# zbMATH — the first resource for mathematics

Terwilliger algebras of wreath products of association schemes. (English) Zbl 1329.05313
Summary: The Terwilliger algebra of an association scheme of order $$n$$ introduced in [P. Terwilliger, J. Algebr. Comb. 1, No. 4, 363–388 (1992; Zbl 0785.05089)] is a subalgebra of the matrix algebra of all $$n \times n$$ matrices. Terwilliger algebras of wreath products of special association schemes are studied in several papers. In this paper we study the Terwilliger algebra of the wreath product $$T \wr S$$ of two arbitrary association schemes $$S$$ and $$T$$. We will express the Terwilliger algebra of $$T \wr S$$ and its primitive central idempotents in terms of the Terwilliger algebras of $$S$$ and $$T$$ and their primitive central idempotents. The known results of A. Hanaki et al. [J. Algebra 343, No. 1, 195–200 (2011; Zbl 1235.05157)], K. Kim [Linear Algebra Appl. 437, No. 11, 2773–2780 (2012; Zbl 1253.05147)], etc. are special cases of our results.

##### MSC:
 05E30 Association schemes, strongly regular graphs 16S50 Endomorphism rings; matrix rings
##### Citations:
Zbl 0785.05089; Zbl 1235.05157; Zbl 1253.05147
Full Text:
##### References:
 [1] Bannai, E.; Munemasa, A., The Terwilliger algebras of group association schemes, Kyushu J. Math., 49, 93-102, (1995) · Zbl 0839.05095 [2] Bhattacharyya, G.; Song, S. Y.; Tanaka, R., Terwilliger algebras of wreath products of one-class association schemes, J. Algebraic Combin., 31, 455-466, (2010) · Zbl 1286.05182 [3] Brouwer, A. E.; Cohen, A. M.; Neumaier, A., Distance-regular graphs, (1989), Springer-Verlag Berlin · Zbl 0747.05073 [4] Evdokimov, S.; Ponomarenko, I., Permutation group approach to association schemes, European J. Combin., 30, 1456-1476, (2009) · Zbl 1228.05311 [5] Guo, K. J., On triply regular graphs [6] Gijswijt, D.; Schrijver, A.; Tanaka, H., New upper bounds for nonbinary codes based on the Terwilliger algebra and semidefinite programming, J. Combin. Theory Ser. A, 113, 1719-1731, (2006) · Zbl 1105.94027 [7] Hanaki, A.; Kim, K.; Maekawa, Y., Terwilliger algebras of direct and wreath products of association schemes, J. Algebra, 343, 195-200, (2011) · Zbl 1235.05157 [8] Hirasaka, M.; Muzychuk, M., On quasi-thin association schemes, J. Combin. Theory Ser. A, 98, 17-32, (2002) · Zbl 0994.05149 [9] Higman, D., Coherent algebras, Linear Algebra Appl., 93, 209-239, (1987) · Zbl 0618.05014 [10] Kim, K., Terwilliger algebras of wreath products by quasi-thin schemes, Linear Algebra Appl., 437, 2773-2780, (2012) · Zbl 1253.05147 [11] Muzychuk, M.; Ponomarenko, I., On quasi-thin association schemes, J. Algebra, 351, 461-489, (2012) · Zbl 1244.05236 [12] Suda, S., Coherent configurations and triply regular association schemes obtained from spherical designs, (2009) [13] Terwilliger, P., The subconstituent algebra of an association scheme, J. Algebraic Combin., 1, 363-388, (1992) · Zbl 0785.05089 [14] Xu, Bangteng, Characterizations of wreath products of one-class association schemes, J. Combin. Theory Ser. A, 118, 7, 1907-1914, (2011) · Zbl 1232.05246 [15] Xu, Bangteng; Zuo, Kezheng, On semisimple varietal Terwilliger algebras whose non-primary ideals are 1-dimensional, J. Algebra, 397, 426-442, (2014) · Zbl 1300.05332 [16] Zieschang, P.-H., Theory of association schemes, (2005), Springer-Verlag Berlin · Zbl 1079.05099
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.