zbMATH — the first resource for mathematics

On Lee association schemes over \(\mathbb{Z}_4\) and their Terwilliger algebra. (English) Zbl 1352.05197
Summary: Let \(F = \{0, 1, 2, 3 \}\) and define the set \(K = \{K_0, K_1, K_2 \}\) of relations on \(F\) such that \((x, y) \in K_i\) if and only if \(x - y \equiv \pm i\pmod 4\). Let \(n\) be a positive integer. We consider the Lee association scheme \(L(n)\) over \(\mathbb{Z}_4\) which is the extension of length \(n\) of the initial scheme \((F, K)\). Let \(\mathcal{T}\) denote the Terwilliger algebra of \(L(n)\) with respect to the zero codeword of length \(n\). We show that \(\mathcal{T}\) is generated by a homomorphic image of the universal enveloping algebra of the Lie algebra \(\mathfrak{sl}_3(\mathbb{C})\) and the center \(Z(\mathcal{T})\). Furthermore, we determine the irreducible modules for \(\mathcal{T}\) using the Schur-Weyl duality.

05E30 Association schemes, strongly regular graphs
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
15A04 Linear transformations, semilinear transformations
20C30 Representations of finite symmetric groups
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
Full Text: DOI
[1] Antoine, J. P.; Speiser, D., Characters and irreducible representations of the simple groups. II. application to the classical groups, J. Math. Phys., 5, 1560-1572, (1964) · Zbl 0134.43704
[2] Bannai, E.; Ito, T., Algebraic combinatorics I: association schemes, Benjamin-Cummings Lecture Note Ser., vol. 58, (1984), Benjamin-Cummings Menlo Park · Zbl 0555.05019
[3] Bender, E. A.; Knuth, D. E., Enumeration of plane partitions, J. Combin. Theory Ser. A, 13, 40-54, (1972) · Zbl 0246.05010
[4] Brouwer, A. E.; Cohen, A. M.; Neumaier, A., Distance-regular graphs, (1989), Springer-Verlag Berlin, Heidelberg · Zbl 0747.05073
[5] Benkart, G. M.; Britten, D. J.; Lemire, F. W., Stability in modules for classical Lie algebras — a constructive approach, Mem. Amer. Math. Soc., 430, (1990) · Zbl 0706.17003
[6] van Dam, E. R.; Koolen, J. H.; Tanaka, H., Distance-regular graphs, Electron. J. Combin., (2016), #DS22, 156 pp. · Zbl 1335.05062
[7] Delsarte, P., An algebraic approach to the association schemes of coding theory, Philips Res. Repts. Suppl., 10, (1973) · Zbl 1075.05606
[8] Etingof, P.; Golberg, O.; Hensel, S.; Liu, T.; Schwendner, A.; Vaintrob, D.; Yudovina, E., Introduction to representation theory, Stud. Math. Libr., vol. 59, (2011), Amer. Math. Soc.
[9] 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
[10] Go, J., The Terwilliger algebra of the hypercube, European J. Combin., 23, 399-429, (2002) · Zbl 0997.05097
[11] Godsil, C. D., Algebraic combinatorics, (1993), Chapman and Hall New York · Zbl 0814.05075
[12] Godsil, C. D., Generalized Hamming schemes, (2010)
[13] Goodman, R.; Wallach, N. R., Symmetry, representations, and invariants, Grad. Texts in Math., vol. 255, (2009), Springer Dordrecht, MR2522486 · Zbl 1173.22001
[14] Hall, B. C., Lie groups, Lie algebras, and representations: an elementary introduction, Grad. Texts in Math., (2003), Springer-Verlag New York · Zbl 1026.22001
[15] Hammons, A. R.; Kumar, P. V.; Calderbank, A. R.; Sloane, N. J.A.; Solé, P., The \(\mathbb{Z}_4\)-linearity of kerdock, preparata, goethals and related codes, IEEE Trans. Inform. Theory, 40, 301-319, (1994) · Zbl 0811.94039
[16] Harada, M.; Lam, C. H.; Munemasa, A., Residue codes of extremal type II-codes and the moonshine vertex operator algebra, Math. Z., 274, 685-700, (2013) · Zbl 1283.94112
[17] Humphreys, J., Introduction to Lie algebras and representation theory, (1972), Springer-Verlag New York · Zbl 0254.17004
[18] Iliev, P., A Lie-theoretic interpretation of multivariate hypergeometric polynomials, Compos. Math., 143, 991-1002, (2012) · Zbl 1248.33028
[19] Iliev, P.; Terwilliger, P., The rahman polynomials and the Lie algebra \(\mathfrak{sl}_3(\mathbb{C})\), Trans. Amer. Math. Soc., 364, 4225-4238, (2012) · Zbl 1353.33007
[20] Leonard, D. A., Orthogonal polynomials, duality and association schemes, SIAM J. Math. Anal., 13, 656-663, (1982) · Zbl 0495.33006
[21] Martin, W. J.; Tanaka, H., Commutative association schemes, European J. Combin., 30, 1497-1525, (2009) · Zbl 1228.05317
[22] Mizukawa, H.; Tanaka, H., \((n + 1, m + 1)\)-hypergeometric functions associated to character algebras, Proc. Amer. Math. Soc., 132, 2613-2618, (2004) · Zbl 1059.33020
[23] Nomura, K.; Terwilliger, P., Krawtchouk polynomials, the Lie algebra \(\mathfrak{sl}_2\), and leonard pairs, Linear Algebra Appl., 437, 345-375, (2012) · Zbl 1261.33001
[24] Sagan, B., The symmetric group, Cole Math. Series, (1991), Wadsworth & Brooks New York
[25] Schrijver, A., New code upper bounds from the Terwilliger algebra and semidefinite programming, IEEE Trans. Inform. Theory, 51, 2859-2866, (2005) · Zbl 1298.94152
[26] Steinberg, B., Representation theory of finite groups: an introductory approach, Universitext, (2012), Springer New York · Zbl 1243.20001
[27] Tanaka, H., New proofs of the assmus-mattson theorem based on the Terwilliger algebra, European J. Combin., 30, 736-746, (2009) · Zbl 1220.94052
[28] Terwilliger, P., Lectures notes on distance-regular graphs given at de la salle university, manila, (2010)
[29] Terwilliger, P., The subconstituent algebra of an association scheme: part I, J. Algebraic Combin., 1, 363-388, (1992) · Zbl 0785.05089
[30] Terwilliger, P., The subconstituent algebra of an association scheme: part II, J. Algebraic Combin., 2, 73-103, (1993) · Zbl 0785.05090
[31] Terwilliger, P., The subconstituent algebra of an association scheme: part III, J. Algebraic Combin., 2, 177-210, (1993) · Zbl 0785.05091
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.