Character formulae for classical groups

In this paper we consider the character of a finite dimensional algebraic representation of $GL_{mn}({\mathbb C})$ restricted to a particular disconnected component of the normalizer of the Levi subgroup $GL_m({\mathbb C})^n$ of $GL_{mn}(\mathbb C)$, generalizing a theorem of Kostant on the character values at the Coxeter element.

Lusztig's algorithm of computing generalized Green functions of reductive groups involves an ambiguity of certain scalars. In this paper, for reductive groups of classical type with arbitrary characteristic, we determine those scalars explicitly, and eliminate the ambiguity. Our results imply that all the generalized Green functions of classical type are computable.

Let G be any of the complex classical groups GL(n), SO(2n+1), Sp(2n), O(2n), let g denote the Lie algebra of G, and let Z(g) denote the subalgebra of G-invariants in the universal enveloping algebra U(g). We derive a Taylor-type expansion for finite-dimensional characters of G (binomial formula) and use it to specify a distinguished linear basis in Z(g). The eigenvalues of the basis elements in highest weight g-modules are certain shifted (or factorial) analogs of Schur funct...

We derive a Murnaghan--Nakayama type formula for the values of unipotent characters of finite classical groups on regular semisimple elements. This relies on Asai's explicit decomposition of Lusztig restriction. We use our formula to show that most complex irreducible characters vanish on some $\ell$-singular element for certain primes $\ell$. As an application we classify the simple endotrivial modules of the finite quasi-simple classical groups. As a further application w...

The Laplacian on the rotation group is invariant by conjugation. Hence, it maps class functions to class functions. A maximal torus consists of block diagonal matrices whose blocks are planar rotations. Class functions are determined by their values of this maximal torus. Hence, the Laplacian induces a second order operator on the maximal torus called the radial Laplacian. In this paper, we derive the expression of the radial Laplacian. Then, we use it to find the eigenvalues...

In this paper, we prove that the dimension of the space spanned by the characters of the symmetric powers of the standard n-dimensional representation of the symmetric group S_n is asymptotic to n^2/2. This is proved by using generating functions to obtain formulas for upper and lower bounds, both asymptotic to n^2/2, for this dimension. In particular, for n>6, these characters do not span the full space of class functions on S_n.

This book describes some computational methods to deal with modular characters of finite groups. It is the theoretical background of the MOC system of the same authors. This system was, and is still used, to compute the modular character tables of sporadic simple groups.

Motivated by statistical applications, this paper introduces Cauchy identities for characters of the compact classical groups. These identities generalize the well-known Cauchy identity for characters of the unitary group, which are Schur functions of symmetric function theory. Application to statistical hypothesis testing is briefly sketched.

This paper is a significant part of a general project aimed to classify all irreducible representations of finite quasi-simple groups over an algebraically closed field, in which the image of at least one element is represented by an almost cyclic matrix. (A square matrix $M$ is called almost cyclic if it is similar to a block-diagonal matrix with two blocks, such that one block is scalar and another block is a matrix whose minimum and characteristic polynomials coincide. Ref...

In a previous paper, "Generalized Green functions and unipotent classes for finite reductive groups, I", we have determined certain unknown scalars involved in the algorithm of computing generalized Green functions in the case of SL_n. In this paper, we consider a similar problem for the case of classical groups Sp_{2n}, SO_N with arbitrary characteristic. Our result enables us, in particular, to compute Green functions in bad characteristic case, which was not known before.