Double Flag Varieties Research Articles

Action of Hecke algebra on the double flag variety of type AIII

Consider a connected reductive algebraic group G and a symmetric subgroup K. Let X=K/BK×G/P be a double flag variety of finite type, where BK is a Borel subgroup of K, and P a parabolic subgroup of G. A general argument shows that the orbit space CX/K inherits a natural action of the Hecke algebra H=H(K,BK) of double cosets via convolutions. However, it is a quite different problem to find out the explicit structure of the Hecke module.In this paper, for the double flag variety of type AIII, we determine the explicit action of H on CX/K in a combinatorial way using graphs. As a by-product, we also get the description of the representation of the Weyl group on CX/K as a direct sum of induced representations.

On generalized Steinberg theory for type AIII

The multiple flag variety 𝔛=Gr(ℂ p+q ,r)×(Fl(ℂ p )×Fl(ℂ q )) can be considered as a double flag variety associated to the symmetric pair (G,K)=(GL p+q (ℂ), GL p (ℂ)×GL q (ℂ)) of type AIII. We consider the diagonal action of K on 𝔛. There is a finite number of orbits for this action, and our first result is a description of these orbits: parametrization (by a certain set of graphs), dimensions, closure relations and cover relations.

Multiple Flag Varieties

This paper is a review of results on multiple flag varieties, i.e., varieties of the form G/P1×· · ·×G/Pr. We provide a classification of multiple flag varieties of complexity 0 and 1 and results on the combinatorics and geometry of B-orbits and their closures in double cominuscule flag varieties. We also discuss questions of finiteness for the number of G-orbits and existence of an open G-orbits on a multiple flag variety.

A Generalization of Steinberg Theory and an Exotic Moment Map

Abstract For a reductive group $G$, Steinberg established a map from the Weyl group to the set of nilpotent $G$-orbits by using moment maps on double flag varieties. In particular, in the case of the general linear group, it provides a geometric interpretation of the Robinson–Schensted correspondence between permutations and pairs of standard tableaux of the same shape. We extend Steinberg’s approach to the case of a symmetric pair $(G, K)$ to obtain two different maps, namely a generalized Steinberg map and an exotic moment map. Although the framework is general, in this paper we focus on the pair $(G,K) = (\textrm{GL}_{2n}({\mathbb{C}}), \textrm{GL}_n({\mathbb{C}}) \times \textrm{GL}_n({\mathbb{C}}))$. Then the generalized Steinberg map is a map from partial permutations to the pairs of nilpotent orbits in $ \mathfrak{gl}_n({\mathbb{C}}) $. It involves a generalization of the classical Robinson–Schensted correspondence to the case of partial permutations. The other map, the exotic moment map, establishes a combinatorial map from the set of partial permutations to that of signed Young diagrams, that is, the set of nilpotent $ K$-orbits in the Cartan space $(\textrm{Lie}(G)/\textrm{Lie}(K))^* $. We explain the geometric background of the theory and combinatorial algorithms, which produce the above-mentioned maps.

Diagonal Orbits in a Type A Double Flag Variety of Complexity One

We continue our study of the inclusion posets of diagonal $SL(n)$-orbit closures in a product of two partial flag varieties. We prove that, if the diagonal action is of complexity one, then the poset is isomorphic to one of the 28 posets that we determine explicitly. Furthermore, our computations show that the number of diagonal $SL(n)$-orbits in any of these posets is at most 10 for any positive integer $n$. This is in contrast with the complexity 0 case, where, in some cases, the resulting posets attain arbitrary heights.

Degenerate 0-Schur Algebras and Nil-Temperley-Lieb Algebras

In Jensen and Su (J. Pure Appl. Algebra 219(2), 277–307 2014) constructed 0-Schur algebras, using double flag varieties. The construction leads to a presentation of 0-Schur algebras using quivers with relations and the quiver presentation naturally gives rise to a new class of algebras, which are introduced and studied in this paper. That is, these algebras are defined on the quivers of 0-Schur algebras with relations modified from the defining relations of 0-Schur algebras by a tuple of parameters t. In particular, when all the entries of t are 1, we recover 0-Schur algebras. When all the entries of t are zero, we obtain a class of basic algebras, which we call the degenerate 0-Schur algebras. We prove that the degenerate algebras are both associated graded algebras and quotients of 0-Schur algebras. Moreover, we give a geometric interpretation of the degenerate algebras using double flag varieties, in the same spirit as Jensen and Su (J. Pure Appl. Algebra 219(2), 277–307 2014), and show how the centralizer algebras are related to nil-Hecke and nil-Temperley-Lieb algebras.

Enhanced zeta distributions and its functional equations

We consider an “enhanced symmetric space”, which is a prehomogeneous vector space. This vector space is intimately related to a double flag variety studied in [1]. On a distinguished open orbit called “enhanced positive cone”, we consider a zeta integral with two complex variables, which is analytically continued to meromorphic family of tempered distributions. One of the main results of this paper is to establish a precise formula for the meromorphic continuation which clarifies the location of poles (and may be useful to obtain residues). We also compute the Fourier transform of the zeta distribution and obtain a functional equation with explicit gamma factors.

A FILTRATION ON EQUIVARIANT BOREL–MOORE HOMOLOGY

Let $G/H$ be a homogeneous variety and let $X$ be a $G$ -equivariant embedding of $G/H$ such that the number of $G$ -orbits in $X$ is finite. We show that the equivariant Borel–Moore homology of $X$ has a filtration with associated graded module the direct sum of the equivariant Borel–Moore homologies of the $G$ -orbits. If $T$ is a maximal torus of $G$ such that each $G$ -orbit has a $T$ -fixed point, then the equivariant filtration descends to give a filtration on the ordinary Borel–Moore homology of $X$ . We apply our findings to certain wonderful compactifications as well as to double flag varieties.

