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Benjamin J. Wyser

Publications and source records attributed to Benjamin J. Wyser.

14 recordsLinked to original sources
arXiv2022

K-orbit closures and Barbasch-Evens-Magyar varieties

We define the Barbasch-Evens-Magyar varieties. We show they are isomorphic to the smooth varieties defined in [D.~Barbasch-S.~Evens '94] that map generically finitely to symmetric orbit closures, thereby giving resolutions of singularities in certain cases. Our definition parallels [P.~Magyar '98]'s construction of the Bott-Samelson varieties [H.~C.~Hansen '73, M.~Demazure '74]. From this alternative viewpoint, one deduces a graphical description in type $A$, stratification into closed subvarieties of the same kind, and determination of the torus-fixed points. Moreover, we explain how these manifolds inherit a natural symplectic structure with Hamiltonian torus action. We then express the moment polytope in terms of the moment polytope of a Bott-Samelson variety.

math.AG↗
arXiv2015

The Bruhat order on clans

We give an explicit description of the closure containment order (or "Bruhat order") on the set of orbits of GL_p \times GL_q on the flag variety GL_{p+q}/B, relative to the parametrization of the orbits by combinatorial objects called "clans". This leads to a corresponding description of the closures of such orbits as sets of flags satisfying certain incidence conditions encoded by the parametrizing clans.

math.CO↗
arXiv2015

K-orbit closures on G/B as universal degeneracy loci for flagged vector bundles splitting as direct sums

We use equivariant localization and divided difference operators to determine formulas for the torus-equivariant fundamental cohomology classes of $K$-orbit closures on the flag variety $G/B$ for various symmetric pairs $(G,K)$. In type $A$, we realize the closures of $K=GL(p,\C) \times GL(q,\C)$-orbits on $GL(p+q,\C)/B$ as universal degeneracy loci for a vector bundle over a variety which is equipped with a single flag of subbundles and which splits as a direct sum of subbundles of ranks $p$ and $q$. The precise description of such a degeneracy locus relies upon knowing a set-theoretic description of $K$-orbit closures, which we provide via a detailed combinatorial analysis of the poset of "$(p,q)$-clans," which parametrize the orbit closures. We describe precisely how our formulas for the equivariant classes of $K$-orbit closures can be interpreted as formulas for the classes of such degeneracy loci in the Chern classes of the bundles involved. In the cases outside of type $A$, we suggest that the orbit closures should parametrize degeneracy loci involving a vector bundle equipped with a non-degenerate symmetric or skew-symmetric bilinear form, a single flag of subbundles which are isotropic or Lagrangian with respect to the form, and a splitting as a direct sum of subbundles with each summand satisfying some property (depending on $K$) with respect to the form. The precise description of such a degeneracy locus is conjectured for all cases in types $B$ and $C$.

math.AG↗
arXiv2014

Polynomials for symmetric orbit closures in the flag variety

In [Wyser-Yong '13] we introduced polynomial representatives of cohomology classes of orbit closures in the flag variety, for the symmetric pair $(GL_{p+q}, GL_p \times GL_q)$. We present analogous results for the remaining symmetric pairs of the form $(GL_n,K)$, i.e., $(GL_n,O_n)$ and $(GL_{2n},Sp_{2n})$. We establish the well-definedness of certain representatives from [Wyser '13]. It is also shown that the representatives have the combinatorial properties of nonnegativity and stability. Moreover, we give some extensions to equivariant $K$-theory.

math.CO↗
arXiv2014

Polynomials for GL_p x GL_q orbit closures in the flag variety

The subgroup K=GL_p x GL_q of GL_{p+q} acts on the (complex) flag variety GL_{p+q}/B with finitely many orbits. We introduce a family of polynomials that specializes to representatives for cohomology classes of the orbit closures in the Borel model. We define and study K-orbit determinantal ideals to support the geometric naturality of these representatives. Using a modification of these ideals, we describe an analogy between two local singularity measures: the H-polynomials and the Kazhdan-Lusztig-Vogan polynomials.

math.RT↗
arXiv2014

Combinatorial results on (1,2,1,2)-avoiding $GL(p,\mathbb{C}) \times GL(q,\mathbb{C})$-orbit closures on $GL(p+q, \mathbb{C})/B$

Using recent results of the second author which explicitly identify the "$(1,2,1,2)$-avoiding" $GL(p,\mathbb{C}) \times GL(q,\mathbb{C})$-orbit closures on the flag manifold $GL(p+q,\mathbb{C})/B$ as certain Richardson varieties, we give combinatorial criteria for determining smoothness, lci-ness, and Gorensteinness of such orbit closures. (In the case of smoothness, this gives a new proof of a theorem of W.M. McGovern.) Going a step further, we also describe a straightforward way to compute the singular locus, the non-lci locus, and the non-Gorenstein locus of any such orbit closure. We then describe a manifestly positive combinatorial formula for the Kazhdan-Lusztig-Vogan polynomial $P_{τ,γ}(q)$ in the case where $γ$ corresponds to the trivial local system on a $(1,2,1,2)$-avoiding orbit closure $Q$ and $τ$ corresponds to the trivial local system on any orbit $Q'$ contained in $\overline{Q}$. This combines the aforementioned result of the second author, results of A. Knutson, the first author, and A. Yong, and a formula of Lascoux and Schützenberger which computes the ordinary (type $A$) Kazhdan-Lusztig polynomial $P_{x,w}(q)$ whenever $w \in S_n$ is cograssmannian.

math.CO↗
arXiv2013

K-orbit closures on G/B as universal degeneracy loci for flagged vector bundles with symmetric or skew-symmetric bilinear form

