Caughman IV, John S.

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Title The Terwilliger Algebra for Bipartite $P$-and $Q$-polynomialSchemes(Groups and Combinatorics)Author(s)Caughman, John S., IVCitation数理解析研究所講究録 (1997), 991: 108-109Issue Date1997-05URL http://hdl.handle.net/2433/61117RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityThe Terwilliger Algebra for BipartiteP- and $Q$-polynomial SchemesJohn S. Caughman, IV 1Extended Abstract.Let $Y=(X, \{\kappa\}_{0\leq i\leq D})$ denote a symmetric association scheme with$D\geq 3$ . Suppose $Y$ is bipartite P- and $Q$-polynomial, and fix any $x\in X$ .Let $T=T(x)$ denote the Terwilligel$\cdot.$$\mathrm{a}(_{J}^{\mathrm{J}\mathrm{b}\mathrm{l}\mathrm{a}}$ for $Y$ with $1^{\cdot}\mathrm{e}_{J}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}$ to $.’\iota$ . Thealgebra $T$ acts on the vector space $V=\mathbb{C}^{X}$ by matrix multiplication, and $V$is referred to as the standard module for T. $V$ is equipped with the standardinner product on $\mathbb{C}^{X}$ . It is known that $T$ is a semisimple matrix algebra,and so by the Wedderburn-Artin theorenl, $V$ decomposes into a direct sumof irreducible $T$-modules. We study the action of $T$ on these modules.Let $E_{0},$ $E_{1,\ldots,D}E$ denote the primitive idempotents for $Y$ and let $E_{0}^{*},$ $E_{1}^{*}$ ,..., $E_{D}^{*}$ denote the dual primitive idempotents for $Y$ with respect to $x$ . Fixany irreducible $T$-module $W\subseteq V$ , and let $r,$ $d,$ $t$ , and $d^{*}$ respectively denotethe endpoint, diameter, dual-endpoint and dual-diameter of $\nu V$ . In othel$\cdot$words, set$r$ $:=$ $?ni\uparrow\iota\{i|E_{i}^{*}W\neq 0\}$ , (1)$d$ $:=$ $|\{i|E_{\dot{i}}^{*}W\neq 0\}|-1$ , (2)$t$ $:=$ $mi?\iota\{i|E_{i}W\neq 0\}$ , (3)$d^{*}$ $:=$ $|\{i|E_{\dot{i}}W\neq 0\}|-1$ . (4)We prove the following theorem.Theorem. With the above notation, let $W$ denote any irreducible T-modulefor Y. Then(i) $W$ must satisfy each of the following$d$ $=$ $d^{*}$ , (5)$2r+d$ $\geq$ $D$ , (6)$2t+d$ $=$ D. (7)(ii) $W$ is thin and dual-thin.1Dept. of Mathematics, University of Wisconsin, 480 Lincoln Dr., Madison, WI 53706.Email: caughman@math.wisc.edu. AMS 1991 Subject Classification: Primary $05\mathrm{E}30$ .991 1997 108-109 108(iii) For any nonzero $v\in E_{t}W$ ,$E_{r}^{*}v,$ $E_{r+1}*\ldots,$$Ev,r+d*v$ is an orthogonal basis for $W$.(iv) For any nonzero $v\in E_{r}^{*}W$ ,$E_{t}v,$ $Et+1v,$$\ldots,$ $Et+dv$ is an orthogonal basis for $\nu V$.We describe the action of $T$ on these bases by generalizing the intersectionand dual-intersection numbers of $Y$ . These constants are then computed fromthe eigenvalues and dual-eigenvalues of $Y$ . Using these expressions, we provethat the isomorphism class of $W$ is deternuined by two $\mathrm{p}\mathrm{a}\mathrm{l}\cdot \mathrm{a}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{e}\Gamma \mathrm{s},$ $r$ and$d$ , the endpoint and diameter of $W,$ and. we obtain $\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{l}\mathrm{p}\mathrm{l}\mathrm{e}$ expressions forthe square-norms of our basis vectors for $W$ . In addition, we show how torecursively compute the multiplicities with which the irreducible T-modulesoccur in the Wedderburn decomposition of $V$ . Finally, we carry out all ofthe above computations for the. bipartite schemes of type I.References.E. Bannai and T. Ito. Algebraic Combinatorics I.. Association Schemes.$\mathrm{B}\mathrm{e}\mathrm{n}\mathrm{j}\mathrm{a}\mathrm{m}\mathrm{i}\mathrm{n}/\mathrm{C}\mathrm{u}\mathrm{m}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{s}$, London, 1984.A. E. Brouwer, A. M. Cohen, and A. Neumaier. Distance-Regular Graphs.Springer-Verlag, Berlin, 1989.P. Terwilliger. The subconstituent algebra of an association scheme, I. $J$.Algebraic Combin., 1 (4) : 363-388, 1992.109

The Terwilliger Algebra for Bipartite $P$-and $Q$-polynomial Schemes(Groups and Combinatorics)

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The Terwilliger Algebra for Bipartite $P$ -and $Q$ -polynomial Schemes(Groups and Combinatorics)

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The Terwilliger Algebra for Bipartite PPP-and QQQ-polynomial Schemes(Groups and Combinatorics)

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The Terwilliger Algebra for Bipartite $P$ -and $Q$ -polynomial Schemes(Groups and Combinatorics)