scholarly journals A Combinatorial Formula for Orthogonal Idempotents in the $0$-Hecke Algebra of $S_N$

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Tom Denton
Keyword(s):  
Symmetric Group ◽  
Hecke Algebra ◽  
Combinatorial Formula ◽  
International Audience

International audience Building on the work of P.N. Norton, we give combinatorial formulae for two maximal decompositions of the identity into orthogonal idempotents in the $0$-Hecke algebra of the symmetric group, $\mathbb{C}H_0(S_N)$. This construction is compatible with the branching from $H_0(S_{N-1})$ to $H_0(S_N)$. En s'appuyant sur le travail de P.N. Norton, nous donnons des formules combinatoires pour deux décompositions maximales de l'identité en idempotents orthogonaux dans l'algèbre de Hecke $H_0(S_N)$ du groupe symétrique à $q=0$. Ces constructions sont compatibles avec le branchement de $H_0(S_{N-1})$ à $H_0(S_N)$.

scholarly journals Combinatorial Gelfand Models

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Ron M. Adin ◽  
Alex Postnikov ◽  
Yuval Roichman
Keyword(s):  
Symmetric Group ◽  
Hecke Algebra ◽  
Hyperoctahedral Group ◽  
International Audience ◽  
Combinatorial Construction

International audience A combinatorial construction of Gelfand models for the symmetric group, for its Iwahori-Hecke algebra and for the hyperoctahedral group is presented.

scholarly journals A Combinatorial Formula for Orthogonal Idempotents in the $0$-Hecke Algebra of the Symmetric Group

10.37236/515 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Tom Denton
Keyword(s):  
Symmetric Group ◽  
Hecke Algebra ◽  
Combinatorial Formula

Building on the work of P.N. Norton, we give combinatorial formulae for two maximal decompositions of the identity into orthogonal idempotents in the $0$-Hecke algebra of the symmetric group, $\mathbb{C}H_0(S_N)$. This construction is compatible with the branching from $S_{N-1}$ to $S_{N}$.

scholarly journals Structure coefficients of the Hecke algebra of $(\mathcal{S}_{2n}, \mathcal{B}_n)$

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Omar Tout
Keyword(s):  
Symmetric Group ◽  
Universal Algebra ◽  
Group Algebra ◽  
Main Tool ◽  
Hecke Algebra ◽  
Natural Analogue ◽  
International Audience

International audience The Hecke algebra of the pair $(\mathcal{S}_{2n}, \mathcal{B}_n)$, where $\mathcal{B}_n$ is the hyperoctahedral subgroup of $\mathcal{S}_{2n}$, was introduced by James in 1961. It is a natural analogue of the center of the symmetric group algebra. In this paper, we give a polynomiality property of its structure coefficients. Our main tool is a combinatorial universal algebra which projects on the Hecke algebra of $(\mathcal{S}_{2n}, \mathcal{B}_n)$ for every $n$. To build it, we introduce new objects called partial bijections. L’algèbre de Hecke de la paire $(\mathcal{S}_{2n}, \mathcal{B}_n)$ , où $\mathcal{B}_n$ est le sous-groupe hyperoctaèdral de $\mathcal{S}_{2n}$, aété introduite par James en 1961. C’est un analogue naturel du centre de l’algèbre du groupe symétrique. Dans ce papier, on donne une propriété de polynomialité de ses coefficients de structure. On utilise une algèbre universelle construite d’une façon combinatoire et qui se projette sur toutes les algèbres de Hecke de $(\mathcal{S}_{2n}, \mathcal{B}_n)$. Pour la construire, on introduit de nouveaux objets appelés bijections partielles.

scholarly journals KAZHDAN–LUSZTIG CELLS AND THE MURPHY BASIS

2006 ◽  
Vol 93 (3) ◽  
pp. 635-665 ◽  
Author(s):  
MEINOLF GECK
Keyword(s):  
Symmetric Group ◽  
Hecke Algebra ◽  
Fundamental Properties ◽  
Algebraic Proofs ◽  
First Time

Let $H$ be the Iwahori–Hecke algebra associated with $S_n$, the symmetric group on $n$ symbols. This algebra has two important bases: the Kazhdan–Lusztig basis and the Murphy basis. We establish a precise connection between the two bases, allowing us to give, for the first time, purely algebraic proofs for a number of fundamental properties of the Kazhdan–Lusztig basis and Lusztig's results on the $a$-function.

