Flag Manifolds Research Articles

Minimal models of oriented partial flag manifolds

Minimal models of oriented partial flag manifolds

Epsilon Dichotomy for Linear Models: The Archimedean Case

Abstract Let $G=\textrm {GL}_{2n}({\mathbb {R}})$ or $G=\textrm {GL}_n({\mathbb {H}})$ and $H=\textrm {GL}_n({\mathbb {C}})$ regarded as a subgroup of $G$. Here, ${\mathbb {H}}$ is the quaternion division algebra over ${\mathbb {R}}$. For a character $\chi $ on ${\mathbb {C}}^\times $, we say that an irreducible smooth admissible moderate growth representation $\pi $ of $G$ is $\chi _H$-distinguished if $\operatorname {Hom}_H(\pi , \chi \circ \det _H)\neq 0$. We compute the root number of a $\chi _H$-distinguished representation $\pi $ twisted by the representation induced from $\chi $. This proves an Archimedean analogue of the conjecture by Prasad and Takloo-Bighash (J. Reine Angew. Math., 2011). The proof is based on the analysis of the contribution of $H$-orbits in a flag manifold of $G$ to the Schwartz homology of principal series representations. A large part of the argument is developed for general real reductive groups of inner type. In particular, we prove that the Schwartz homology $H_\ast (H, \pi \otimes \chi )$ is finite-dimensional and hence it is Hausdorff for a reductive symmetric pair $(G, H)$ and a finite-dimensional representation $\chi $ of $H$.

On the Geometry of the Flag Varieties of Orthogonal group and Symplectic group

The aim of this paper is to discuss an important discourse, as most literatures highlighted the dimension of partial flag in a certain manner, according to a fixed definition that does not include all possible cases. In the present paper the existence of two different cases of flags in the partial flag varieties of the groups and is explained wholly. In addition to this we will proceed by calculating the dimensions for all the given cases after pertaining the signature of these flags that are studied.

Equivariant K-theory of the semi-infinite flag manifold as a nil-DAHA module

The equivariant K-theory of the semi-infinite flag manifold, as developed recently by Kato, Naito, and Sagaki, carries commuting actions of the nil-double affine Hecke algebra (nil-DAHA) and a q-Heisenberg algebra. The action of the latter generates a free submodule of rank |W|, where W is the (finite) Weyl group. We show that this submodule is stable under the nil-DAHA, which enables one to express the nil-DAHA action in terms of |W|times |W| matrices over the q-Heisenberg algebra. Our main result gives an explicit algebraic construction of these matrices as a limit from the (non-nil) DAHA in simply-laced type. This construction reveals that multiplication by equivariant scalars, when expressed in terms of the Heisenberg algebra, is given by the nonsymmetric q-Toda system introduced by Cherednik and the author.

On finitely nondegenerate closed homogeneous CR manifolds

A complex flag manifold $$\textsf {F}{=}{{\textbf {G}}}/{{\textbf {Q}}}$$ decomposes into finitely many real orbits under the action of a real form $${{\textbf {G}}}^\upsigma $$ of $${{\textbf {G}}}$$ . Their embedding into $$\textsf {F}$$ defines on them CR manifold structures. We characterize and list all the closed real orbits which are finitely nondegenerate.

Symmetric Toda, gradient flows, and tridiagonalization

The Toda lattice (1967) is a Hamiltonian system given by n points on a line governed by an exponential potential. Flaschka (1974) showed that the Toda lattice is integrable by interpreting it as a flow on the space of symmetric tridiagonal n×n matrices, while Moser (1975) showed that it is a gradient flow on a projective space. The symmetric Toda flow of Deift, Li, Nanda, and Tomei (1986) generalizes the Toda lattice flow from tridiagonal to all symmetric matrices. They showed the flow is integrable, in the classical sense of having d integrals in involution on its 2d-dimensional phase space. The system may be viewed as integrable in other ways as well. Firstly, Symes (1980, 1982) solved it explicitly via QR-factorization and conjugation. Secondly, Deift, Li, Nanda, and Tomei (1986) ‘tridiagonalized’ the system into a family of tridiagonal Toda lattices which are solvable and integrable. In this paper we derive their tridiagonalization procedure in a natural way using the fact that the symmetric Toda flow is diffeomorphic to a twisted gradient flow on a flag variety, which may then be decomposed into flows on a product of Grassmannians. These flows may in turn be embedded into projective spaces via Plücker embeddings, and mapped back to tridiagonal Toda lattice flows using Moser’s construction. In addition, we study the tridiagonalized flows projected onto a product of permutohedra, using the twisted moment map of Bloch, Flaschka, and Ratiu (1990). These ideas are facilitated in a natural way by the theory of total positivity, building on our previous work (2023).

