Tridiagonal pairs, alternating elements, and distance-regular graphs

Abstract

The positive part Uq+ of Uq(slˆ2) has a presentation with two generators W0, W1 and two relations called the q-Serre relations. The algebra Uq+ contains some elements, said to be alternating. There are four kinds of alternating elements, denoted {W−k}k∈N, {Wk+1}k∈N, {Gk+1}k∈N, {G˜k+1}k∈N. The alternating elements of each kind mutually commute. A tridiagonal pair is an ordered pair of diagonalizable linear maps A,A⁎ on a nonzero, finite-dimensional vector space V, that each act in a (block) tridiagonal fashion on the eigenspaces of the other one. Let A, A⁎ denote a tridiagonal pair on V. Associated with this pair are six well-known direct sum decompositions of V; these are the eigenspace decompositions of A and A⁎, along with four decompositions of V that are often called split. In our main results, we assume that A, A⁎ has q-Serre type. Under this assumption A, A⁎ satisfy the q-Serre relations, and V becomes an irreducible Uq+-module on which W0=A and W1=A⁎. We describe how the alternating elements of Uq+ act on the above six decompositions of V. We show that for each decomposition, every alternating element acts in either a (block) diagonal, (block) upper bidiagonal, (block) lower bidiagonal, or (block) tridiagonal fashion. We investigate two special cases in detail. In the first case the eigenspaces of A and A⁎ all have dimension one. In the second case A and A⁎ are obtained by adjusting the adjacency matrix and a dual adjacency matrix of a distance-regular graph that has classical parameters and is formally self-dual.