## The flag major index and group actions on polynomial rings.(English)Zbl 1058.20031

Summary: A new extension of the major index, defined in terms of Coxeter elements, is introduced. For the classical Weyl groups of type $$B$$, it is equidistributed with length. For more general wreath products it appears in an explicit formula for the Hilbert series of the (diagonal action) invariant algebra.

### MSC:

 20F55 Reflection and Coxeter groups (group-theoretic aspects) 20F05 Generators, relations, and presentations of groups 20P05 Probabilistic methods in group theory 05E15 Combinatorial aspects of groups and algebras (MSC2010)
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### References:

 [1] R. M. Adin, Y. Roichman, Proc. of 11-th Conference on Formal Power Series and Algebraic Combinatorics, 1999, Universitat Politècnica de Catalunya, Barcelona [2] Carlitz, L., q -Bernoulli and Eulerian numbers, Trans. am. math. soc., 76, 332-350, (1954) · Zbl 0058.01204 [3] Clarke, R.J.; Foata, D., Eulerian calculus. I. univariable statistics, Europ. J. combinatorics, 15, 345-362, (1994) · Zbl 0811.05069 [4] Clarke, R.J.; Foata, D., Eulerian calculus. II. an extension of han’s fundamental transformation, Europ. J. combinatorics, 16, 221-252, (1995) · Zbl 0822.05066 [5] Clarke, R.J.; Foata, D., Eulerian calculus. III. the ubiquitous Cauchy formula, Europ. J. combinatorics, 16, 329-355, (1995) · Zbl 0826.05058 [6] Foata, D., On the netto inversion number of a sequence, Proc. am. math. soc., 19, 236-240, (1968) · Zbl 0157.03403 [7] Foata, D.; Schützenberger, M.P., Major index and inversion number of permutations, Math. nachr., 83, 143-159, (1978) · Zbl 0319.05002 [8] Garsia, A.; Gessel, I., Permutation statistics and partitions, Adv. math., 31, 288-305, (1979) · Zbl 0431.05007 [9] Gordon, B., Two theorems on multipartite partitions, J. London math. soc., 38, 459-464, (1963) · Zbl 0119.04105 [10] Goulden, I.P., A bijective proof of stanley’s shuffling theorem, Trans. am. math. soc., 57, 147-160, (1985) · Zbl 0535.05002 [11] Haiman, M., Conjectures on the quotient ring by diagonal invariants, J. algebr. comb., 3, 17-76, (1994) · Zbl 0803.13010 [12] Humphreys, J.E., Reflection groups and Coxeter groups, (1992), Cambridge University Press · Zbl 0768.20016 [13] W. Kraskiewicz, J. Weyman [14] Lal, A.K., Some games related to permutation group statistics and their type B analogues, Ars comb., 45, 276-286, (1997) · Zbl 0934.91006 [15] Macdonald, I.G., Symmetric functions and Hall polynomials, (1995), Oxford University Press Oxford · Zbl 0487.20007 [16] P. A. MacMahon [17] Reiner, V., Signed permutation statistics, Europ. J. combinatorics, 14, 553-567, (1993) · Zbl 0793.05005 [18] Reiner, V., Signed permutation statistics and cycle type, Europ. J. combinatorics, 14, 569-579, (1993) · Zbl 0793.05006 [19] Reutenauer, C., Free Lie algebras, (1993), Oxford University Press · Zbl 0798.17001 [20] Roichman, Y., some combinatorial properties of the coinvariant algebra, Proceedings of FPSAC-98, (1998), p. 529-537 [21] Roichman, Y., Schubert polynomials, kazhdan – lusztig basis and characters, Discrete math., 217, 353-365, (2000) · Zbl 0969.05066 [22] Y. Roichman [23] Steingrimsson, E., Permutation statistics of indexed permutations, Europ. J. combinatorics, 15, 187-205, (1994) · Zbl 0790.05002 [24] Stembridge, J., On the eigenvalues of representations of reflection groups and wreath products, Pac. J. math., 140, 359-396, (1989) · Zbl 0641.20011
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