The Terwilliger algebra of an almost-bipartite P- and Q-polynomial association scheme

Abstract

Let Y denote a D-class symmetric association scheme with D ≥ 3, and suppose Y is almostbipartite P-and Q-polynomial. Let x denote a vertex of Y and let T = T (x) denote the corresponding Terwilliger algebra. We prove that any irreducible T -module W is both thin and dual thin in the sense of Terwilliger. We produce two bases for W and describe the action of T on these bases. We prove that the isomorphism class of W as a T -module is determined by two parameters, the dual endpoint and diameter of W . We find a recurrence which gives the multiplicities with which the irreducible T -modules occur in the standard module. We compute this multiplicity for those irreducible T -modules which have diameter at least D − 3.

The Terwilliger algebra of the incidence graphs of Johnson geometry, II

“…For more information, see [4][5][6]. The Terwilliger algebras of group schemes were discussed in [1,2].…”

New code upper bounds for the folded n-cube

“…By the multiplicity with which W appears in V , we mean the number of irreducible T -modules in this sum which are isomorphic to W . See [3,4,9,10] for more information on the Terwilliger algebra. Lemma 2.1.…”

“…It follows from Proposition 4.6(i)-(iv) that for each triple (i, j, t) ∈ I the matrix M t i,j := U T 2 M t i,j U 2 is in block diagonal form: for each even 0 ≤ r ≤ D − 1 there are 2D Proof. From [4] we known that W is thin, r + d * = D and the isomorphism class of W is determined by its dual endpoint and d * . By [1, pp.…”

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“…They have been studied by many researchers; for example, see [3][4][5], etc. Recently, Kurihara and Nozaki [7] obtained a necessary and sufficient condition for a symmetric association scheme to be a Q-polynomial association scheme.…”