Jose Simental Rodriguez: Representations of Hecke algebras and lattice-path counting

Abstract

The branching rule for representations of the symmetric group tells us that, over a field of characteristic zero, the dimension of the irreducible representation indexed by a partition \lambda is given by the number of directed lattice paths from \lambda (though of as an integer vector) to the origin that stay inside the dominant Weyl chamber. Over a field of positive characteristic (or for Hecke algebras at roots of unity) this is no longer true, and the dimension of an irreducible is in general unknown. We will see, however, that there is a nice class of irreducibles (called calibrated, completely splittable or tame by different authors) whose dimension is given by the number of lattice paths to the origin that stay within a dilation of the fundamental alcove. Then we will provide an explicit resolution of these modules by Specht modules, whose dimensions are as in the characteristic zero case. This gives a representation-theoretic interpretation of combinatorial results of Filasetta, Krattenthaler and others. Based on joint work with C. Bowman and E. Norton.