Character formulae for classical groups

We estimate the proportion of several classes of elements in finite classical groups which are readily recognised algorithmically, and for which some power has a large fixed point subspace and acts irreducibly on a complement of it. The estimates are used in complexity analyses of new recognition algorithms for finite classical groups in arbitrary characteristic.

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.

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.

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...

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 presents a powerfull method to integrate general monomials on the classical groups with respect to their invariant (Haar) measure. The method has first been applied to the orthogonal group in [J. Math. Phys. 43, 3342 (2002)], and is here used to obtain similar integration formulas for the unitary and the unitary symplectic group. The integration formulas turn out to be of similar form. They are all recursive, where the recursion parameter is the number of column (r...

In this investigation of character tables of finite groups we study basic sets and associated representation theoretic data for complementary sets of conjugacy classes. For the symmetric groups we find unexpected properties of characters on restricted sets of conjugacy classes, like beautiful combinatorial determinant formulae for submatrices of the character table and Cartan matrices with respect to basic sets; we observe that similar phenomena occur for the transition matri...