scholarly journals A type B analog of the Lie representation

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Andrew Berget
Keyword(s):  
Symmetric Group ◽  
Weyl Group ◽  
Regular Representation ◽  
Type B ◽  
International Audience

International audience We describe a type B analog of the much studied Lie representation of the symmetric group. The nth Lie representation of Sn restricts to the regular representation of Sn−1, and our generalization mimics this property. Specifically, we construct a representation of the type B Weyl group Bn that restricts to the regular representation of Bn−1. We view both of these representations as coming from the internal zonotopal algebra of the Gale dual of the corresponding reflection arrangements.

scholarly journals Matrix-Ball Construction of affine Robinson-Schensted correspondence

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Michael Chmutov ◽  
Pavlo Pylyavskyy ◽  
Elena Yudovina
Keyword(s):  
Symmetric Group ◽  
Weyl Group ◽  
Type A ◽  
Group Symmetry ◽  
Dominant Weight ◽  
Dominance Condition ◽  
The Matrix ◽  
International Audience ◽  
Affine Type

International audience In his study of Kazhdan-Lusztig cells in affine type A, Shi has introduced an affine analog of Robinson- Schensted correspondence. We generalize the Matrix-Ball Construction of Viennot and Fulton to give a more combi- natorial realization of Shi's algorithm. As a biproduct, we also give a way to realize the affine correspondence via the usual Robinson-Schensted bumping algorithm. Next, inspired by Honeywill, we extend the algorithm to a bijection between extended affine symmetric group and triples (P, Q, ρ) where P and Q are tabloids and ρ is a dominant weight. The weights ρ get a natural interpretation in terms of the Affine Matrix-Ball Construction. Finally, we prove that fibers of the inverse map possess a Weyl group symmetry, explaining the dominance condition on weights.

Fischer Matrices for Projective Representations of Generalized Symmetric Groups

2009 ◽  
Vol 16 (03) ◽  
pp. 449-462
Author(s):  
Mohammed S. Almestady ◽  
Alun O. Morris
Keyword(s):  
Symmetric Group ◽  
Weyl Group ◽  
Symmetric Groups ◽  
Type B ◽  
Projective Representations ◽  
Ordinary Case ◽  
Covering Groups

The aim of this work is to calculate the Fischer matrices for the covering groups of the Weyl group of type Bn and the generalized symmetric group. It is shown that the Fischer matrices are the same as those in the ordinary case for the classes of Sn which correspond to partitions with all parts odd. For the classes of Sn which correspond to partitions in which no part is repeated more than m times, the Fischer matrices are shown to be different from the ordinary case.

scholarly journals Double Schubert polynomials for the classical Lie groups

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Takeshi Ikeda ◽  
Leonardo Mihalcea ◽  
Hiroshi Naruse
Keyword(s):  
Weyl Group ◽  
Lie Groups ◽  
Equivariant Cohomology ◽  
Natural Extension ◽  
Type B ◽  
Infinite Rank ◽  
Schubert Polynomials ◽  
International Audience ◽  
Schubert Classes ◽  
Double Schubert Polynomials

International audience For each infinite series of the classical Lie groups of type $B$, $C$ or $D$, we introduce a family of polynomials parametrized by the elements of the corresponding Weyl group of infinite rank. These polynomials represent the Schubert classes in the equivariant cohomology of the corresponding flag variety. They satisfy a stability property, and are a natural extension of the (single) Schubert polynomials of Billey and Haiman, which represent non-equivariant Schubert classes. When indexed by maximal Grassmannian elements of the Weyl group, these polynomials are equal to the factorial analogues of Schur $Q$- or $P$-functions defined earlier by Ivanov. Pour chaque série infinie des groupe de Lie classiques de type $B$,$C$ ou $D$, nous présentons une famille de polynômes indexées par de éléments de groupe de Weyl correspondant de rang infini. Ces polynômes représentent des classes de Schubert dans la cohomologie équivariante des variétés de drapeaux. Ils ont une certain propriété de stabilité, et ils étendent naturellement des polynômes Schubert (simples) de Billey et Haiman, que représentent des classes de Schubert dans la cohomologie non-équivariante. Quand ils sont indexées par des éléments Grassmanniennes de groupes de Weyl, ces polynômes sont égaux à des analogues factorielles de fonctions $Q$ et $P$ de Schur, étudiées auparavant par Ivanov.

