scholarly journals Stability properties of Plethysm: new approach with combinatorial proofs (Extended abstract)

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Laura Colmenarejo
Keyword(s):  
Symmetric Functions ◽  
Symmetric Groups ◽  
Linear Groups ◽  
New Approach ◽  
Stability Properties ◽  
Integer Points ◽  
General Linear Groups ◽  
International Audience ◽  
Geometric Techniques ◽  
Homogeneous Basis

International audience Plethysm coefficients are important structural constants in the theory of symmetric functions and in the representations theory of symmetric groups and general linear groups. In 1950, Foulkes observed stability properties: some sequences of plethysm coefficients are eventually constants. Such stability properties were proven by Brion with geometric techniques and by Thibon and Carré by means of vertex operators. In this paper we present a newapproach to prove such stability properties. This new proofs are purely combinatorial and follow the same scheme. We decompose plethysm coefficients in terms of other plethysm coefficients (related to the complete homogeneous basis of symmetric functions). We show that these other plethysm coefficients count integer points in polytopes and we prove stability for them by exhibiting bijections between the corresponding sets of integer points of each polytope. Les coefficients du pléthysme sont des constantes de structure importantes de la théorie des fonctions symétriques, ainsi que de la théorie de la représentation des groupes symétriques et des groupes généraux linéaires. En 1950, Foulkes a observé pour ces coefficients de phénomènes de stabilité: certaines suites de coefficients du pléthysme sont stationnaires. De telles propriétés ont été démontrées par Brion, au moyen de techniques géométriques, et par Thibon et Carré, au moyen d’opérateurs vertex. Dans ce travail, nous présentons une nouvelle approche, purement combinatoire, pour démontrer des propriétés de stabilité de ce type. Nous décomposons les coefficients du pléthysme comme somme alternées de coefficients de pléthysme d’un autre type (liés à la base des fonctions symétriques sommes complètes), qui comptent les points entiers dans des polytopes. Nous démontrons la stabilité des suites de ces coefficients en exhibant des bijections entres les ensembles de points entiers des polytopes correspondants.

scholarly journals Young modules for symmetric groups

2001 ◽  
Vol 71 (2) ◽  
pp. 201-210 ◽  
Author(s):  
Karin Erdmann
Keyword(s):  
Representation Theory ◽  
Finite Groups ◽  
Symmetric Groups ◽  
General Linear ◽  
Linear Groups ◽  
Schur Algebras ◽  
General Linear Groups ◽  
Green Correspondence

AbstractLet K be a field of characteristic p. The permutation modules associated to partitions of n, usually denoted as Mλ, play a central role not only for symmetric groups but also for general linear groups, via Schur algebras. The indecomposable direct summands of these Mλ were parametrized by James; they are now known as Young modules; and Klyachko and Grabmeier developed a ‘Green correspondence’ for Young modules. The original parametrization used Schur algebras; and James remarked that he did not know a proof using only the representation theory of symmetric groups. We will give such proof, and we will at the same time also prove the correspondence result, by using only the Brauer construction, which is valid for arbitrary finite groups.

scholarly journals Dual combinatorics of zonal polynomials

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Valentin Féray ◽  
Piotr Sniady
Keyword(s):  
Irreducible Character ◽  
Symmetric Functions ◽  
Symmetric Groups ◽  
Combinatorial Formula ◽  
Zonal Polynomials ◽  
Power Sums ◽  
Power Sum ◽  
Recent Developments ◽  
International Audience ◽  
Special Case

International audience In this paper we establish a new combinatorial formula for zonal polynomials in terms of power-sums. The proof relies on the sign-reversing involution principle. We deduce from it formulas for zonal characters, which are defined as suitably normalized coefficients in the expansion of zonal polynomials in terms of power-sum symmetric functions. These formulas are analogs of recent developments on irreducible character values of symmetric groups. The existence of such formulas could have been predicted from the work of M. Lassalle who formulated two positivity conjectures for Jack characters, which we prove in the special case of zonal polynomials. Dans cet article, nous établissons une nouvelle formule combinatoire pour les polynômes zonaux en fonction des fonctions puissance. La preuve utilise le principe de l'involution changeant les signes. Nous en déduisons des formules pour les caractères zonaux, qui sont définis comme les coefficients des polynômes zonaux écrits sur la base des fonctions puissance, normalisés de manière appropriée. Ces formules sont des analogues de développements récents sur les caractères du groupe symétrique. L'existence de telles formules aurait pu être prédite à partir des travaux de M. Lassalle, qui a proposé deux conjectures de positivité sur les caractères de Jack, que nous prouvons dans le cas particulier des polynômes zonaux.

scholarly journals MODULAR DECOMPOSITION NUMBERS OF CYCLOTOMIC HECKE AND DIAGRAMMATIC CHEREDNIK ALGEBRAS: A PATH THEORETIC APPROACH

