The Clebsch--Gordan rule and the Hamming graphs

Abstract

Let D ≥ 1 and q ≥ 3 be two integers. Let H(D) = H(D, q) denote the Ddimensional Hamming graph over a q-element set. Let T (D) denote the Terwilliger algebra of H(D). In this paper we apply the Clebsch-Gordan rule for U (sl 2 ) to decompose the standard T (D)-module into the direct sum of irreducible T (D)-modules.

The Terwilliger algebra of symplectic dual polar graphs, the subspace lattices and U(sl2)

“…such that V is both a T -and a G-module. Using ( 2) and (10), one finds that ρ(g) commutes with the matrices in the Bose-Mesner algebra, i.e.…”

“…In recent years, much efforts have been dedicated to the decomposition of the standard module of distance-regular graphs in irreducible submodules of their Terwilliger algebra. The Hamming [2,9,10,16] and Johnson 1 [3,7,15,18,22] cases have been worked out in great details. Distance-regular graphs associated to certain q-polynomials of the Askey-scheme have also received some attention.…”

The Terwilliger algebra of symplectic dual polar graphs, the subspace lattices and $U_q(sl_2)$

“…In recent years, much efforts have been dedicated to the decomposition of the standard module of distance-regular graphs in irreducible submodules of their Terwilliger algebra. The Hamming [2,9,10,16] and Johnson [3,7,15,18,22] cases have been worked out in great details. Distance-regular graphs associated to certain q-polynomials of the Askey-scheme have also received some attention.…”

The Terwilliger algebra of the halved cube

“…Note that H(D, 2) is an example of the D-dimensional Hamming graphs. In the case of the D-dimensional Hamming graphs, the decomposition formula for the standard modules was recently given in [5,Theorem 1.7] and [2, p. 17 By Lemma 3.2(i) we have A 2 = v 2 (A). Combined with Lemma 4.1 it follows that the eigenvalues of A are v 2 (θ ) = θ i for i = 0, 1, .…”