Parametrizations of Irreducible Rational Representations of Coxeter Groups

Abstract

We develop two closely related methods for parametrizing the rational irreducible characters of an arbitrary finite Coxeter group W. The goal is to provide a uniform approach to such a parametrization, independent of Coxeter type. The two methods generalize two approaches to describing the irreducible representations of the symmetric groups, which coincide in type A but do not coincide generally. Our methods associate characters to pairs of reflection subgroups, in one case by considering common constituents of permutation and signed permutation modules, and in the other case by a generalization of the Specht modules. We ask whether, using either method, we can identify a set of subgroup pairs for which the matrix of multiplicities of rational irreducibles in the characters associated to these subgroup pairs is unitriangular. Such a unitriangular matrix provides a parametrization of the irreducible rational characters. For the noncrystallographic types H and I, we give a positive answer to this question. In type H, we show computationally that we can parametrize the irreducible rational characters of H4 using generalized Specht modules, and we can parametrize the irreducible characters of H3 using both methods. Moreover, we give an explicit decomposition of the generalized common constituents for the dihedral groups I2(n) for all n, and we prove that we can always exhibit a unitriangular multiplicity matrix using generalized common constituents. In type A our theory coincides with the classical theory of Specht modules and does not give any new information. In type B the approach we take is closely related to an existing parametrization of the irreducible characters, but it appears to have some novel elements.