## A generalization of the Terwilliger algebra.(English)Zbl 0960.05108

A character algebra $$C=\langle X_0,\dots{},X_d\rangle$$ is a finite dimensional associative commutative $$\mathbb{C}$$-algebra with a basis $$1=X_0,\dots{},X_d$$ having the following three properties: (1) $$X_iX_j=\sum_{h=0}^d p_{ij}^h X_h$$ for some real numbers $$p_{ij}^h$$ (structure constants of $$C$$). (2) There exist a permutation $$i\mapsto i'$$ of $$\{0,\dots{},d\}$$ and positive numbers $$k_i$$ (valencies of $$C$$) such that $$p_{ij}^0=\delta_{ji'}k_i$$. (3) The linear map $$\tau: C\to C$$ with $$\tau(X_i)=X_{i'}$$ is a $$\mathbb{C}$$-algebra isomorphism and the linear map $$\pi_0: C\to {\mathbb{C}}$$ with $$\pi_0(X_i)=k_{i}$$ is a $$\mathbb{C}$$-algebra homomorphism. The scalar $$N=\sum_{i=0}^d k_i$$ is called the size of $$C$$. There exists a basis $$E_0,\dots{},E_d$$ of $$C$$ (unique up to a permutation of $$E_1,\dots{},E_d$$) such that $$E_iE_j=\delta_{ij}E_i$$, $$X_0=\sum_{i=0}^d E_i$$ and $$E_0=N^{-1}\sum_{i=0}^d X_i$$. We refer to the elements $$E_0,\dots{},E_d$$ as the primitive idempotents of $$C$$. Let $$X_i=\sum_{j=0}^d p_i(j)E_j$$ and $$E_i=N^{-1}\sum_{j=0}^d q_i(j)X_i$$. Then $$P=(P_{ij})$$ with $$P_{ij}=p_j(i)$$ is called the matrix of eigenvalues of $$C$$ (associated with the ordering $$E_0,\dots{},E_d$$). Suppose $$C=\langle X_0,\dots{},X_d\rangle$$ and $$C^*=\langle X_0^*,\dots{},X_d^* \rangle$$ are character algebras. Fix an ordering $$E_0,\dots{},E_d$$ of the primitive idempotents of $$C$$ with associated matrix of eigenvalues $$P$$ and an ordering $$E_0^*,\dots{},E_d^*$$ of the primitive idempotents of $$C^*$$ with associated matrix of eigenvalues $$P^*$$. We say that $$C$$ and $$C^*$$ are dual whenever $$PP^*\in \text{Span}\{I\}$$. In this case the size of $$C$$ and the size of $$C^*$$ coincide and $$PP^*=NI$$. Suppose $$C=\langle X_0,\dots{},X_d\rangle$$ and $$C^*=\langle X_0^*,\dots{},X_d^* \rangle$$ are character algebras which are dual with respect to the orderings $$E_0,\dots{},E_d$$ and $$E_0^*,\dots{},E_d^*$$ of their primitive idempotents. The Terwilliger algebra $$\mathcal T$$ is the associative $$\mathbb{C}$$-algebra with 1 which is generated by $$x_0,\dots{},x_d,x_0^*,\dots{},x_d^*$$ subject to relations $$x_0=x_0^*$$, $$x_ix_j=\sum_{h=0}^d p_{ij}^h x_h$$, $$x_i^*x_j^*=\sum_{h=0}^d p_{ij}^{h*} x_h^*$$, $$e_h^*x_ie_j^*=0$$ if $$p_{ij}^h=0$$, $$e_hx_i^*e_j=0$$ if $$p_{ij}^{h*}=0$$, where $$p_{ij}^h$$ and $$p_{ij}^{h*}$$ are structure constants of $$C$$ and $$C^*$$ respectively, $$e_i=N^{-1}\sum_{j=0}^d q_i(j)x_j$$ and $$e_i^*=N^{-1} \sum_{j=0}^d q_i^*(j)x_j^*$$. The element $$u_0=N \sum_{r=0}^d k_r^{-1}e_r^*e_0e_r^*$$ is a central idempotent of $$\mathcal T$$ and $$\mathcal T$$ is a direct sum of two sided ideals $${\mathcal T}u_0$$ and $${\mathcal T}(1-u_0)$$. The following results are obtained in this paper. The algebra $${\mathcal T}u_0$$ is isomorphic to $$M_{d+1}({\mathbb{C}})$$. There exists a unique irreducible $$\mathcal T$$-module $$V$$ with $$e_0V\neq 0$$ (Proposition 8.4). We refer to $$V$$ as the primary module of $$\mathcal T$$. Two bases of the primary module are found, one diagonalizes $$C$$ and the other diagonalizes $$C^*$$.

### MSC:

 5e+30 Association schemes, strongly regular graphs

### Keywords:

Terwilliger algebra; character algebra
Full Text:

### References:

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