The generalized Terwilliger algebra and its finite-dimensional modules when \(d=2\)
A character algebra \(C=\langle X_0,\dots ,X_d\rangle\) is a finite-dimensional associative commutative \(\mathbf C\)-algebra together with a basis \(X_0,\dots ,X_d\) having the following properties: (1) \(X_0\) is the multiplicative identity element of \(C\); (2) \(X_iX_j=\sum_{h=0}^d p_{ij}^hX_h\) for \(p_{ij}^h\in {\mathbf R}\); (3) there exist an involution \(i\mapsto i'\) of \(0,\dots ,d\) and natural numbers \(k_i\) such that \(p_{ij}^0=\delta_{ji'}k_i\); (4) the linear map \(\tau: C\to C\), which satisfies \(\tau(X_i)=X_{i'}\) is a \(\mathbf C\)-algebra isomorphism; (5) the linear map \(\pi_0: C\to {\mathbf C}\), which satisfies \(\pi_0(X_i)=k_i\) is a \(\mathbf C\)-algebra homomorphism. The number \(N=\sum_{i=0}^d k_i\) is called the size of \(C\). There exists a basis \(E_0,\dots ,E_d\) of \(C\) (unique up to a permutation of \(E_0,\dots ,E_d\)) 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 \(C\). If \(X_i=\sum_{r=0}^d P_{ri}E_r\), then the matrix \(P\) is called the matrix of eigenvalues for \(C\). Let \(C^*=\langle X_0^*,\dots ,X_d^*\rangle\) be a character algebra of the size \(N^*\) with primitive idempotents \(E_0^*,\dots ,E_d^*\) and the matrix of eigenvalues \(P^*\). We say \(C\) and \(C^*\) are dual, if \(PP^*=N I\). If \(C\) and \(C^*\) are dual, then \(N^*=N\). Let \(C\) and \(C^*\) are dual character algebras. The generalized Terwilliger algebra \(\mathcal T\) is the associative \(\mathbf C\)-algebra with 1 which is generated by the symbols \(x_0,\dots ,x_d,x_0^*,\dots ,x_d^*\) subject to relations (T1) \(x_0=x_0^*\); (T2) \(x_ix_j=\sum_{h=0}^d p_{ij}^hx_h\); (T2*) \(x_i^*x_j^*=\sum_{h=0}^d p_{ij}^{h*}x_h^*\); (T3) \(e_h^*x_ie_j^*=0\), if \(p_{ij}^h=0\); (T3*) \(e_hx_i^*e_j=0\), if \(p_{ij}^{h*}=0\), where \(e_i=N^{-1}\sum_{j=0}^d P_{ji}^*x_j\), \(e_i^*=N^{-1}\sum_{j=0}^d P_{ji}x_j^*\). In this paper the finite-dimensional indecomposable and the finite-dimensional irreducible \(\mathcal T\)-modules are classified for \(d=2\).
association schemes, Terwilliger algebra, character algebra
scientific article; zbMATH DE number 1762095