Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Character Polynomials, their q-analogs and the Kronecker product. | Character Polynomials their q-analogs and the Kronecker product A. M. Garsia and A. Goupil Department of Mathematics University of California San Diego California USA garsia@ Departement de mathematiques et d informatique Universite du Quebec a Trois-Rivières Trois-Rivières Quebec Canada Submitted Sep 22 2008 Accepted Jul 25 2009 Published Jul 31 2009 Mathematics Subject Classifications 20C30 20C08 05E05 05A18 05A15 Dedicated to Anders Bjorner on the occasion of his sixtieth birthday. Abstract The numerical calculation of character values as well as Kronecker coefficients can efficently be carried out by means of character polynomials. Yet these polynomials do not seem to have been given a proper role in present day literature or software. To show their remarkable simplicity we give here an umbral version and a recursive combinatorial construction. We also show that these polynomials have a natural counterpart in the standard Hecke algebra Hn q . Their relation to Kronecker products is brought to the fore as well as special cases and applications. This paper may also be used as a tutorial for working with character polynomials in the computation of Kronecker coefficients. I. Introduction We recall that the value xa of the irreducible Sn character indexed by a partition A Al . Afc at a permutation of Sn with cycle structure a 1ai2a2 nan is given by the Frobenius formula xa A x p Ai n 1 A2 n 2 x1 x2 An n n x n Work supported by a grant from NSF. Work partially supported by a grant from NSERC. THE ELECTRONIC JOURNAL OF COMBINATORICS 16 2 2009 R19 1 where A x A x1 . xn and pa pa x1 . xn denote respectively the Vandermonde determinant and the power sums symmetric functions. The character polynomial qM x1 x2 . . xn is the unique polynomial in Q x1 x2 . . xn with the property that for all partitions p h k and A n k p with n k p1 we have n-fc A1 1 2a2 .nan qM ai a2 . . an . Moreover with an appropriate change of sign and .
