Weyl Group Element Research Articles

Shapovalov elements for U q ( s l ( N + 1 ) )

For a simple Lie algebra, Shapovalov elements give rise to highest weight vectors in Verma modules. The usual construction of these elements uses induction on the length of a certain Weyl group element. If g = s l ( N + 1 ) explicit expressions for Shapovalov elements were given in I. M. Musson [Explicit expressions for Shapovalov elements in type A, J. Algebra 623 (2023), 358–394]. Here we adapt the argument to the quantized enveloping algebra of g .

Join operation for the Bruhat order and Verma modules

We observe that the join operation for the Bruhat order on a Weyl group agrees with the intersections of Verma modules in type A. The statement is not true in other types, and we propose a weaker correspondence. Namely, we introduce distinguished subsets of the Weyl group on which the join operation conjecturally agrees with the intersections of Verma modules. We also relate our conjecture with a statement about the socles of the cokernels of inclusions between Verma modules. The latter determines the first Ext space between a simple module and a Verma module. We give a conjectural complete description of such socles which we verify in a number of cases. Along the way, we determine the poset structure of the join-irreducible elements in Weyl groups and obtain closed formulae for certain families of Kazhdan–Lusztig polynomials.

MV polytopes and reduced double Bruhat cells

When G is a complex reductive algebraic group, MV polytopes are in bijection with the non-negative tropical points of the unipotent group of G . By fixing w from the Weyl group, we can define MV polytopes whose highest vertex is labelled by w . We show that these polytopes are in bijection with the non-negative tropical points of the reduced double Bruhat cell labelled by w^{-1} . To do this, we define a collection of generalized minor functions \Delta_{\gamma}^{\mathrm{new}} which tropicalize on the reduced Bruhat cell to the BZ data of an MV polytope of highest vertex w . We also describe the combinatorial structure of MV polytopes of highest vertex w . We explicitly describe the map from the Weyl group to the subset of elements bounded by w in the Bruhat order which sends u \mapsto v if the vertex labelled by u coincides with the vertex labelled by v for every MV polytope of highest vertex w . As a consequence of this map, we prove that these polytopes have vertices labelled by Weyl group elements less than w in the Bruhat order.

Equivariant K-Theory and Tangent Spaces to Schubert Varieties

Tangent spaces to Schubert varieties of type A were characterized by Lakshmibai and Seshadri [LS84]. This result was extended to the other classical types by Lakshmibai [Lak95, Lak00b], and [Lak00a]. We give a uniform characterization of tangent spaces to Schubert varieties in cominuscule G/P. Our results extend beyond cominuscule G/P; they describe the tangent space to any Schubert variety in G/B at a point xB, where x is a cominuscule Weyl group element in the sense of Peterson. Our results also give partial information about the tangent space to any Schubert variety at any point. Our method is to describe the tangent spaces of Kazhdan–Lusztig varieties, and then to recover results for Schubert varieties. Our proof uses a relationship between weights of the tangent space of a variety with torus action and factors of the class of the variety in torus equivariant K-theory. The proof relies on a formula for Schubert classes in equivariant K-theory due to Graham [Gra02] and Willems [Wil06] and on a theorem on subword complexes due to Knutson and Miller [KM04, KM05].

Signed alternating descent enumeration in classical Weyl groups

The alternating descent statistic on permutations was introduced by Chebikin as a variant of the descent statistic. In this paper, we get a formula for the signed enumeration of alternating descents and in our proof we need a signed convolution type identity involving the Eulerian polynomials. When n is even, we give a more general multivariate version and we also get a formula for the signed enumeration of the alternating major index. We generalize our results to the case when alternating descents are summed up with sign over the elements in classical Weyl groups.

PBW filtration and monomial bases for Demazure modules in types A and C

We characterise the symplectic Weyl group elements such that the FFLV basis is compatible with the PBW filtration on symplectic Demazure modules, extending type A results by the second author. Surprisingly, the number of such elements depends not on the type A or C of the Lie algebra but on the rank only.

