Diagonal invariants and the refined multimahonian distribution
Fabrizio Caselli
DOI: 10.1007/s10801-008-0159-7
Abstract
Combinatorial aspects of multivariate diagonal invariants of the symmetric group are studied. As a consequence we deduce the existence of a multivariate extension of the classical Robinson-Schensted correspondence. Further byproducts are a purely combinatorial algorithm to describe the irreducible decomposition of the tensor product of two irreducible representations of the symmetric group, and new symmetry results on permutation enumeration with respect to descent sets.
Pages: 193–213
Keywords: keywords diagonal invariants; symmetric groups; descent sets; Hilbert series; Kronecker coefficients
Full Text: PDF
References
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2. Adin, R., Brenti, F., Roichman, Y.: Descent representations and multivariate statistics. Trans. Amer. Math. Soc. 357(8), 3051-3082 (2005)
3. Allen, E.E.: The descent monomials and a basis for the diagonally symmetric polynomials. J. Algebraic Combin. 3, 5-16 (1994)
4. Bagno, E., Biagioli, R.: Colored-descent representations of complex reflection groups G(r, p, n). Is- rael J. Math. 160, 317-347 (2007)
5. Barcelo, H., Reiner, V., Stanton, D.: Bimahonian distributions. J. London Math. Soc. 77, 627-646 (2008)
6. Bergeron, F., Lamontagne, F.: Decomposition of the diagonal action of Sn on the coinvariant space of Sn \times Sn. Sém. Lothar. Combin. 52, B2e2 (2004/2007)
7. Bessenrodt, C., Kleshchev, A.: On Kronecker products of complex representations of the symmetric and alternating groups. Pacific J. Math. 190(2), 201-223 (1999)
8. Biagioli, R., Caselli, F.: Invariant algebras and major indices for classical Weyl groups. Proc. London Math. Soc. (3) 88(3), 603-631 (2004)
9. Biagioli, R., Caselli, F.: A descent basis for the coinvariant algebra of type D. J. Algebra 275, 517-539 (2004)
10. Chevalley, C.: Invariants of finite groups generated by reflections. Amer. J. Math. 77, 778-782 (1955) 11. de G. Robinson, B.: On the representations of the symmetric group. Amer. J. Math. 60(3), 745-760 (1938)
12. Désarménien, J., Foata, D.: Fonctions symétriques et séries hypergéométriques basiques multivariées. Bull. Soc. Math. France 113(1), 3-22 (1985)
13. Dvir, Y.: On the Kronecker product of Sn characters. J. Algebra 154(1), 125-140 (1993)
14. Derksen, H.: Computing with characters of the symmetric groups, preprint available at
15. Foata, D., Schützenberger, M.P.: Major index and inversion number of permutations. Math. Nachr. 83, 143-159 (1978)
16. Garsia, A., Gessel, I.: Permutation statistics and partitions. Adv. Math. 31, 288-305 (1979)
17. Garsia, A., McLarnan, T.: Relations between Young's natural and the Kazhdan-Lusztig representations of Sn. Adv. Math. 69, 32-92 (1988)
18. Garsia, A., Stanton, D.: Group actions of Stanley-Reisner rings and invariants of permutations groups. Adv. Math. 51, 107-201 (1984)
19. Gessel, I.: Multipartite P-partitions and inner products of skew Schur functions, Combinatorics and algebra. Contemp. Math., vol. 34, pp. 289-301. Amer. Math. Soc., Providence (1984)
20. Gordon, B.: Two theorems on multipartite partitions. J. London Math. Soc. 38, 459-464 (1963)
21. Kraśkiewicz, W., Weyman, J.: Algebra of coinvariants and the action of a Coxeter element. Bayreuth. Math. Schr. No. 63, 265-284 (2001)
22. MacMahon, P.A.: Combinatory analysis, vol.
1. Cambridge University Press, London (1915)
23. Regev, A.: On the height of the Kronecker product of Sn characters. Israel J. Math. 42(1-2), 60-64 (1982)
24. Reiner, V.: Quotients of Coxeter complexes and P -partitions. Mem. Amer. Math. Soc. 95(460) (1992)
25. Rosas, M.H.: The Kronecker product of Schur functions indexed by two-row shapes or hook shapes. J. Algebraic Combin. 14, 153-173 (2001)
26. Schensted, C.: Longest increasing and decreasing subsequences. Canad. J. Math. 13, 179-191 (1961)
27. Schützenberger, M.P.: Quelques remarques sur une construction de Schensted. Math. Scand. 12, 117- 128 (1963)
28. Shephard, G.C., Todd, J.A.: Finite unitary reflection groups. Canadian J. Math. 6, 274-304 (1954)
29. Solomon, L.: Partition identities and invariants of finite groups. J. Combin. Theory Ser. A 23, 148-175 (1977)
30. Stanley, R.P.: Invariants of finite groups and their applications to combinatorics. Bull. Amer. Math.