Katriel's Operators for Products of Conjugacy Classes of S n
Alain Goupil
, Dominique Poulalhon2
and Gilles Schaeffer2
2dagger
DOI: 10.1007/s10801-005-6904-2
Abstract
We define a family of differential operators indexed with fixed point free partitions. When these differential operators act on normalized power sum symmetric functions q 
(x), the coefficients in the decomposition of this action in the basis q (x) are precisely those of the decomposition of products of corresponding conjugacy classes of the symmetric group S n. The existence of such operators provides a rigorous definition of Katriel 
s elementary operator representation of conjugacy classes and allows to prove the conjectures he made on their properties.
(x), the coefficients in the decomposition of this action in the basis q (x) are precisely those of the decomposition of products of corresponding conjugacy classes of the symmetric group S n. The existence of such operators provides a rigorous definition of Katriel s elementary operator representation of conjugacy classes and allows to prove the conjectures he made on their properties.Pages: 137–146
Keywords: keywords conjugacy classes; symmetric functions; operator; structure constants
Full Text: PDF
References
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2. S. Corteel, A. Goupil, and G. Schaeffer, “Content evaluation and class symmetric functions,” Adv. in Math. 188(2), 315-336.
3. H.K. Farahat and G. Higman, “The centres of symmetric group rings,” Proc. Roy. Soc. London Ser. A 250 (1959), 212-221.
4. A. Garsia and A. Goupil, “Explicit computation of characters's polynomials,” (in preparation).
5. I.P. Goulden and D.M. Jackson, “Transitive factorizations in the symmetric group, and combinatorial aspects of singularity theory,” preprint of Dept. of Combinatorics and optimization, University of Waterloo, March 1999.
6. A. Goupil and C. Chauve, “Combinatorial operators for Kronecker powers of representations of Sn,” submitted, Seminaire Lotharingien de Combinatoire, March 2005.
7. A. Goupil and G. Schaeffer, “Factoring N -cycles and counting maps of given genus,” Europ. J. Combinatorics 19 (1998), 819-834
8. A. Goupil, D. Poulalhon, and G. Schaeffer, Central Characters and Conjugacy Classes of the Symmetric Group or on some Conjectures of J. Katriel, Formal Power series and Algebraic combinatorics, (Moscow, 2000), Springer, Berlin 2000, pp. 238-249.
9. J. Katriel, “Some useful results concerning the representation theory of the symmetric group,” J. Phys. A 24(22) (1991), 5227-5234.
10. J. Katriel, “A conjecture concerning the evaluation of products of class-sums of the symmetric group.,” in Groups '93 Galway/St. Andrews' conference, C.M. Campbell, et al. (Eds.), Lond. Math. Soc. Lect. Note Ser. 212 (1995), 322-332.
11. J. Katriel “Explicit expressions for the central characters of the symmetric group,” Discrete Applied Math. 67, (1996), 149-156.
12. J. Katriel, “Class-sum products in the symmetric group: Minimally connected reduced class coefficients,” Combinatorics and physics (Marseille, 1995). Math. Comput. Modelling 26(8-10) (1997), 161-167.
13. J. Katriel, “The class algebra of the symmetric group,” in FPSAC 10th Conference, Toronto, 1998, pp. 401-410.
14. A. Lascoux and J.-Y. Thibon, “Vertex Operators and the class algebras of symmetric groups,” Zap. Nauchn. Sem. S. Petersburg. Otdel. Mat. Inst. Steklov. (POMI) 283 (2001), Teor. Predst. Din. Sist. Komb. i Algoritm. Metody 156-177 6 261.
15. I.G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd ed., Clarendon Press, Oxford, 1995.
16. P.A. MacMahon, Combinatory Analysis, vol. I, II, Chelsea, New York, 1960.
17. D. Poulalhon and G. Schaeffer, “Factorizations of large cycles in the symmetric group,” Discrete Math., 254(1-3) (2002), 433-458.