Cluster-Permutohedra and Submanifolds of Flag Varieties with Torus Actions

Abstract In this paper, we describe a relation between the notion of graphicahedron, introduced by Araujo-Pardo, Del Río-Francos, López-Dudet, Oliveros, and Schulte in 2010 and toric topology of manifolds of sparse isospectral Hermitian matrices. More precisely, we recall the notion of a cluster-permutohedron, a certain finite poset defined for a simple graph $\Gamma $. This poset is build as a combination of cosets of the symmetric group, and the geometric lattice of the graphical matroid of $\Gamma $. This poset is similar to the graphicahedron of $\Gamma $, in particular, 1-skeleta of both posets are isomorphic to Cayley graphs of the symmetric group. We describe the relation between cluster-permutohedron and graphicahedron using Galois connection and the notion of a core of a finite topology. We further prove that the face poset of the natural torus action on the manifold of isospectral $\Gamma $-shaped Hermitian matrices is isomorphic to the cluster-permutohedron. Using recent results in toric topology, we show that homotopy properties of graphicahedra may serve an obstruction to equivariant formality of isospectral matrix manifolds. We introduce a generalization of a cluster-permutohedron and describe the combinatorial structure of a large family of manifolds with torus actions, including Grassmann manifolds and partial flag manifolds.

Inverse K-Chevalley formulas for semi-infinite flag manifolds, II: Arbitrary weights in ADE type

We continue the study, begun in [11], of inverse Chevalley formulas for the equivariant K-group of semi-infinite flag manifolds. Using the language of alcove paths, we reformulate and extend our combinatorial inverse Chevalley formula to arbitrary weights in all simply-laced types (conjecturally also in type E8).

Arbeitsgemeinschaft: Geometric Representation Theory

Our understanding of algebraic representations of reductive algebraic groups in positive characteristic has seen big advances in the last years and has been largely transformed into the geometric theory of studying parity sheaves on affine Grassmannians and affine flag varieties or, equivalently and more combinatorially, the diagrammatic Hecke category. This has led, among other things, to a geometric proof of the linkage principle and a greatly simplified proof of Lusztig’s character formula for large characteristics.

Newton–Okounkov polytopes of flag varieties and marked chain-order polytopes

Marked chain-order polytopes are convex polytopes constructed from a marked poset. They give a discrete family relating a marked order polytope with a marked chain polytope. In this paper, we consider the Gelfand–Tsetlin poset of type A A , and realize the associated marked chain-order polytopes as Newton–Okounkov bodies of the flag variety. Our realization connects previous realizations of Gelfand–Tsetlin polytopes and Feigin–Fourier–Littelmann–Vinberg polytopes as Newton–Okounkov bodies in a uniform way. As an application, we prove that the flag variety degenerates into the irreducible normal projective toric variety corresponding to a marked chain-order polytope. We also construct a specific basis of an irreducible highest weight representation. The basis is naturally parametrized by the set of lattice points in a marked chain-order polytope.

Motzkin Combinatorics in Linear Degenerations of the Flag Variety

Abstract We establish an explicit combinatorial/homological characterization of supports for linear degenerations of flag varieties. For such purpose, we introduce the concept of an excessive multisegment. It provides a new class of combinatorial objects counted by Motzkin numbers.

A cluster structure on the coordinate ring of partial flag varieties

The main goal of this paper is to show that the (multi-homogeneous) coordinate ring of a partial flag variety C[G/PK−] contains a cluster algebra if G is any semisimple complex algebraic group. We use derivation properties and a special lifting map to prove that the cluster algebra structure A of the coordinate ring C[NK] of a Schubert cell constructed by Goodearl and Yakimov can be lifted, in an explicit way, to a cluster structure Aˆ living in the coordinate ring of the corresponding partial flag variety. Then we use a minimality condition to prove that the cluster algebra Aˆ is equal to C[G/PK−] after localizing some special minors.

Geometric properties of the Kazhdan–Lusztig Schubert basis

We study classes determined by the Kazhdan-Lusztig basis of the Hecke algebra in the $K$-theory and hyperbolic cohomology theory of flag varieties. We first show that, in $K$-theory, the two different choices of Kazhdan-Lusztig bases produce dual bases, one of which can be interpreted as characteristic classes of the intersection homology mixed Hodge modules. In equivariant hyperbolic cohomology, we show that if the Schubert variety is smooth, then the class it determines coincides with the class of the Kazhdan-Lusztig basis; this was known as the Smoothness Conjecture. For Grassmannians, we prove that the classes of the Kazhdan-Lusztig basis coincide with the classes determined by Zelevinsky's small resolutions. These properties of the so-called KL-Schubert basis show that it is the closest existing analogue to the Schubert basis for hyperbolic cohomology; the latter is a very useful testbed for more general elliptic cohomologies.

