## 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:

 [1] Arad, Z.; Blau, H.I., An infinite family of nonabelian simple table algebras not induced by finite nonabelian simple groups, Groups St. Andrews 1989, London mathematical society lecture note series, 159, (1991), Cambridge Univ. Press Cambridge, p. 29-37 · Zbl 0731.20008 [2] Arad, Z.; Blau, H.I., On table algebras and applications to finite group theory, J. algebra, 138, 137-185, (1991) · Zbl 0790.20015 [3] Arad, Z.; Blau, H.I.; Erez, J.; Fisman, E., Real table algebras and applications to finite groups of extended camina-Frobenius type, J. algebra, 168, 615-647, (1994) · Zbl 0818.20006 [4] Arad, Z.; Blau, H.I.; Fisman, E., Table algebras and applications to products of characters in finite groups, J. algebra, 138, 186-194, (1991) · Zbl 0790.05097 [5] Arad, Z.; Fisman, E., Table algebras, C-algebras, and applications to finite group theory, Comm. algebra, 19, 2955-3009, (1991) · Zbl 0790.20016 [6] Z. Arad, E. Fisman, V. Miloslavsky, and, M. Muzychuk, On antisymmetric homogeneous integral table algebras of degree three, in, Homogeneous Integral Table Algebras of Degree Three: A Trilogy, preprint. · Zbl 0958.20010 [7] Arad, Z.; Fisman, E.; Scoppola, C.M., Table algebras of extended Gagola type and applications to finite group theory, Groups ’93 Galway/St. Andrews, London mathematical society lecture note series, 211, (1995), Cambridge Univ. Press Cambridge, p. 13-22 · Zbl 0851.20004 [8] Balmaceda, J.M.P.; Oura, M., The Terwilliger algebras of the group association schemes of S5 and A5, Kyushu J. math., 48, 221-231, (1994) · Zbl 0821.05059 [9] Bannai, E.; Ito, T., Algebraic combinatorics I: association schemes, (1984), Benjamin/Cummings Menlo Park · Zbl 0555.05019 [10] Bannai, E.; Munemasa, A., The Terwilliger algebras of group association schemes, Kyushu J. math., 49, 93-102, (1995) · Zbl 0839.05095 [11] H. I. Blau, Homogeneous integral table algebras of degree three with no nontrivial linear elements, in, Homogeneous Integral Table Algebras of Degree Three: A Trilogy, preprint. · Zbl 0958.20010 [12] Blau, H.I., Integral table algebras, affine diagrams, and the analysis of degree two, J. algebra, 178, 872-918, (1995) · Zbl 0924.20004 [13] Blau, H.I., Quotient structures in C-algebras, J. algebra, 175, 24-64, (1995) · Zbl 0895.20005 [14] Blau, H.I., Integral table algebras of degree two, Algebra colloq., 4, 393-408, (1997) · Zbl 0894.20004 [15] H. I. Blau, and, B. Xu, Homogeneous integral table algebras of degree three with a faithful real element, in, Homogeneous Integral Table Algebras of Degree Three: A Trilogy, preprint. · Zbl 0958.20010 [16] Blau, H.I.; Xu, B., On homogeneous integral table algebras, J. algebra, 199, 142-168, (1998) · Zbl 0948.20004 [17] Brouwer, A.E.; Cohen, A.M.; Neumaier, A., Distance-regular graphs, (1989), Springer-Verlag Berlin · Zbl 0747.05073 [18] Caughman, J.S., The Terwilliger algebra for bipartite P- and Q-polynomial association schemes, Discrete math., 196, 65-95, (1999) · Zbl 0924.05067 [19] B. V. C. Collins, The Terwilliger algebra of an almost-bipartite distance-regular graph and its antipodal cover, preprint. · Zbl 0955.05113 [20] Collins, B.V.C., The girth of a thin distance-regular graph, Graphs combin., 13, 21-34, (1997) [21] Curtin, B., 2-homogeneous bipartite distance-regular graphs, Discrete math., 187, 