scholarly journals On C-Bases, Partition Pairs and Filtrations for Induced or Restricted Specht Modules

2019 ◽  
Vol 26 (01) ◽  
pp. 161-180
Author(s):  
Christos A. Pallikaros
Keyword(s):  
Symmetric Group ◽  
Hecke Algebra ◽  
Specht Module ◽  
Specht Modules

We obtain alternative explicit Specht filtrations for the induced and the restricted Specht modules in the Hecke algebra of the symmetric group (defined over the ring A = ℤ[q1/2, q−1/2], where q is an indeterminate) using C-bases for these modules. Moreover, we provide a link between a certain C-basis for the induced Specht module and the notion of pairs of partitions.

scholarly journals $(\ell,0)$-Carter Partitions, their Crystal-Theoretic Behavior and Generating Function

10.37236/854 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Chris Berg ◽  
Monica Vazirani
Keyword(s):  
Symmetric Group ◽  
Generating Function ◽  
Hecke Algebra ◽  
Specht Module ◽  
Combinatorial Description ◽  
Basic Representation ◽  
Crystal Graph ◽  
Generating Series ◽  
The One ◽  
Regular Partitions

In this paper we give an alternate combinatorial description of the "$(\ell,0)$-Carter partitions". The representation-theoretic significance of these partitions is that they indicate the irreducibility of the corresponding specialized Specht module over the Hecke algebra of the symmetric group. Our main theorem is the equivalence of our combinatoric and the one introduced by James and Mathas, which is in terms of hook lengths. We use our result to find a generating series which counts such partitions, with respect to the statistic of a partition's first part. We then apply our description of these partitions to the crystal graph $B(\Lambda_0)$ of the basic representation of $\widehat{{sl}_{\ell}}$, whose nodes are labeled by $\ell$-regular partitions. Here we give a fairly simple crystal-theoretic rule which generates all $(\ell,0)$-Carter partitions in the graph $B(\Lambda_0)$.

On a conjecture of Carter concerning irreducible Specht modules

1978 ◽  
Vol 83 (1) ◽  
pp. 11-17 ◽  
Author(s):  
G. D. James
Keyword(s):  
Irreducible Representation ◽  
Symmetric Group ◽  
Characteristic Zero ◽  
Irreducible Representations ◽  
Specht Module ◽  
Specht Modules ◽  
Definition Of

We study the question: Which ordinary irreducible representations of the symmetric group remain irreducible modulo a prime p?Let Sλ be the Specht module corresponding to the partition λ of n. The definition of Sλ is ‘independent of the field we are working over’. When the field has characteristic zero, Sλ is irreducible, and gives the ordinary irreducible representation of corresponding to the partition λ. Thus we are interested in the problem of whether or not Sλ is irreducible over a field of characteristic p.

scholarly journals A Specht Module Analog for the Rook Monoid

10.37236/1619 ◽  
2001 ◽  
Vol 9 (1) ◽  
Author(s):  
Cheryl Grood
Keyword(s):  
Representation Theory ◽  
Symmetric Group ◽  
Algebraic Structure ◽  
Irreducible Representations ◽  
Specht Module ◽  
Specht Modules ◽  
Rook Monoid ◽  
Combinatorial Construction

The wealth of beautiful combinatorics that arise in the representation theory of the symmetric group is well-known. In this paper, we analyze the representations of a related algebraic structure called the rook monoid from a combinatorial angle. In particular, we give a combinatorial construction of the irreducible representations of the rook monoid. Since the rook monoid contains the symmetric group, it is perhaps not surprising that the construction outlined in this paper is very similar to the classic combinatorial construction of the irreducible $S_n$-representations: namely, the Specht modules.

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 Murnaghan-Nakayama Rules for Characters of Iwahori-Hecke Algebras of the Complex Reflection Groups G(r, p, n)

1998 ◽  
Vol 50 (1) ◽  
pp. 167-192 ◽  
Author(s):  
Tom Halverson ◽  
Arun Ram
Keyword(s):  
Symmetric Group ◽  
Infinite Series ◽  
Hecke Algebras ◽  
Hecke Algebra ◽  
Diagonal Matrix ◽  
Weyl Groups ◽  
Reflection Groups ◽  
Irreducible Characters ◽  
Complex Reflection ◽  
Complex Reflection Groups

AbstractIwahori-Hecke algebras for the infinite series of complex reflection groups G(r, p, n) were constructed recently in the work of Ariki and Koike [AK], Broué andMalle [BM], and Ariki [Ari]. In this paper we give Murnaghan-Nakayama type formulas for computing the irreducible characters of these algebras. Our method is a generalization of that in our earlier paper [HR] in whichwe derivedMurnaghan-Nakayama rules for the characters of the Iwahori-Hecke algebras of the classical Weyl groups. In both papers we have been motivated by C. Greene [Gre], who gave a new derivation of the Murnaghan-Nakayama formula for irreducible symmetric group characters by summing diagonal matrix entries in Young's seminormal representations. We use the analogous representations of the Iwahori-Hecke algebra of G(r, p, n) given by Ariki and Koike [AK] and Ariki [Ari].

