Character formulae for classical groups

We derive several identities that feature irreducible characters of the general linear, the symplectic, the orthogonal, and the special orthogonal groups. All the identities feature characters that are indexed by shapes that are "nearly" rectangular, by which we mean that the shapes are rectangles except for one row or column that might be shorter than the others. As applications we prove new results in plane partitions and tableaux enumeration, including new refinements of t...

The aim of this paper is to give another proof of a theorem of D.Prasad, which calculates the character of an irreducible representation of $\text{GL}(mn,\mathbb{C})$ at the diagonal elements of the form $\underline{t} \cdot c_n$, where $\underline{t}=(t_1,t_2,\cdots,t_m)$ $\in$ $(\mathbb{C}^*)^{m}$ and $c_n=(1,\omega_n,\omega_n^{2},\cdots,\omega_n^{n-1})$, where $\omega_n=e^{\frac{2\pi \imath}{n}}$, and expresses it as a product of certain characters for $\text{GL}(m,\mathbb...

We give an $AB$-factorization of the supercharacter table of the group of $n\times n$ unipotent upper triangular matrices over $\FF_q$, where $A$ is a lower-triangular matrix with entries in $\ZZ[q]$ and $B$ is a unipotent upper-triangular matrix with entries in $\ZZ[q^{-1}]$. To this end we introduce a $q$ deformation of a new power-sum basis of the Hopf algebra of symmetric functions in noncommutative variables. The factorization is obtain from the transition matrices betwe...

This survey article is an introduction to some of Lusztig's work on the character theory of a finite group of Lie type $G(F_q)$, where $q$ is a power of a prime~$p$. It is partly based on two series of lectures given at the Centre Bernoulli (EPFL) in July 2016 and at a summer school in Les Diablerets in August 2015. Our focus here is on questions related to the parametrization of the irreducible characters and on results which hold without any assumption on~$p$ or~$q$.

We provide an elementary and self-contained derivation of formulae for products and ratios of characteristic polynomials from classical groups using classical results due to Weyl and Littlewood.

We determine invariants like the Smith normal form and the determinant for certain integral matrices which arise from the character tables of the symmetric groups S_n and their double covers. In particular, we give a simple computation, based on the theory of Hall-Littlewood symmetric functions, of the determinant of the regular character table of S_n with respect to an integer r>1. This result had earlier been proved by Olsson in a longer and more indirect manner. As a conse...

This paper describes how to use subgroups to parameterize unipotent classes in the classical algebraic group in characteristic 2. These results can be viewed as an extension of the Bala-Carter Theorem, and give a convenient way to compare unipotent classes in a group $G$ with unipotent classes of a subgroup $X$ where $G$ is exceptional and $X$ is a Levi subgroup of classical type.

The paper studies how to compute irreducible characters of the generalized symmetric group $C_k\wr{S}_n$ by iterative algorithms. After reproving the Murnaghan-Nakayama rule by vertex algebraic method, we formulate a new iterative formula for characters of the generalized symmetric group. As applications, we find a numerical relation between the character values of $C_k\wr S_n$ and modular characters of $S_{kn}$.

We prove a determinantal type formula to compute the characters for a class of irreducible representations of the general Lie superalgebra $\mathfrak{gl}(m|n)$ in terms of the characters of the symmetric powers of the fundamental representation and their duals. This formula was conjectured by J. van der Jeugt and E. Moens and was generalized the well-known Jacobi-Trudi formula.

Let $\text{U}(n,\mathbb{F}_{q^2})$ denote the subgroup of unitary matrices of the general linear group $\text{GL}(n,\mathbb{F}_{q^2})$ which fixes a Hermitian form and $M\geq 2$ an integer. This is a companion paper to the previous works where the elements of the groups $\text{GL}(n,\mathbb{F}_{q})$, $\text{Sp}(2n,\mathbb{F}_{q})$, $\text{O}^{\pm}(2n,\mathbb{F}_{q})$ and $\text{O}(2n+1,\mathbb{F}_{q})$ which has an $M$-th root in the concerned group, have been described. Here...