On permutation statistics and Hecke algebra characters (Q2741039)

This paper is primarily a survey on the interplay between the characters of Hecke algebras of type \(A\) and certain permutation statistics, in particular the length function, the descent number and the major index. However, the authors also prove some new results in the course of the paper.NEWLINENEWLINENEWLINE\S 1 contains a concise but informative introduction. In \S 2, permutation statistics are introduced, as are the Hecke algebra and coinvariant algebra associated to the symmetric group. Some of the relationships between permutation statistics, the Robinson-Schensted-Knuth correspondence and Kostka-Foulkes polynomials are stated. \S 3 is devoted to a study of the characters of the symmetric group and its coinvariant algebra, both in terms of permutation statistics; the Hecke algebra is similarly treated in \S 4. Both \S 3 and \S 4 apply results on characters to derive purely combinatorial identities on permutation statistics. The final section, \S 5, discusses the situation for arbitrary Coxeter groups and states some open problems.NEWLINENEWLINENEWLINEThere is a long list of references, but a relevant paper that is missing from the list is \textit{S. Ariki}'s paper [Adv. Stud. Pure Math. 28, 1-20 (2000; Zbl 0986.05097)]. That paper contains a direct proof of the result on Kazhdan-Lusztig cells and Knuth classes mentioned in the last paragraph of section 2.2.1.NEWLINENEWLINEFor the entire collection see [Zbl 0963.00024].