Newton–Okounkov bodies for categories of modules over quiver Hecke algebras

We show that for a finite-type Lie algebra $\mathfrak{g}$, Kang-Kashiwara-Kim-Oh's monoidal categorification $\mathcal{C}_w$ of the quantum coordinate ring $\mathcal{A}_{\mathfrak{g}}(\mathfrak{n}(w))$ provides a natural framework for the construction of Newton-Okounkov bodies. In particular, this yields for every seed $\mathcal{S}$ of $\mathcal{A}_{\mathfrak{g}}(\mathfrak{n}(w))$ a simplex $\Delta_{\mathcal{S}}$ of codimension $1$ in $\mathbb{R}^{l(w)}$. We exhibit various geometric and combinatorial properties of these simplices by characterizing their rational points, their normal fans, and their volumes. The key tool is provided by the explicit description in terms of root partitions of the determinantial modules of a certain seed in $\mathcal{S}$ of $\mathcal{A}_{\mathfrak{g}}(\mathfrak{n}(w))$ constructed in \cite{GLS,KKKO}. This is achieved using the recent results of Kashiwara-Kim \cite{KK}. As an application, we prove an equality of rational functions involving root partitions for cluster variables. It implies an expression of the Peterson-Proctor hook formula in terms of heights of monoidal cluster variables in $\mathcal{C}_w$, suggesting further connections between cluster theory and the combinatorics of fully-commutative elements of Weyl groups.

Weyl Groups and Cluster Structures of Families of Log Calabi–Yau Surfaces

Abstract Given a generic Looijenga pair $(Y,D)$ together with a toric model $\rho :(Y,D)\rightarrow (\overline {Y},\overline {D})$, one can construct a seed $\textbf {s}$ such that the corresponding $\mathcal {X}$-cluster variety $\mathcal {X}_{\textbf {s}}$ can be viewed as the universal family of the log Calabi–Yau surface $U=Y\setminus D$. In cases where $(Y,D)$ is positive and $\mathcal {X}_{\textbf {s}}$ is not acyclic, we describe the action of the Weyl group of $(Y,D)$ on the scattering diagram $\mathfrak {D}_{\textbf {s}}$. Moreover, we show that there is a Weyl group element $\textbf {w}$ of order $2$ that either agrees with or approximates the Donaldson–Thomas transformation $\textrm {DT}_{\mathcal {X}_{\textbf {s}}}$ of $\mathcal {X}_{\textbf {s}}$. As a corollary, $\textrm {DT}_{\mathcal {X}_{\textbf {s}}}$ is cluster. In positive non-acyclic cases, we also apply the folding technique as developed in [20] and construct a maximally folded new seed $\overline {\textbf {s}}$ from $\textbf {s}$. The $\mathcal {X}$-cluster variety $\mathcal {X}_{\overline {\textbf {s}}}$ is a locally closed subvariety of $\mathcal {X}_{\textbf {s}}$ and corresponds to the maximally degenerate subfamily in the universal family. We show that the action of the special Weyl group element $\textbf {w}$ on $\mathfrak {D}_{\textbf {s}}$ descends to $\mathfrak {D}_{\overline {\textbf {s}}}$ and permutes distinct subfans in $\mathfrak {D}_{\overline {\textbf {s}}}$, generalizing the well-known case of the Markov quiver.

Curve Neighborhoods of Schubert Varieties in the Odd Symplectic Grassmannian

Let IG(k,2n + 1) be the odd symplectic Grassmannian. It is a quasi-homogeneous space with homogeneous-like behavior. A very limited description of curve neighborhoods of Schubert varieties in IG(k,2n + 1) was used by Mihalcea and the second named author to prove an (equivariant) quantum Chevalley rule. In this paper, we give a full description of the irreducible components of curve neighborhoods in terms of the Hecke product of (appropriate) Weyl group elements, k-strict partitions, and BC-partitions. The latter set of partitions respect the Bruhat order with inclusions. Our approach follows the philosophy of Buch and Mihalcea’s curve neighborhood calculations of Schubert varieties in the homogeneous cases.

Gelfand–Kirillov Dimensions and Associated Varieties of Highest Weight Modules

Abstract In this paper, we present a uniform formula of Lusztig’s $ \textbf {a}$-functions on classical Weyl groups. Then we obtain an efficient algorithm for the Gelfand–Kirillov dimensions of simple highest weight modules of classical Lie algebras, whose highest weight is not necessarily regular or integral. To deal with type $ D $, we prove an interesting property about domino tableaux associated with Weyl group elements by introducing an invariant, called the hollow tableau. As an application, the associated varieties of all the simple highest weight Harish–Chandra modules are explicitly determined, including the exceptional cases.