39-70, (1998) · Zbl 0958.05143 [22] Curtin, B., Bipartite distance-regular graphs, Graphs combin., 15, 143-158, (1999) · Zbl 0927.05083 [23] Curtin, B., Distance-regular graphs which support a spin model are thin, Discrete math., 197/198, 205-216, (1999) · Zbl 0929.05095 [24] Curtin, B., The local structure of a bipartite distance-regular graph, European J. combin., 20, 739-758, (1999) · Zbl 0940.05074 [25] B. Curtin, and, K. Nomura, Distance-regular graphs related to U_{q}(sl(2)), submitted for publication. · Zbl 0967.05067 [26] Curtis, C.; Reiner, I., Representation theory of finite groups and associative algebras, (1962), Interscience New York · Zbl 0131.25601 [27] G. Dickie, A note on bipartite P-polynomial association schemes and dual-bipartite Q-polynomial association schemes, preprint. · Zbl 0898.05084 [28] Dickie, G., Twice Q-polynomial distance-regular graphs are thin, European J. combin., 16, 555-560, (1995) · Zbl 0852.05085 [29] J. Go, The Terwilliger algebra of the hypercube Q_{D}, submitted for publication. [30] Hoheisel, G., Über charaktere, Monatsch. math. phys., 48, 448-456, (1939) · JFM 65.1121.01 [31] Ishibashi, H., The Terwilliger algebras of certain association schemes over the Galois rings of characteristic 4, Graphs combin., 12, 39-54, (1996) · Zbl 0852.05081 [32] Jaeger, F.; Matsumoto, M.; Nomura, K., Bose – mesner algebras related to type II matrices and spin models, J. algebraic combin., 8, 39-72, (1998) · Zbl 0974.05084 [33] Kawada, Y., Über den dualitätssatz der charaktere nichtcommutativer gruppen, Proc. physico-math. soc. Japan, 24, 97-109, (1942) · Zbl 0063.03172 [34] Koppinen, M., On algebras with two multiplications, including Hopf algebras and Bose mesner algebras, J. algebra, 182, 256-273, (1996) · Zbl 0897.16022 [35] McMullen, J.R., An algebraic theory of hypergroups, Bull. austral. math. soc., 20, 35-55, (1979) · Zbl 0399.20062 [36] McMullen, J.R.; Price, J.F., Duality for finite abelian hypergroups over splitting fields, Bull. austral. math. soc., 20, 57-70, (1979) · Zbl 0399.20063 [37] Passman, D.S., A course in ring theory, (1991), Brooks/Cole Pacific Grove · Zbl 0783.16001 [38] Suzuki, H., Association schemes with multiple Q-polynomial structures, J. algebraic combin., 7, 181-196, (1998) · Zbl 0974.05082 [39] Suzuki, H., Imprimitive Q-polynomial association schemes, J. algebraic combin., 7, 165-180, (1998) · Zbl 0974.05083 [40] Tanabe, K., The irreducible modules of the Terwilliger algebras of Doob schemes, J. algebraic combin., 6, 173-195, (1997) · Zbl 0868.05056 [41] Terwilliger, P.M., The subconstituent algebra of an association scheme I, J. algebraic combin., 1, 363-388, (1992) · Zbl 0785.05089 [42] Terwilliger, P.M., The subconstituent algebra of an association scheme II, J. algebraic combin., 2, 73-103, (1993) · Zbl 0785.05090 [43] Terwilliger, P.M., The subconstituent algebra of an association scheme III, J. algebraic combin., 2, 177-210, (1993) · Zbl 0785.05091 [44] Tomiyama, M.; Yamazaki, N., The subconstituent algebra of a strongly regular graph, Kyushu J. math., 48, 323-334, (1994) · Zbl 0842.05098 [45] Xu, B., On a class of integral table algebras, J. algebra, 178, 760-781, (1995) · Zbl 0839.05096 [46] Xu, B., Polynomial table algebras and their covering numbers, J. algebra, 176, 504-527, (1995) · Zbl 0836.20006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.