Some explicit expressions for the structure coefficients of the center of the symmetric group algebra involving cycles of length three

Year 2019, Volume: 6 Issue: 2, 53 - 62, 07.05.2019

Omar Tout

https://doi.org/10.13069/jacodesmath.560404

Cited By: 1

Abstract

We use the combinatorial way to give an explicit expression for the product of the class of cycles of length three with an arbitrary class of cycles. In addition, an explicit formula for the coefficient of an arbitrary class in the expansion of the product of an arbitrary class by the class of cycles of length three is given.

Keywords

The center of the symmetric group algebra, Structure coefficients, Product of conjugacy classes of the symmetric group, Representation theory of the symmetric group

References

• [1] Z. Arad, M. Herzog (Eds.), Products of Conjugacy Classes in Groups, Lecture Notes in Math., vol. 1112, Springer-Verlag, Berlin, 1985.
• [2] E. A. Bertram, V. K. Wei, Decomposing a permutation into two large cycles: An enumeration, SIAM J. Algebraic Discrete Methods 1(4)(1980) 450–461.
• [3] G. Boccara, Nombre de representations d’une permutation comme produit de deux cycles de longueurs donnees, Discrete Math. 29(2) (1980) 105–134.
• [4] R. Cori, Un code pour les graphes planaires et ses applications, Astérisque 27 (1975) 169pp.
• [5] H. K. Farahat, G. Higman, The centres of symmetric group rings, Proc. Roy. Soc. London Ser. A 250(1261) (1959) 212–221.
• [6] A. Goupil, G. Schaeffer, Factoring n-cycles and counting maps of given genus, European J. Combin. 19(7) (1998) 819–834.
• [7] D. M. Jackson, T. I. Visentin, Character theory and rooted maps in an orientable surface of given genus: Face-colored maps, Trans. Amer. Math. Soc. 322(1) (1990) 365–376.
• [8] J. Katriel, J. Paldus, Explicit expression for the product of the class of transpositions with an arbitrary class of the symmetric group, R Gilmore (Ed.), Group Theoretical Methods in Physics, World Scientific, Singapore (1987) 503–506.
• [9] R. P. Stanley, Factorization of permutations into n-cycles, Discrete Math. 37(2-3) (1981) 255–262.
• [10] W. A. Stein et al., Sage Mathematics Software (Version 4.8), The Sage Development Team, http://www.sagemath.org.
• [11] O. Tout, Polynomialité des coefficients de structure des algèbres de doubles-classes. Ph.D thesis, Université de Bordeaux, 2014.
• [12] O. Tout, A general framework for the polynomiality property of the structure coefficients of double– class algebras, J. Algebr. Comb. 45(4) (2017) 1111–1152.