数理解析研究所講究録
991 巻 1997 年 108-109

108
The Terwilliger Algebra for Bipartite
P- and -polynomial Schemes$Q$

John S. Caughman, IV 1

Extended Abstract.

Let $Y=(X, \{\kappa\}_{0\leq i\leq D})$ denote a symmetric association scheme with
$D\geq 3$ . Suppose
is bipartite P- and -polynomial, and fix any $x\in X$ .
to . Thefor with
Let $T=T(x)$ denote the Terwilligel
by matrix multiplication, andalgebra acts on the vector space
is equipped with the standard
is referred to as the standard module for T.
is a semisimple matrix algebra,. It is known thatinner product on
decomposes into a direct sum
and so by the Wedderburn-Artin theorenl,of irreducible -modules. We study the action of on these modules.
,and let$E_{1,\ldots,D}E$ denote the primitive idempotents forLetwith respect to . Fix...,denote the dual primitive idempotents forrespectively denote, and, and let
any irreducible -module
the endpoint, diameter, dual-endpoint and dual-diameter of . In othelwords, set
$Q$

$Y$

$Y$

$\cdot.$$\mathrm{a}(_{J}^{\mathrm{J}\mathrm{b}\mathrm{l}\mathrm{a}}$$1^{\cdot}\mathrm{e}_{J}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}$$.’\iota$$V$$V=\mathbb{C}^{X}$

$T$

$V$

$T$

$\mathbb{C}^{X}$

$V$

$T$

$T$

$Y$

$E_{0},$$E_{0}^{*},$$Y$$E_{D}^{*}$$T$$W\subseteq V$

$r,$

$d,$

$E_{1}^{*}$$x$$d^{*}$$t$$\nu V$

(1)

,$r$

$:=$

$?ni\uparrow\iota\{i|E_{i}^{*}W\neq 0\}$

$d$

$:=$

$|\{i|E_{\dot{i}}^{*}W\neq 0\}|-1$

$t$

$:=$

$mi?\iota\{i|E_{i}W\neq 0\}$$d^{*}$$:=$$|\{i|E_{\dot{i}}W\neq 0\}|-1$$\cdot$

,

,
.

(2)

(3)
(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$d^{*}$,$\geq$$D$,$=$

D.

$d$

$=$

$2r+d$$2t+d$

(5)
(6)
(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$05\mathrm{E}30$.

109(iii) For any nonzero $v\in E_{t}W$ ,$E_{r}^{*}v,$$Ev,r+d*v$ is an orthogonal basis for $W$.

$E_{r+1}*\ldots,$(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 on these bases by generalizing the intersectionand dual-intersection numbers of . These constants are then computed fromthe eigenvalues and dual-eigenvalues of . Using these expressions, we proveandthat the isomorphism class of $W$ is deternuined by twoexpressions for, the endpoint and diameter of $W,$ and. we obtainthe square-norms of our basis vectors for $W$ . In addition, we show how torecursively compute the multiplicities with which the irreducible T-modules
occur in the Wedderburn decomposition of . Finally, we carry out all of
the above computations for the. bipartite schemes of type I.
$T$$Y$$Y$$\mathrm{p}\mathrm{a}\mathrm{l}\cdot \mathrm{a}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{e}\Gamma \mathrm{s},$

$d$

$r$

$\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{l}\mathrm{p}\mathrm{l}\mathrm{e}$$V$

References.
E. Bannai and T. Ito.
$\mathrm{B}\mathrm{e}\mathrm{n}\mathrm{j}\mathrm{a}\mathrm{m}\mathrm{i}\mathrm{n}/\mathrm{C}\mathrm{u}\mathrm{m}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{s}$Algebraic Combinatorics I.. Association Schemes.
, 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.
Algebraic Combin., 1 (4) : 363-388, 1992.

.

$J$