Equigeodesics on Generalized Flag Manifolds with Four Isotropy Summands

Equigeodesics on Generalized Flag Manifolds with Four Isotropy Summands

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.

Conjectural Positivity of Chern–Schwartz–MacPherson Classes for Richardson Cells

Abstract Following some work of Aluffi–Mihalcea–Schürmann–Su for the CSM classes of Schubert cells and some elaborate computer calculations by Rimanyi and Mihalcea, I conjecture that the CSM classes of the Richardson cells expressed in the Schubert basis have nonnegative coefficients. This conjecture was principally motivated by a new product $\square $ coming from the Segre-SM classes in the cohomology of flag varieties (such that the associated Gr of this product is the standard cup product) and the conjecture that the structure constants of this new product $\square $ in the standard Schubert basis have alternating sign behavior. I prove that this conjecture on the sign of the structure constants of $\square $ would follow from my above positivity conjecture about the CSM classes of Richardson cells.

Nearby cycles on Drinfeld–Gaitsgory–Vinberg interpolation Grassmannian and long intertwining functor

Let G be a reductive group, and let U, U− be the unipotent radicals of a pair of opposite parabolic subgroups P, P−. We prove that the DG categories of U((t))-equivariant and U−((t))-equivariant D-modules on the affine Grassmannian GrG are canonically dual to each other. We show that the unit object witnessing this duality is given by nearby cycles on the Drinfeld–Gaitsgory–Vinberg interpolation Grassmannian defined in a recent work by Finkelberg, Krylov, and Mirković. We study various properties of the mentioned nearby cycles, and in particular compare them with the nearby cycles studied in works by Schieder. We also generalize our results to the Beilinson–Drinfeld Grassmannian GrG,XI and to the affine flag variety FlG. This version of the paper contains fewer appendices than the version submitted to arXiv.

Relative Richardson varieties

AbstractA Richardson variety in a flag variety is an intersection of two Schubert varieties defined by transverse flags. We define and study relative Richardson varieties, which are defined over a base scheme with a vector bundle and two flags. To do so, we generalise transversality of flags to a relative notion, versality, that allows the flags to be non-transverse over some fibers. Relative Richardson varieties share many of the geometric properties of Richardson varieties. We generalise several geometric and cohomological facts about Richardson varieties to relative Richardson varieties. We also prove that the local geometry of a relative Richardson variety is governed, in a precise sense, by the two intersecting Schubert varieties, giving a generalisation, in the flag variety case, of a theorem of Knutson–Woo–Yong; we also generalise this result to intersections of arbitrarily many relative Schubert varieties. We give an application to Brill–Noether varieties on elliptic curves, and a conjectural generalisation to higher genus curves.

A Tableau Formula for Vexillary Schubert Polynomials in Type C

Ikeda-Mihalcea-Naruse's double Schubert polynomials [Adv. Math. 226 (2011)] represent the equivariant cohomology classes of Schubert varieties in the type C flag varieties. The goal of this paper is to obtain a new tableau formula of these polynomials associated to vexillary signed permutations introduced by Anderson-Fulton. To achieve that goal, we introduce flagged factorial (Schur) $Q$-functions, combinatorially defined functions in terms of marked shifted tableaux for flagged strict partitions, and prove their Schur-Pfaffian formula. As an application, we also obtain a new combinatorial formula of factorial $Q$-functions of Ivanov in which monomials bijectively corresponds to flagged marked shifted tableaux.

Transfer Matrices of Rational Spin Chains via Novel BGG-Type Resolutions

We obtain BGG-type formulas for transfer matrices of irreducible finite-dimensional representations of the classical Lie algebras $\mathfrak{g}$, whose highest weight is a multiple of a fundamental one and which can be lifted to the representations over the Yangian $Y(\mathfrak{g})$. These transfer matrices are expressed in terms of transfer matrices of certain infinite-dimensional highest weight representations (such as parabolic Verma modules and their generalizations) in the auxiliary space. We further factorise the corresponding infinite-dimensional transfer matrices into the products of two Baxter $Q$-operators, arising from our previous study (arXiv:2001.04929, arXiv:2104.14518) of the degenerate Lax matrices. Our approach is crucially based on the new BGG-type resolutions of the finite-dimensional $\mathfrak{g}$-modules, which naturally arise geometrically as the restricted duals of the Cousin complexes of relative local cohomology groups of ample line bundles on the partial flag variety $G/P$ stratified by $B_{-}$-orbits.