Parametrization of Kloosterman sets and SL3-Kloosterman sums

We stratify the SL3 big cell Kloosterman sets using the reduced word decomposition of the Weyl group element, inspired by the Bott-Samelson factorization. Thus the SL3 long word Kloosterman sum is decomposed into finer parts, and we write it as a finite sum of a product of two classical Kloosterman sums. The fine Kloosterman sums end up being the correct pieces to consider in the Bruggeman-Kuznetsov trace formula on the congruence subgroup Γ0(N)⊆SL3(Z). Another application is a new explicit formula, expressing the triple divisor sum function in terms of a double Dirichlet series of exponential sums, generalizing Ramanujan's formula.

W-translated Schubert divisors and transversal intersections

We study the toric degeneration of Weyl group translated Schubert divisors of a partial flag variety \(F{\ell _{{n_1}, \ldots,{n_k};n}}\) via Gelfand-Cetlin polytopes. We propose a conjecture that Schubert varieties of appropriate dimensions intersect transversally up to translation by Weyl group elements, and verify it in various cases, including the complex Grassmannian Gr(2, n) and the complete flag variety Fℓ1,2,3,4.

Eulerian central limit theorems and Carlitz identities in positive elements of classical Weyl groups

Central Limit Theorems are known for the Eulerian statistic descent (or excedance) in the symmetric group $\SSS_n$. Recently, Fulman, Kim, Lee and Petersen gave a Central Limit Theorem for descent over the alternating group $\AAA_n$ and also gave a Carlitz identity in $\AAA_n$ using descents. In this paper, we give a Central Limit Theorem in $\AAA_n$ involving excedances. We extend these to the positive elements in type B and type D Coxeter groups. Boroweic and Mlotkowski enumerated type B descents over $\DD_n$, the type D Coxeter group and gave similar results. We refine their results for both the positive and negative part of $\DD_n$. Our results are a consequence of signed enumeration over these subsets.

Strong minuscule elements in the finite Weyl groups

We introduce the notion of a strong minuscule element, which is a dominant minuscule element $w$ in the Weyl group for which there exists a unique (dominant) integral weight $\Lambda$ such that $w$ is $\Lambda$-minuscule. We prove that the dominant integral weight associated to a strong minuscule element is the fundamental weight corresponding to a short simple root (in this paper, all simple roots in the simply-laced cases are treated as short roots). In addition, we enumerate the strong minuscule elements explicitly. As an application of this enumeration, we determine the dimension of certain Demazure module in the finite-dimensional irreducible module whose highest weight is a minuscule weight.

Combinatorics of canonical bases revisited: type A

We initiate a new approach to the study of the combinatorics of several parametrizations of canonical bases. In this work we deal with Lie algebras of type A. Using geometric objects called rhombic tilings we derive a “Crossing Formula” for the action of the crystal operators on Lusztig data for an arbitrary reduced word of the longest Weyl group element. We provide the following three applications of this result. Using the tropical Chamber Ansatz of Berenstein–Fomin–Zelevinsky we prove an enhanced version of the Anderson–Mirković conjecture for the crystal structure on MV polytopes. We establish a duality between Kashiwara’s string and Lusztig’s parametrization, revealing that each of them is controlled by the crystal structure of the other. We identify the potential functions of the unipotent radical of a Borel subgroup of \(SL_n\) defined by Berenstein–Kazhdan and Gross–Hacking–Keel–Kontsevich, respectively, with a function arising from the crystal structure on Lusztig data.

Bigrassmannian permutations and Verma modules

We show that bigrassmannian permutations determine the socle of the cokernel of an inclusion of Verma modules in type A. All such socular constituents turn out to be indexed by Weyl group elements from the penultimate two-sided cell. Combinatorially, the socular constituents in the cokernel of the inclusion of a Verma module indexed by win S_n into the dominant Verma module are shown to be determined by the essential set of w and their degrees in the graded picture are shown to be computable in terms of the associated rank function. As an application, we compute the first extension from a simple module to a Verma module.