scholarly journals Cohomology Classes of Interval Positroid Varieties and a Conjecture of Liu

10.37236/6960 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Brendan Pawlowski
Keyword(s):  
Symmetric Group ◽  
Finite Subset ◽  
Upper Bound ◽  
Cohomology Class ◽  
Symmetric Functions ◽  
A Priori ◽  
Specht Module ◽  
Group A ◽  
Schubert Classes ◽  
Stanley Symmetric Functions

To each finite subset of $\mathbb{Z}^2$ (a diagram), one can associate a subvariety of a complex Grassmannian (a diagram variety), and a representation of a symmetric group (a Specht module). Liu has conjectured that the cohomology class of a diagram variety is represented by the Frobenius characteristic of the corresponding Specht module. We give a counterexample to this conjecture.However, we show that for the diagram variety of a permutation diagram, Liu's conjectured cohomology class $\sigma$ is at least an upper bound on the actual class $\tau$, in the sense that $\sigma - \tau$ is a nonnegative linear combination of Schubert classes. To do this, we exhibit the appropriate diagram variety as a component in a degeneration of one of Knutson's interval positroid varieties (up to Grassmann duality). A priori, the cohomology classes of these interval positroid varieties are represented by affine Stanley symmetric functions. We give a different formula for these classes as ordinary Stanley symmetric functions, one with the advantage of being Schur-positive and compatible with inclusions between Grassmannians.

scholarly journals $(\ell, 0)$-Carter Partitions and their crystal theoretic interpretation

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Chris Berg ◽  
Monica Vazirani
Keyword(s):  
Hecke Algebra ◽  
Specht Module ◽  
Regular Partition ◽  
Basic Representation ◽  
Crystal Graph ◽  
Generating Series ◽  
International Audience ◽  
The One ◽  
Regular Partitions ◽  
Nous Utilisons

International audience In this paper we give an alternate combinatorial description of the "$(\ell,0)$-Carter partitions''. Our main theorem is the equivalence of our combinatoric and the one introduced by James and Mathas ($\textit{A q-analogue of the Jantzen-Schaper theorem}$). The condition of being an $(\ell,0)$-Carter partition is fundamentally related to the hook lengths of the partition. The representation-theoretic significance of their combinatoric on an $\ell$-regular partition is that it indicates the irreducibility of the corresponding Specht module over the finite Hecke algebra. We use our result to find a generating series which counts the number of such partitions, with respect to the statistic of a partition's first part. We then apply our description of these partitions to the crystal graph $B(\Lambda_0)$ of the basic representation of $\widehat{\mathfrak{sl}_{\ell}}$, whose nodes are labeled by $\ell$-regular partitions. Here we give a fairly simple crystal-theoretic rule which generates all $(\ell,0)$-Carter partitions in the graph of $B(\Lambda_0)$. Dans cet article, nous donnons une description combinatoire alternative des partitions "$(\ell,0)$-Carter". Notre théorème principal est une équivalence entre notre combinatoire et celle introduite par James et Mathas ($\textit{A q-analogue of the Jantzen-Schaper theorem}$). La propriété $(\ell,0)$-Carter est fondamentalement liée aux longueurs des équerres de la partition. En terme de théorie des représentations, leur combinatoire pour une partition $\ell$-régulière permet de déterminer l'irréducibilité du module de Specht spécialisé sur l’algèbre de Hecke finie. Nous utilisons notre résultat pour déterminer leur série génératrice en fonction de la taille de la première part. Nous utilisons ensuite notre description de ces partitions au graphe cristallin $B(\Lambda _0)$ de la représentation basique de $\widehat{\mathfrak{sl}_{\ell}}$, dont les nœuds sont étiquetés par les partitions $\ell$-régulières. Nous donnons une règle cristalline relativement simple permettant d'engendrer toutes les partitions $\ell$-régulières $(\ell,0)$-Carter dans le graphe de $B(\Lambda _0)$.

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

10.37236/3592 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Omar Tout
Keyword(s):  
Symmetric Group ◽  
Group Algebra ◽  
Main Tool ◽  
Hecke Algebra ◽  
Natural Analogue ◽  
New Class ◽  
Finite Dimensional ◽  
Finite Dimensional Algebras ◽  
Combinatorial Algebra

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 algebra which projects onto the Hecke algebra of $(\mathcal{S}_{2n},\mathcal{B}_n)$ for every $n$. To build it, by using partial bijections we introduce and study a new class of finite dimensional algebras.

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