scholarly journals The quasiinvariants of the symmetric group

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Jason Bandlow ◽  
Gregg Musiker
Keyword(s):  
Symmetric Group ◽  
Polynomial Ring ◽  
Coxeter Group ◽  
Regular Representation ◽  
Constant Term ◽  
Left Regular Representation ◽  
Left Regular ◽  
Isotypic Component ◽  
International Audience ◽  
The Ideal

International audience For $m$ a non-negative integer and $G$ a Coxeter group, we denote by $\mathbf{QI_m}(G)$ the ring of $m$-quasiinvariants of $G$, as defined by Chalykh, Feigin, and Veselov. These form a nested series of rings, with $\mathbf{QI_0}(G)$ the whole polynomial ring, and the limit $\mathbf{QI}_{\infty}(G)$ the usual ring of invariants. Remarkably, the ring $\mathbf{QI_m}(G)$ is freely generated over the ideal generated by the invariants of $G$ without constant term, and the quotient is isomorphic to the left regular representation of $G$. However, even in the case of the symmetric group, no basis for $\mathbf{QI_m}(G)$ is known. We provide a new description of $\mathbf{QI_m}(S_n)$, and use this to give a basis for the isotypic component of $\mathbf{QI_m}(S_n)$ indexed by the shape $[n-1,1]$. Pour $m$ un entier positif ou nul et $G$ un groupe de Coxeter, nous notons $\mathbf{QI_m}(G)$ l'anneau des quasiinvariants définis par Chalykh, Feigin et Veselov. On obtient ainsi une série d'anneaux emboités, $\mathbf{QI_0}(G)$ étant l'anneau des polynômes, et la limite $\mathbf{QI}_{\infty}(G)$ l'anneau des invariants usuels. Il est remarquable que l'anneau $\mathbf{QI_m}(G)$ est librement généré sur l'idéal engendré par les invariants de $G$ sans terme constant, et le quotient est isomorphe à la représentation régulière à gauche de $G$. Cependant, même dans le cas du groupe symétrique, aucune base de $\mathbf{QI_m}(G)$ n'est connue. Nous donnons une nouvelle description de $\mathbf{QI_m}(G)$ et l'utilisons pour obtenir une base du composant isotypique de $\mathbf{QI_m}(S_n)$ indexée par la partition $(n-1,1)$.

scholarly journals Irreducible Modular Representations of the Reflection GroupG(m,1,n)

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
José O. Araujo ◽  
Tim Bratten ◽  
Cesar L. Maiarú
Keyword(s):  
Symmetric Group ◽  
Weyl Group ◽  
Reflection Group ◽  
Geometric Construction ◽  
Weyl Groups ◽  
Modular Representations ◽  
Complex Reflection ◽  
Complex Reflection Group ◽  
One Year ◽  
Combinatorial Methods

In an article published in 1980, Farahat and Peel realized the irreducible modular representations of the symmetric group. One year later, Al-Aamily, Morris, and Peel constructed the irreducible modular representations for a Weyl group of typeBn. In both cases, combinatorial methods were used. Almost twenty years later, using a geometric construction based on the ideas of Macdonald, first Aguado and Araujo and then Araujo, Bigeón, and Gamondi also realized the irreducible modular representations for the Weyl groups of typesAnandBn. In this paper, we extend the geometric construction based on the ideas of Macdonald to realize the irreducible modular representations of the complex reflection group of typeG(m,1,n).

scholarly journals A Diagrammatic Temperley-Lieb Categorification

2010 ◽  
Vol 2010 ◽  
pp. 1-47 ◽  
Author(s):  
Ben Elias
Keyword(s):  
Symmetric Group ◽  
Weyl Group ◽  
Planar Graphs ◽  
Monoidal Category ◽  
Hecke Algebra ◽  
Coxeter Complex ◽  
Quotient Category ◽  
Irreducible Modules ◽  
Soergel Bimodules