Integral quantum cluster structures

We prove a general theorem for constructing integral quantum cluster algebras over Z[q±1/2], namely, that under mild conditions the integral forms of quantum nilpotent algebras always possess integral quantum cluster algebra structures. These algebras are then shown to be isomorphic to the corresponding upper quantum cluster algebras, again defined over Z[q±1/2]. Previously, this was only known for acyclic quantum cluster algebras. The theorem is applied to prove that, for every symmetrizable Kac–Moody algebra g and Weyl group element w, the dual canonical form Aq(n+(w))Z[q±1] of the corresponding quantum unipotent cell has the property that Aq(n+(w))Z[q±1]⊗Z[q±1]Z[q±1/2] is isomorphic to a quantum cluster algebra over Z[q±1/2] and to the corresponding upper quantum cluster algebra over Z[q±1/2].

Local Poisson groupoids over mixed product Poisson structures and generalised double Bruhat cells

Given a standard complex semisimple Poisson Lie group $(G, \pi_{st})$, generalised double Bruhat cells $G^{u, v}$ and generalised Bruhat cells $O^u$ equipped with naturally defined holomorphic Poisson structures, where u, v are finite sequences of Weyl group elements, were defined and studied by Jiang Hua Lu and the author. We prove in this paper that $G^{u,u}$ is naturally a Poisson groupoid over $O^u$, extending a result from the aforementioned authors about double Bruhat cells in $(G, \pi_{st})$. Our result on $G^{u,u}$ is obtained as an application of a construction interesting in its own right, of a local Poisson groupoid over a mixed product Poisson structure associated to the action of a pair of Lie bialgebras. This construction involves using a local Lagrangian bisection in a double symplectic groupoid closely related to the global R-matrix studied by Weinstein and Xu, to twist a direct product of Poisson groupoids.

Gamma Positivity of the Excedance-Based Eulerian Polynomial in Positive Elements of Classical Weyl Groups

The Eulerian polynomial $$ \mathrm {AExc}_n(t)$$ enumerating excedances in the symmetric group $$\mathfrak {S}_n$$ is known to be gamma positive for all n. When enumeration is done over the type B and type D Coxeter groups, the type B and type D Eulerian polynomials are also gamma positive for all n. We consider $$ \mathrm {AExc}_n^+(t)$$ and $$ \mathrm {AExc}_n^-(t)$$ , the polynomials which enumerate excedance in the alternating group $$\mathcal {A}_n$$ and in $$\mathfrak {S}_n - \mathcal {A}_n$$ , respectively. We show that $$ \mathrm {AExc}_n^+(t)$$ is gamma positive iff $$n \ge 5$$ is odd. When $$n \ge 4$$ is even, $$ \mathrm {AExc}_n^+(t)$$ is not even palindromic, but we show that it is the sum of two gamma positive summands. An identical statement is true about $$ \mathrm {AExc}_n^-(t)$$ . We extend similar results to the excedance based Eulerian polynomial when enumeration is done over the positive elements in both type B and type D Coxeter groups. Gamma positivity results are known when excedance is enumerated over derangements in $$\mathfrak {S}_n$$ . We extend some of these to the case when enumeration is done over even and odd derangements in $$\mathfrak {S}_n$$ .

Separable elements and splittings of Weyl groups

We continue the study of separable elements in finite Weyl groups, introduced in [11]. These elements generalize the well-studied class of separable permutations. We show that the multiplication map W/U×U→W is a length-additive bijection or splitting of the Weyl group W when U is an order ideal in right weak order generated by a separable element, where W/U denotes the generalized quotient. This generalizes a result for the symmetric group, answering an open problem of Wei [17].For a generalized quotient of the symmetric group, we show that this multiplication map is a bijection if and only if U is an order ideal in right weak order generated by a separable element, thereby classifying those generalized quotients which induce splittings of the symmetric group, resolving a problem of Björner and Wachs from 1988 [4]. We also prove that this map is always surjective when U is an order ideal in right weak order. Interpreting these sets of permutations as linear extensions of 2-dimensional posets gives the first direct combinatorial proof of an inequality due originally to Sidorenko in 1991, answering an open problem Morales, Pak, and Panova [13]. We also prove a new q-analog of Sidorenko's formula. All of these results are conjectured to extend to arbitrary finite Weyl groups.Finally, we show that separable elements in W are in bijection with the faces of all dimensions of several copies of the graph associahedron of the Dynkin diagram of W. This correspondence associates to each separable element w a certain nested set; we give product formulas for the rank generating functions of the principal upper and lower order ideals generated by w in terms of these nested sets, generalizing several known formulas.