Real double flag varieties for the symplectic group

In this paper we study a key example of a Hermitian symmetric space and a natural associated double flag variety, namely for the real symplectic group G and the symmetric subgroup L, the Levi part of the Siegel parabolic PS. We give a detailed treatment of the case of the maximal parabolic subgroups Q of L corresponding to Grassmannians and the product variety of G/PS and L/Q; in particular we classify the L-orbits here, and find natural explicit integral transforms between degenerate principal series of L and G.

Invariants of the Cox rings of double flag varieties of low complexity for exceptional groups

We find the algebras of unipotent invariants of the Cox rings for all double flag varieties of complexity and for the exceptional simple algebraic groups; namely, we obtain presentations of these algebras in terms of generators and relations. It is well known that such an algebra is free in the case of complexity . In this paper, we show that, in the case of complexity , the algebra in question is either a free algebra or a hypersurface. A similar result for classical groups was previously obtained by the author. Knowing the structure of this algebra enables one to decompose tensor products of some irreducible representations effectively into irreducible summands and to obtain some branching rules. Bibliography: 10 titles.

Invariants of the Cox rings of double flag varieties of low complexity for exceptional groups

Найдены алгебры унипотентных инвариантов колец Кокса всех двойных многообразий флагов сложностей $0$ и $1$ для особых простых алгебраических групп - получено их задание с помощью образующих и соотношений. Известно, что в случае сложности $0$ указанная алгебра свободна (как для особых, так и для классических групп). В работе показано, что в случае сложности $1$ рассматриваемая алгебра свободна или является гиперповерхностью. Аналогичный результат для классических групп был получен автором ранее. Знание структуры данной алгебры позволяет эффективно раскладывать на неприводимые слагаемые тензорные произведения некоторых неприводимых представлений и получать некоторые правила ветвления. Библиография: 10 названий.

Invariants of the Cox rings of low-complexity double flag varieties for classical groups

Invariants of the Cox rings of low-complexity double flag varieties for classical groups

Cell Decompositions of Double Bott–Samelson Varieties

Let $G$ be a connected complex semisimple Lie group. Webster and Yakimov have constructed partitions of the double flag variety $G/B \times G/B_-$ , where $(B, B_-)$ is a pair of opposite Borel subgroups of $G$ , generalizing the Deodhar decompositions of $G/B$ . We show that these partitions can be better understood by constructing cell decompositions of a product of two Bott–Samelson varieties $Z_{ {\bf u}, {\bf v} }$ , where $ {\bf u}$ and $ {\bf v}$ are sequences of simple reflections. We construct coordinates on each cell of the decompositions and in the case of a positive subexpression, we relate these coordinates to regular functions on a particular open subset of $Z_{ {\bf u}, {\bf v} }$ . Our motivation for constructing cell decompositions of $Z_{ {\bf u}, {\bf v} }$ was to study a certain natural Poisson structure on $Z_{ {\bf u}, {\bf v} }$ .

On Orbits in Double Flag Varieties for Symmetric Pairs

Let G be a connected, simply connected semisimple algebraic group over the complex number field, and let K be the fixed point subgroup of an involutive automorphism of G so that (G, K) is a symmetric pair. We take parabolic subgroups P of G and Q of K, respectively, and consider the product of partial flag varieties G/P and K/Q with diagonal K-action, which we call a double flag variety for a symmetric pair. It is said to be of finite type if there are only finitely many K-orbits on it. In this paper, we give a parametrization of K-orbits on G/P × K/Q in terms of quotient spaces of unipotent groups without assuming the finiteness of orbits. If one of P ⊂ G or Q ⊂ K is a Borel subgroup, the finiteness of orbits is closely related to spherical actions. In such cases, we give a complete classification of double flag varieties of finite type, namely, we obtain classifications of K-spherical flag varieties G/P and G-spherical homogeneous spaces G/Q.

Classification of double flag varieties of complexity 0 and 1

We obtain a classification of double flag varieties of complexity 0 and 1 and consider an application to the problem of decomposing tensor products of irreducible representations of semisimple Lie groups.

On quantum [formula omitted

The quantum GL(n) of Faddeev, Reshetikhin, and Takhtajan, and that of Dipper and Donkin are realized geometrically by using double partial flag varieties. As a consequence, the difference of these two Hopf algebras is caused by a twist of a cocycle in the multiplication.

Double-partition quantum cluster algebras

A family of quantum cluster algebras is introduced and studied. In general, these algebras are new, but sub-classes have been studied previously by other authors. The algebras are indexed by double-partitions or double flag varieties. Equivalently, they are indexed by broken lines L. By grouping together neighboring mutations into quantum line mutations we can mutate from the cluster algebra of one broken line to another. Compatible pairs can be written down. The algebras are equal to their upper cluster algebras. The variables of the quantum seeds are given by elements of the dual canonical basis.

A Deodhar-type stratification on the double flag variety

We describe a stratification on the double flag variety $G/B^+\times G/B^-$ of a complex semisimple algebraic group $G$ analogous to the Deodhar stratification on the flag variety $G/B^+$, which is a refinement of the stratification into orbits both for $B^+\times B^-$ and for the diagonal action of $G$, just as Deodhar's stratification refines the orbits of $B^+$ and $B^-$. We give a coordinate system on each stratum, and show that all strata are coisotropic subvarieties. Also, we discuss possible connections to the positive and cluster geometry of $G/B^+\times G/B^-$, which would generalize results of Fomin and Zelevinsky on double Bruhat cells and Marsh and Rietsch on double Schubert cells.