The monoidal category of Soergel bimodules categorifies the Hecke algebra of a finite Weyl group. In the case of the symmetric group, morphisms in this category can be drawn as graphs in the plane. We define a quotient category, also given in terms of planar graphs, which categorifies the Temperley-Lieb algebra. Certain ideals appearing in this quotient are related both to the 1-skeleton of the Coxeter complex and to the topology of 2D cobordisms. We demonstrate how further subquotients of this category will categorify the irreducible modules of the Temperley-Lieb algebra.

scholarly journals Representations of Weyl groups of type B induced from centralisers of involutions

1991 ◽  
Vol 44 (2) ◽  
pp. 337-344 ◽  
Author(s):  
Philip D. Ryan
Keyword(s):  
Weyl Group ◽  
Symmetric Groups ◽  
Conjugacy Classes ◽  
Weyl Groups ◽  
Type B ◽  
Complex Character ◽  
Linear Character ◽  
Complex Linear

Let G be a Weyl group of type B, and T a set of representatives of the conjugacy classes of self-inverse elements of G. For each t in T, we construct a (complex) linear character πt of the centraliser of t in G, such that the sum of the characters of G induced from the πt contains each irreducible complex character of G with multiplicity precisely 1. For Weyl groups of type A (that is, for the symmetric groups), a similar result was published recently by Inglis, Richardson and Saxl.

scholarly journals Affine descents and the Steinberg torus

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Kevin Dilks ◽  
T. Kyle Petersen ◽  
John R. Stembridge
Keyword(s):  
Weyl Group ◽  
Generating Functions ◽  
Cell Complex ◽  
Premier Lieu ◽  
Coxeter Complex ◽  
Affine Weyl Group ◽  
Nous Montrons ◽  
International Audience ◽  
Translation Subgroup

International audience Let $W \ltimes L$ be an irreducible affine Weyl group with Coxeter complex $\Sigma$, where $W$ denotes the associated finite Weyl group and $L$ the translation subgroup. The Steinberg torus is the Boolean cell complex obtained by taking the quotient of $\Sigma$ by the lattice $L$. We show that the ordinary and flag $h$-polynomials of the Steinberg torus (with the empty face deleted) are generating functions over $W$ for a descent-like statistic first studied by Cellini. We also show that the ordinary $h$-polynomial has a nonnegative $\gamma$-vector, and hence, symmetric and unimodal coefficients. In the classical cases, we also provide expansions, identities, and generating functions for the $h$-polynomials of Steinberg tori. Nous considérons un groupe de Weyl affine irréductible $W \ltimes L$ avec complexe de Coxeter $\Sigma$, où $W$ désigne le groupe de Weyl fini associé et $L$ le sous-groupe des translations. Le tore de Steinberg est le complexe cellulaire Booléen obtenu comme le quotient de $\Sigma$ par $L$. Nous montrons que les $h$-polynômes, ordinaires et de drapeaux, du tore de Steinberg (sans la face vide) sont des fonctions génératrices sur $W$ pour une statistique de type descente, étudiée en premier lieu par Cellini. Nous montrons également qu'un $h$-polynôme ordinaire possède un $\gamma$-vecteur positif, et par conséquent, a des coefficients symétriques et unimodaux. Dans les cas classiques, nous donnons également des développements, des identités et des fonctions génératrices pour les $h$-polynômes des tores de Steinberg.

scholarly journals Minimal factorizations of a cycle: a multivariate generating function

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Philippe Biane ◽  
Matthieu Josuat-Vergès
Keyword(s):  
Symmetric Group ◽  
Generating Function ◽  
Simple Formula ◽  
Hook Length ◽  
Noncrossing Partitions ◽  
Long Cycle ◽  
Number Of Factors ◽  
Multivariate Generalization ◽  
International Audience

International audience It is known that the number of minimal factorizations of the long cycle in the symmetric group into a product of k cycles of given lengths has a very simple formula: it is nk−1 where n is the rank of the underlying symmetric group and k is the number of factors. In particular, this is nn−2 for transposition factorizations. The goal of this work is to prove a multivariate generalization of this result. As a byproduct, we get a multivariate analog of Postnikov's hook length formula for trees, and a refined enumeration of final chains of noncrossing partitions.

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