Dual polar graphs, the quantum algebra [formula omitted], and Leonard systems of dual q-Krawtchouk type

Abstract

In this paper we consider how the following three objects are related: (i) the dual polar graphs; (ii) the quantum algebra Uq(sl2); (iii) the Leonard systems of dual q-Krawtchouk type. For convenience we first describe how (ii) and (iii) are related. For a given Leonard system of dual q-Krawtchouk type, we obtain two Uq(sl2)-module structures on its underlying vector space. We now describe how (i) and (iii) are related. Let Γ denote a dual polar graph. Fix a vertex x of Γ and let T=T(x) denote the corresponding subconstituent algebra. By definition T is generated by the adjacency matrix A of Γ and a certain diagonal matrix A*=A*(x) called the dual adjacency matrix that corresponds to x. By construction the algebra T is semisimple. We show that for each irreducible T-module W the restrictions of A and A* to W induce a Leonard system of dual q-Krawtchouk type. We now describe how (i) and (ii) are related. We obtain two Uq(sl2)-module structures on the standard module of Γ. We describe how these two Uq(sl2)-module structures are related. Each of these Uq(sl2)-module structures induces a C-algebra homomorphism Uq(sl2)→T. We show that in each case T is generated by the image together with the center of T. Using the combinatorics of Γ we obtain a generating set L,F,R,K of T along with some attractive relations satisfied by these generators.