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The Terwilliger algebras of bipartite \(P\)- and \(Q\)-polynomial schemes. (English) Zbl 0924.05067
Let \(Y=(X,\{R_i\}_{0\leq i\leq D})\) be a bipartite \(P\)- and \(Q\)-polynomial association scheme. Then \(A_0,\dots,A_D\) form a basis for the Bose-Mesner algebra \(M\) of \(Y\) (\(A_i\) is the \(i\)th associate matrix of \(Y\)). We now recall the dual Bose-Mesner algebra \(M^*\) of \(Y\). Fix any \(x\in X\). For integer \(i\) (\(0\leq i\leq D\)) let \(E_i^*=E_i^*(x)\) denote the diagonal matrix with the rows and columns indexed by \(X\) and \((E_i^*)yy=1\), if \(xy\in R_i\), \((E_i^*)yy=0\), if \(xy\notin R_i\). We refer to \(E_i^*\) as the \(i\)th dual idempotent of \(Y\) with respect to \(x\). It follows that the matrices \(E_0^*,\dots,E_D^*\) form a basis for a algebra \(M^*=M^*(x)\) known as the dual Bose-Mesner algebra of \(Y\) with respect to \(x\). The subalgebra \(T=T(x)\) of \(\text{Mat}_X({\mathbb{C}})\) generated by \(M\) and \(M^*\) is called the Terwilliger algebra of \(Y\) with respect to \(x\). Let \(V\) denote the vector space \({\mathbb{C}}^X\) (column vectors). Then \(\text{Mat}_X({\mathbb{C}})\) acts on \({\mathbb{C}}^X\) by left multiplication. By a \(T\)-module we mean a subspace \(W\) of \(V\) such that \(TW\subseteq W\). The author proves that any irreducible \(T\)-module \(W\) is both thin and dual thin in the sense of Terwilliger. The isomorphism class of \(W\) as a \(T\)-module is determined by two parameters, the endpoint and diameter of \(W\). A recurrence which gives the multiplicities with which the irreducible \(T\)-modules occur in standard module \(V\) is found.

MSC:
05E30 Association schemes, strongly regular graphs
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