on. resolution of the singularities of a Schubert cycle; Grassmannian; strict transforms of the Schubert cycles Boudhraa, Z.: Resolution of singularities of Schubert cycles. J. pure appl. Algebra 90, No. 2, 105-113 (1993) Global theory and resolution of singularities (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds Resolution of singularities of Schubert cycles | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Fix an integer \(n>0\) and a non-degenerate quadratic form \(q\) on \(\mathbb {C}^{2n}\). Let \(\mathbb {G}\) be the orthogonal variety of all maximal isotropic subspaces of \((\mathbb {C}^{2n},q)\). \(\mathbb {G}\) has two connected components and any of them, \(X\), is called a spinor variety. \(X\) is a smooth rational homogeneous space of dimension \(n(n-1)/2\) with Picard group \(\mathbb {Z}\). Fix an effective class \(\alpha \in A_1(X)\) and call \(d\) its degree with respect to the ample generator of \(\mathrm{Pic}(X)\). Fix an elliptic curve \(C\) and let \({\mathbf {Hom}}_{\alpha }(C,X)\) be the scheme parametrizing all morphisms \(f: C \to X\) with \(f_\ast [C] =\alpha\). In this paper the author proves that \({\mathbf {Hom}}_{\alpha }(C,X)\) is irreducible for all \(d\geq 2\) (it has dimension \(2(n-1)d\) if \(d\geq n-1\) and dimension \(2(n-1)d + (n-1)(n-d-1)/2\) if \(2 \leq d \leq n-2\)). This type of results arises for low \(n\) in the study of vector bundles [\textit{A. Iliev} and \textit{D. Markushevich}, Asian J. Math. 11, No. 3, 427--458 (2007; Zbl 1136.14031)]. For maps from \(\mathbb {P}^1\) to homogeneous spaces, see the author in [Ann. Inst. Fourier 52, No. 1, 105--132 (2002; Zbl 1037.14021), J. Algebra 294, No. 2, 431--462 (2005; Zbl 1093.14070)]. The irreducibility for elliptic curves was proved (also for a few other homogeneous varieties) with a different method in [\textit{B. Pasquier} and \textit{N. Perrin}, Doc. Math. J. DMV 18, 679--704 (2013; Zbl 1299.14041)]. spinor variety; orthogonal Grassmannian; elliptic curve; quantum cohomology; Chow group Homogeneous spaces and generalizations, Grassmannians, Schubert varieties, flag manifolds, Special algebraic curves and curves of low genus, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Elliptic curves on spinor 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 Young's raising operators to introduce and study \textit{double theta polynomials}, which specialize to both the theta polynomials of Buch, Kresch, and Tamvakis, and to double (or factorial) Schur \(S\)-polynomials and \(Q\)-polynomials. These double theta polynomials give Giambelli formulas which represent the equivariant Schubert classes in the torus-equivariant cohomology ring of symplectic Grassmannians, and we employ them to obtain a new presentation of this ring in terms of intrinsic generators and relations. Tamvakis, H.; Wilson, E., \textit{double theta polynomials and equivariant Giambelli formulas}, Math. Proc. Cambridge Philos. Soc., 160, 353-377, (2016) Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations, Classical problems, Schubert calculus, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Double theta polynomials and equivariant Giambelli formulas | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials For the review see C. R. Acad. Sci., Paris, Sér. I 312, No. 3, 255-258 (1991; Zbl 0721.58021). flag manifold; Toda lattice; Bruhat cells; semisimple complex Lie algebra Flaschka, H.; Haine, L.: Variété de drapeaux et réseaux de Toda. Math. Z. 208, 545-556 (1991) Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Simple, semisimple, reductive (super)algebras, Grassmannians, Schubert varieties, flag manifolds Variétés de drapeaux et réseaux de Toda. (Flag manifolds and Toda lattices) | 0 |
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