Terwilliger algebras of wreath products of one-class association schemes
Gargi Bhattacharyya
, Sung Y. Song
and Rie Tanaka
DOI: 10.1007/s10801-009-0196-x
Abstract
In this paper, we study the wreath product of one-class association schemes K n = H(1, n) for n\geq 2. We show that the d-class association scheme K n 1\wr K n 2\wr \frac{1}{4} \wr K n d K_{n_{1}}\wr K_{n_{2}}\wr \cdots \wr K_{n_{d}} formed by taking the wreath product of K n i K_{n_{i}} (for n i \geq 2) has the triple-regularity property. Then based on this fact, we determine the structure of the Terwilliger algebra of K n 1\wr K n 2\wr \frac{1}{4} \wr K n d K_{n_{1}}\wr K_{n_{2}}\wr \cdots \wr K_{n_{d}} by studying its irreducible modules. In particular, we show that every non-primary module of this algebra is 1-dimensional.
Pages: 455–466
Keywords: keywords commutative association schemes; Terwilliger algebra; wreath product
Full Text: PDF
References
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2. Bannai, E., Ito, T.: Algebraic Combinatorics. I. Association schemes. The Benjamin/Cummings Publishing Co., Inc., Menlo Park (1984)
3. Bannai, E., Munemasa, A.: The Terwilliger algebra of group association scheme. Kyushu J. Math. 49, 93-102 (1995)
4. Bhattacharyya, G.: Terwilliger algebras of wreath products of association schemes. Ph.D. Dissertation, Iowa State University (2008)
5. Brouwer, A., Cohen, A.M., Neumaier, A.: Distancce-Regular Graphs. Springer, Berlin (1989)
6. Caughman, IV, J.S.: The Terwilliger algebra of bipartite P - and Q-polynomial schemes. Discrete Math. 196, 65-95 (1999)
7. Drozd, Yu.A., Kirichenko, V.V.: Finite Dimensional Algebras. Springer, Berlin, Heidelberg (1994)
8. Egge, E.: A generalization of the Terwilliger algebra. J. Algebra 233, 213-252 (2000)
9. Go, J.: The Terwilliger algebra of the hypercube. Europ. J. Combin. 23(4), 399-429 (2002)
10. Jaeger, F.: On spin models, triply regular association schemes, and duality. J. Algebraic Combin. 2, 103-144 (1995)
11. Levstein, F., Maldonado, C., Penazzi, D.: The Terwilliger algebra of a Hamming scheme H (d, q). Europ. J. Combin. 27, 1-10 (2006)
12. Martin, W.J., Stinton, D.R.: Association schemes for ordered orthogonal arrays and (T , M, S)-nets. Canad. J. Math. 51, 326-346 (1999)
13. Munemasa, A.: An application of Terwilliger algebra. Unpublished preprint 1993. (Preprint found on:
14. Song, S.Y.: Fusion relation in products of association schemes. Graphs and Combin. 18, 655-665 (2002)
15. Tanabe, K.: The irreducible modules of the Terwilliger algebras of Doob schemes. J. Algebraic Combin. 2, 173-195 (1997)
16. Terwilliger, P.: The subconstituent algebra of an association scheme (Part I). J. Algebraic Combin. 1, 363-388 (1992); (Part II; III). J. Algebraic Combin. 2, 73-103; 177-210 (1993)
17. Terwilliger, P.: Algebraic combinatorics. Course lecture notes at University of Wisconsin (1996)
18. Tomiyama, M., Yamazaki, N.: The subconstituent algebra of a strongly regular graph. Kyushu J. Math.
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