We use equivariant localization and divided difference operators to determine formulas for the torus-equivariant fundamental cohomology classes of $K$-orbit closures on the flag variety $G/B$, where $G = GL(n,\C)$, and where $K$ is one of the symmetric subgroups $O(n,\C)$ or $Sp(n,\C)$. We realize these orbit closures as universal degeneracy loci for a vector bundle over a variety equipped with a single flag of subbundles and a nondegenerate symmetric or skew-symmetric bilinear form taking values in the trivial bundle. We describe how our equivariant formulas can be interpreted as giving formulas for the classes of such loci in terms of the Chern classes of the various bundles.

math.AG↗
arXiv2013

Schubert calculus of Richardson varieties stable under spherical Levi subgroups

We observe that the expansion in the basis of Schubert cycles for $H^*(G/B)$ of the class of a Richardson variety stable under a spherical Levi subgroup is described by a theorem of Brion. Using this observation, along with a combinatorial model of the poset of certain symmetric subgroup orbit closures, we give positive combinatorial descriptions of certain Schubert structure constants on the full flag variety in type $A$. Namely, we describe $c_{u,v}^w$ when $u$ and $v$ are inverse to Grassmannian permutations with unique descents at $p$ and $q$, respectively. We offer some conjectures for similar rules in types $B$ and $D$, associated to Richardson varieties stable under spherical Levi subgroups of $SO(2n+1,\C)$ and $SO(2n,\C)$, respectively.

math.CO↗
arXiv2012

GL(p) x GL(q)-orbit closures on the flag variety and Schubert structure constants for (p,q)-pairs

We give positive combinatorial descriptions of Schubert structure constants $c_{u,v}^w$ for the full flag variety in type $A_{n-1}$ when $u$ and $v$ form what we refer to as a "$(p,q)$-pair" ($p+q=n$). The key observation is that a certain subset of the $GL(p,\mathbb{C}) \times GL(q,\mathbb{C})$-orbit closures on the flag variety (those satisfying an easily stated pattern avoidance condition) are Richardson varieties. The result on structure constants follows when one combines this observation with a theorem of Brion concerning intersection numbers of spherical subgroup orbit closures and Schubert varieties.

math.CO↗
arXiv2012

Symmetric subgroup orbit closures on flag varieties: Their equivariant geometry, combinatorics, and connections with degeneracy loci

We give explicit formulas for torus-equivariant fundamental classes of closed $K$-orbits on the flag variety $G/B$ when $G$ is one of the classical groups $SL(n,\C)$, $SO(n,\C)$, or $Sp(2n,\C)$, and $K$ is a symmetric subgroup of $G$. We describe parametrizations of each orbit set and the combinatorics of its weak order, allowing us to compute formulas for the equivariant classes of all remaining orbit closures using divided difference operators. In each of the cases in type A, we realize the $K$-orbit closures as universal degeneracy loci of a certain type, involving a vector bundle $V$ over a scheme $X$ equipped with a flag of subbundles and a further structure determined by $K$. We describe how our equivariant formulas can be interpreted as formulas for such loci in the Chern classes of the various bundles on $X$.

math.AG↗
arXiv2012

K-orbits on G/B and Schubert constants for pairs of signed shuffles in types C and D

We give positive descriptions for certain Schubert structure constants $c_{u,v}^w$ for the full flag variety in Lie types $C$ and $D$. This is accomplished by first observing that a number of the $K=GL(n,\C)$-orbit closures on these flag varieties coincide with Richardson varieties, and then applying a theorem of M. Brion on the decomposition of such an orbit closure in the Schubert basis in terms of paths in the weak order graph.

math.CO↗
arXiv2011

Second cohomology for finite groups of Lie type

Let $G$ be a simple, simply-connected algebraic group defined over $\mathbb{F}_p$. Given a power $q = p^r$ of $p$, let $G(\mathbb{F}_q) \subset G$ be the subgroup of $\mathbb{F}_q$-rational points. Let $L(λ)$ be the simple rational $G$-module of highest weight $λ$. In this paper we establish sufficient criteria for the restriction map in second cohomology $H^2(G,L(λ)) \rightarrow H^2(G(\mathbb{F}_q),L(λ))$ to be an isomorphism. In particular, the restriction map is an isomorphism under very mild conditions on $p$ and $q$ provided $λ$ is less than or equal to a fundamental dominant weight. Even when the restriction map is not an isomorphism, we are often able to describe $H^2(G(\mathbb{F}_q),L(λ))$ in terms of rational cohomology for $G$. We apply our techniques to compute $H^2(G(\mathbb{F}_q),L(λ))$ in a wide range of cases, and obtain new examples of nonzero second cohomology for finite groups of Lie type.

math.RT↗
arXiv2011

First cohomology for finite groups of Lie type: simple modules with small dominant weights

Let $k$ be an algebraically closed field of characteristic $p > 0$, and let $G$ be a simple, simply connected algebraic group defined over $\mathbb{F}_p$. Given $r \geq 1$, set $q=p^r$, and let $G(\mathbb{F}_q)$ be the corresponding finite Chevalley group. In this paper we investigate the structure of the first cohomology group $H^1(G(\mathbb{F}_q),L(λ))$ where $L(λ)$ is the simple $G$-module of highest weight $λ$. Under certain very mild conditions on $p$ and $q$, we are able to completely describe the first cohomology group when $λ$ is less than or equal to a fundamental dominant weight. In particular, in the cases we consider, we show that the first cohomology group has dimension at most one. Our calculations significantly extend, and provide new proofs for, earlier results of Cline, Parshall, Scott, and Jones, who considered the special case when $λ$ is a minimal nonzero dominant weight.

math.GR↗