On characters of Weyl groups

Functional Analysis and Its Applications, 1982

2012

This paper describes a permutation notation for the Weyl groups of type F 4 and G 2. The image in the permutation group is presented as well as an analysis of the structure of the group. This description enables faster computations in these Weyl groups which will prove useful for a variety of applications.

Cornell University - arXiv, 2022

In this paper, we define a mixed-base number system over a Weyl group of type D, the group even-signed permutations. We introduce one-to-one correspondence between positive integers and elements of Weyl groups of type D after constructed the subexceedant function associating to the group. Thus, the integer representations of all classical Weyl groups are now completed.

Acta Polytechnica, 2016

Weyl group orbit functions are defined in the context of Weyl groups of simple Lie algebras. They are multivariable complex functions possessing remarkable properties such as (anti)invariance with respect to the corresponding Weyl group, continuous and discrete orthogonality. A crucial tool in their definition are so-called sign homomorphisms, which coincide with one-dimensional irreducible representations. In this work we generalize the definition of orbit functions using characters of irreducible representations of higher dimensions. We describe their properties and give examples for Weyl groups of rank 2 and 3.

Journal of Combinatorial Theory, Series A, 2001

A MacMahon symmetric function is a formal power series in a finite number of alphabets that is invariant under the diagonal action of the symmetric group. In this article, we show that the MacMahon symmetric functions are the generating functions for the orbits of sets of functions indexed by partitions under the diagonal action of a Young subgroup of a symmetric group. We define a MacMahon chromatic symmetric function that generalizes Stanley's chromatic symmetric function. Then, we study some of the properties of this new function through its connection with the noncommutative chromatic symmetric function of Gebhard and Sagan.

arXiv (Cornell University), 2020

Physica D: Nonlinear Phenomena, 1982

Consideration of a classification of the number of parlitions of a natural number according to the members of sub-partitions differing from unity leads to a non-recursive formula for the number of irreducible representations of the symmetric group 5%.

Journal of Mathematical Sciences, 2005

The present paper is a revised Russian translation of the paper "A new approach to representation theory of symmetric groups," Selecta Math., New Series, 2, No. 4, 581-605 (1996). Numerous modifications to the text were made by the first author for this publication. Bibliography: 35 titles. To the memory of D. Coxeter Preface This paper is a revised Russian translation of a paper by the same authors (see the reference below) and is devoted to a nontraditional approach to the representation theory of the symmetric groups (and, more generally, to the representation theory of Coxeter and local groups). The translation was prepared for the Russian edition of the book W. Fulton, Young Tableaux. With Applications to Representation Theory and Geometry, Cambridge Univ. Press, Cambridge, 1997, which, hopefully, will appear sooner or later. In the editor's preface to the Russian translation of the book it is explained what is the drawback of the conventional approach to the representation theory of the symmetric groups: it does not take into account important properties of these groups, namely, that they are Coxeter groups, and that they form an inductive chain, which implies that the theory must be constructed inductively. A direct consequence of these drawbacks is, in particular, that Young diagrams and tableaux appear ad hoc; there presence in the theory is justified only after the proof of the branching theorem. The theory described in this paper is intended to correct these defects. The first attempt in this direction was the paper [30] by the first author, in which it was proved that if we assume that the branching graph of irreducible complex representations of the symmetric groups is distributive, then it must be the Young graph. As it turned out, this a priori assumption is superfluous-the distributivity follows directly from the fact that S n is a Coxeter group if we involve remarkable generators of the Gelfand-Tsetlin subalgebra of the group algebra C[S n ], namely, the Young 1-Jucys 2-Murphy generators (see [19, 30]). But all numerous later expositions, including the very good book by Fulton, followed the classical version of the theory, which goes back to Frobenius, Schur, and Young; although some nice simplifications were made, such as von Neumann's lemma, Weyl's lemma, the notion of tabloids, etc., but the general scheme of the construction of the theory remained the same. 3 The reader can find references to the books on the representation theory of the symmetric groups in the monograph by James and Kerber [18], in the book by James [17], which was translated into Russian, and in earlier textbooks. The key point of our approach, which explains the appearance of Young tableaux as well as the general idea of our method, is that the points of the spectrum of the Gelfand-Tsetlin algebra with respect to the Young-Jucys-Murphy generators are so-called content vectors, i.e., integer vectors in R n that satisfy certain simple conditions, which follow from the Coxeter relations, and the coordinates of these integer vectors are the so-called contents of the boxes of Young tableaux (see Sec. 6); since the content vector uniquely determines a Young tableau, it follows that the points of the spectrum are precisely Young tableaux. The corresponding eigenvectors determine a basis in each representation, and the set of vectors corresponding to tableaux with a given diagram form a basis of the irreducible representation of S n (the Young-Gelfand-Tsetlin basis). Thus the correspondence "diagrams" ↔ "irreducible representations" obtains a natural (one might say, spectral) explanation.

