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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.

05E30 Association schemes, strongly regular graphs
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[1] Balmaceda, J.M.P.; Oura, M., The Terwilliger algebras of the group association schemes of S5 and A5, Kyushu J. math., 48, 221-231, (1994) · Zbl 0821.05059
[2] Bannai, E.; Ito, T., Algebraic combinatorics I: association schemes, (1984), Benjamin/Cummings London · Zbl 0555.05019
[3] Bannai, E.; Munemasa, A., The Terwilliger algebras of group association schemes, Kyushu J. math., 49, 93-102, (1995) · Zbl 0839.05095
[4] Brouwer, A.E., On the uniqueness of a certain thin near octagon (or partial 2-geometry, or parallelism) derived from the binary golay code, IEEE trans. inform. theory, IT-29, 370-371, (1983) · Zbl 0505.94014
[5] Brouwer, A.E.; Cohen, A.M.; Neumaier, A., Distance-regular graphs, (1989), Springer-Verlag Berlin · Zbl 0747.05073
[6] J.S. Caughman IV, Spectra of bipartite P- and Q-polynomial association schemes, Graphs Combin., to appear. · Zbl 1054.05101
[7] B.V.C. Collins, The Terwilliger algebra of an almost-bipartite distance-regular graph and its antipodal cover, Preprint. · Zbl 0955.05113
[8] Collins, B.V.C., The girth of a thin distance-regular graph, Graphs combin., 13, 21-34, (1997)
[9] B. Curtin, Bipartite distance-regular graphs, Preprint. · Zbl 0939.05088
[10] Curtis, C.W.; Reiner, I., Representation theory of finite groups and associative algebras, (1962), Interscience New York · Zbl 0131.25601
[11] G. Dickie, A note on bipartite P-polynomial association schemes and dual-bipartite Q-polynomial association schemes, Preprint. · Zbl 0898.05084
[12] Dickie, G., Twice Q-polynomial distance-regular graphs are thin, European, J. combin., 16, 555-560, (1995) · Zbl 0852.05085
[13] Egawa, Y., Characterization of H(n, q) by the parameters, J. combin. theory ser. A, 31, 108-125, (1981) · Zbl 0472.05056
[14] S.A. Hobart, T. Ito, The structure of nonthin irreducible T-modules: Ladder bases and classical parameters, J. Algebraic Combin., to appear. · Zbl 0911.05059
[15] Ishibashi, H., The Terwilliger algebras of certain association schemes over the Galois rings of characteristic 4, Graphs combin., 12, 39-54, (1996) · Zbl 0852.05081
[16] Tanabe, K., The irreducible modules of the Terwilliger algebras of Doob schemes, J. algebraic combin., 6, 173-195, (1997) · Zbl 0868.05056
[17] Terwilliger, P., P- and Q-polynomial schemes with q = −1, J. combin. theory ser. B, 42, 64-67, (1987) · Zbl 0675.05016
[18] Terwilliger, P., The classification of distance-regular graphs of type IIB, Combinatorica, 8, 125-132, (1988) · Zbl 0727.05050
[19] Terwilliger, P., The subconstituent algebra of an association scheme, I, J. algebraic combin., 1, 4, 363-388, (1992) · Zbl 0785.05089
[20] Terwilliger, P., The subconstituent algebra of an association scheme, II, J. algebraic combin., 2, 1, 73-103, (1993) · Zbl 0785.05090
[21] Terwilliger, P., The subconstituent algebra of an association scheme, III, J. algebraic combin., 2, 2, 177-210, (1993) · Zbl 0785.05091
[22] Tomiyama, M.; Yamazaki, N., The subconstitutent algebra of a strongly regular graph, Kyushu J. math., 48, 323-334, (1994) · Zbl 0842.05098
[23] Watatani, Y., Association schemes, Terwilliger algebras, and takesaki duality, (), 19-31, (Japanese)
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