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 Peak algebras, paths in the Bruhat graph and Kazhdan-Lusztig polynomials

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Francesco Brenti ◽  
Fabrizio Caselli
Keyword(s):  
Combinatorial Formula ◽  
Quasisymmetric Functions ◽  
International Audience ◽  
Bruhat Graph ◽  
Lusztig Polynomials

International audience We obtain a nonrecursive combinatorial formula for the Kazhdan-Lusztig polynomials which holds in complete generality and which is simpler and more explicit than any existing one, and which cannot be linearly simplified. Our proof uses a new basis of the peak subalgebra of the algebra of quasisymmetric functions. On montre une formule combinatoire pour les polynômes de Kazhdan-Lusztig qui est valable en toute généralité. Cette formule est plus simple et plus explicite que toutes les autres formules connues; de plus, elle ne peut pas être simplifiée linéairement. La preuve utilise une nouvelle base pour la sous-algèbre des sommets de l’algèbre des fonctions quasi-symmetriques.

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.

scholarly journals 0-Hecke algebra actions on coinvariants and flags

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Jia Huang
Keyword(s):  
Generating Functions ◽  
Hecke Algebra ◽  
Flag Variety ◽  
Major Index ◽  
Inversion Number ◽  
International Audience ◽  
Descent Set ◽  
Complete Flag ◽  
Coinvariant Algebra

International audience By investigating the action of the 0-Hecke algebra on the coinvariant algebra and the complete flag variety, we interpret generating functions counting the permutations with fixed inverse descent set by their inversion number and major index. En étudiant l'action de l'algèbre de 0-Hecke sur l'algèbre coinvariante et la variété de drapeaux complète, nous interprétons les fonctions génératrices qui comptent les permutations avec un ensemble inverse de descentes fixé, selon leur nombre d'inversions et leur "major index''.

scholarly journals Power Sum Expansion of Chromatic Quasisymmetric Functions

10.37236/4761 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Christos A. Athanasiadis
Keyword(s):  
Symmetric Group ◽  
Symmetric Function ◽  
Symmetric Functions ◽  
Unit Interval ◽  
Quasisymmetric Function ◽  
Interval Order ◽  
Combinatorial Formula ◽  
Natural Unit ◽  
Power Sum ◽  
Chromatic Symmetric Function

The chromatic quasisymmetric function of a graph was introduced by Shareshian and Wachs as a refinement of Stanley's chromatic symmetric function. An explicit combinatorial formula, conjectured by Shareshian and Wachs, expressing the chromatic quasisymmetric function of the incomparability graph of a natural unit interval order in terms of power sum symmetric functions, is proven. The proof uses a formula of Roichman for the irreducible characters of the symmetric group.

scholarly journals Intervals and factors in the Bruhat order

2015 ◽  
Vol Vol. 17 no. 1 (Combinatorics) ◽  
Author(s):  
Bridget Eileen Tenner
Keyword(s):  
Symmetric Group ◽  
Bruhat Order ◽  
Order Ideal ◽  
Principal Order ◽  
International Audience ◽  
Reduced Words

Combinatorics International audience In this paper we study those generic intervals in the Bruhat order of the symmetric group that are isomorphic to the principal order ideal of a permutation w, and consider when the minimum and maximum elements of those intervals are related by a certain property of their reduced words. We show that the property does not hold when w is a decomposable permutation, and that the property always holds when w is the longest permutation.

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