2018 ◽  
Vol 6 ◽  
Author(s):  
C. BOWMAN ◽  
A. G. COX
Keyword(s):  
Representation Theory ◽  
Symmetric Groups ◽  
Upper Bounds ◽  
Linear Groups ◽  
Theoretic Approach ◽  
Strong Linkage ◽  
Cherednik Algebras ◽  
Arbitrary Characteristic ◽  
General Linear Groups ◽  
Decomposition Numbers

We introduce a path theoretic framework for understanding the representation theory of (quantum) symmetric and general linear groups and their higher-level generalizations over fields of arbitrary characteristic. Our first main result is a ‘super-strong linkage principle’ which provides degree-wise upper bounds for graded decomposition numbers (this is new even in the case of symmetric groups). Next, we generalize the notion of homomorphisms between Weyl/Specht modules which are ‘generically’ placed (within the associated alcove geometries) to cyclotomic Hecke and diagrammatic Cherednik algebras. Finally, we provide evidence for a higher-level analogue of the classical Lusztig conjecture over fields of sufficiently large characteristic.

scholarly journals The generalized Gelfand–Graev characters of GLn(Fq)

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Scott Andrews ◽  
Nathaniel Thiem
Keyword(s):  
Finite Groups ◽  
Type A ◽  
General Linear ◽  
Linear Groups ◽  
General Linear Groups ◽  
Representations Of Finite Groups ◽  
International Audience ◽  
Unipotent Representations ◽  
Groups Of Lie Type

International audience Introduced by Kawanaka in order to find the unipotent representations of finite groups of Lie type, gener- alized Gelfand–Graev characters have remained somewhat mysterious. Even in the case of the finite general linear groups, the combinatorics of their decompositions has not been worked out. This paper re-interprets Kawanaka's def- inition in type A in a way that gives far more flexibility in computations. We use these alternate constructions to show how to obtain generalized Gelfand–Graev representations directly from the maximal unipotent subgroups. We also explicitly decompose the corresponding generalized Gelfand–Graev characters in terms of unipotent representations, thereby recovering the Kostka–Foulkes polynomials as multiplicities.

scholarly journals Estimating deep Littlewood-Richardson Coefficients

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Hariharan Narayanan
Keyword(s):  
Representation Theory ◽  
Approximation Scheme ◽  
General Linear ◽  
Linear Groups ◽  
General Linear Groups ◽  
International Audience ◽  
Structure Constants ◽  
Strongly Polynomial

International audience Littlewood Richardson coefficients are structure constants appearing in the representation theory of the general linear groups $(GL_n)$. The main results of this paper are: 1. A strongly polynomial randomized approximation scheme for Littlewood-Richardson coefficients corresponding to indices sufficiently far from the boundary of the Littlewood Richardson cone. 2. A proof of approximate log-concavity of the above mentioned class of Littlewood-Richardson coefficients. Coefficients de Littlewood Richardson sont des constantes de structure apparaissant dans la théorie de la représentation des groupes linéaires généraux $(GL_n)$. Les principaux résultats de cette étude sont les suivants: 1. Un schéma d’approximation polynomiale randomisée fortement pour des coefficients de Littlewood-Richardson correspondant aux indices suffisamment loin de la limite du cône Littlewood Richardson. 2. Une preuve de l’approximatif log-concavité de la classe de coefficients de Littlewood-Richardson mentionné ci-dessus.

A note on decomposition numbers for general linear groups and symmetric groups

1985 ◽  
Vol 97 (1) ◽  
pp. 57-62 ◽  
Author(s):  
S. Donkin
Keyword(s):  
Chevalley Group ◽  
Reductive Group ◽  
Symmetric Groups ◽  
General Linear ◽  
Linear Groups ◽  
Reductive Groups ◽  
Weyl Module ◽  
General Linear Groups ◽  
Group I ◽  
Decomposition Numbers

In [5] James proved theorems on the decomposition numbers, for the general linear groups and symmetric groups, involving the removal of the first row or column from partitions. In [1] we gave different proofs of these theorems based on a result valid for the decomposition numbers of any reductive group. (I am grateful to J. C. Jantzen for pointing out that the Theorem in [1] may also be derived from the universal Chevalley group case, which follows from the proof of 1 ·18 Satz of [6] – the analogue of equation (1) of the proof being obtained by means of the natural isometry (with respect to contra-variant forms) between a certain sum of weight spaces of a Weyl module V(λ) of highest weight λ and the Weyl module corresponding to λ for the Chevalley group determined by the subset of the base involved.) However, we have recently noticed that this result for reductive groups, even when specialized to the case of GLn, gives a substantial generalization of James's Theorems. This generalization, which we give here, is an expression for the decomposition number [λ: μ] for a pair of partitions λ, μ whose diagrams can be simultaneously cut by a horizontal (or vertical) line so as to leave the same number of nodes above the line (or to the left of the line for a vertical cut) in both cases. Cutting between the first and second rows gives James's principal of row removal ([5], theorem 1) and cutting between the first and second column gives his principle of column removal ([5], theorem 2). Another special case of our horizontal result, involving the removal of bottom rows of a pair of partitions, is stated in [7], Satz 8.

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