European Journal of Combinatorics, 1990

The purpose of this note is to define a certain regular graph (Q, r) on 56 points and show that its automorphism group is the Weyl group of type E 7 , isomorphic to Z2 x Sp(6, 2) acting imprimitively with blocks of cardinality 2. It is well known that Weyl (E7) has an imprimitive permutation representations of degree 56, but the construction here is somewhat novel, beginning as it does, with the symmetric group of degree 8, acting on two sets of cardinality 28. This does not appear to be in the literature. There is a further reason for this construction. Application of the graph and the identification of its automorphism is made in [1], where a four homogeneous polynomial function I on a 56-dimensional space M over a field K is constructed, and it is shown that the group of isometries of I, O(M,f) = {g E GL(M) I/(gm) = I(m) for all m EM} is the universal Chevalley group E7(K). The organization of this paper is as follows: in Section 2 we define the graph and derive some initial properties; in Section 3 we identify the automorphism group; and in Section 4 we derive some further properties useful for [1].

Discrete Mathematics, 1978

Advances in Mathematics, 2008

We study asymptotics of an irreducible representation of the symmetric group S n corresponding to a balanced Young diagram λ (a Young diagram with at most O( √ n) rows and columns) in the limit as n tends to infinity. We find an optimal asymptotic bound for characters χ λ (π). Our main achievement is that-contrary to previous results in this direction-we do not assume that the length |π| of the permutation is small in comparison to n. Our main tool is an analogue of Frobenius character formula which holds true not only for cycles but for arbitrary permutations.

Journal of Algebra, 2002

Each symplectic group over the field of two elements has two exceptional doubly transitive actions on sets of quadratic forms on the defining symplectic vector space. This paper studies the associated 2-modular permutation modules. Filtrations of these modules are constructed which have subquotients which are modules for the symplectic group over an algebraically closed field of characteristic 2 and which, as such, have filtrations by Weyl modules and dual Weyl modules having fundamental highest weights. These Weyl modules have known submodule structures. It is further shown that the submodule structures of the Weyl modules are unchanged when restricted to the finite subgroups Sp(2n, 2) and O ± (2n, 2).

European Journal of Combinatorics, 2006

The generating functions of the major index and of the flag-major index, with each of the one-dimensional characters over the symmetric and hyperoctahedral group, respectively, have simple product formulas. In this paper, we give a factorial-type formula for the generating function of the D-major index with sign over the Weyl groups of type D. This completes a picture which is now known for all the classical Weyl groups.

Discrete applied mathematics, 1996

A conjecture concerning the construction of explicit expressions for the central characters of the symmetric group S, is presented. The expression for the central characters corresponding to a class with a given cycle structure (l)"(2)" . (n)'" is a polynomial in the symmetric power sums over the "contents" of the Young diagram specifying the irreducible representation of interest. Each term in this polynomial is a product of symmetric power sums multiplied by a polynomial in n. Both the degrees of the polynomials in n and the powers of the various symmetric power sums appearing in each term are specified by an algorithm involving sets of partitions of integers associated with the lengths of the nontrivial cycles specifying the class of interest and combinations thereof. The coefficients of the polynomials in n are obtained by solving a system of linear equations which arise from the evaluation of the proposed expression for the central characters with respect to Young diagrams whose number of boxes is less than EYE, if,, and equating to zero. 0166-218X/96/$15.00 0 1996 Elsevier Science B.V. All rights reserved SSDI 0166-218X(95)00016-X

Annals of Combinatorics, 2010

We determine all the multiplicity-free representations of the symmetric group. This project is motivated by a combinatorial problem involving systems of set-partitions with a specific pattern of intersection.