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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:

05E30 Association schemes, strongly regular graphs
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