A Solomon descent theory for the wreath products $G\wr\mathfrak S_n$

Abstract

Abstract. We propose an analogue of Solomon's descent theory for the case of a wreath product G S n , where G is a finite abelian group. Our construction mixes a number of ingredients: Mantaci-Reutenauer algebras, Specht's theory for the representations of wreath products, Okada's extension to wreath products of the Robinson-Schensted correspondence, and Poirier's quasisymmetric functions. We insist on the functorial aspect of our definitions and explain the relation of our results with previous work concerning the hyperoctaedral group.

Colored-descent representations of complex reflection groups G(r, p, n)

“…The representations R D,C , called colored-descent representations, generalize to all groups G(r, p, n), the descent representations introduced for S n and B n by Adin, Brenti and Roichman in [3], see also [8] for the case of D n . In the case of G(r, n), a Solomon's descent algebra approach to these representations has been done by Baumann and Hohlweg [6]. Since their study is restricted to wreath products, it will be interesting to extend their results to all complex reflection groups, thus getting characters of all our modules as images of a particular class of elements of the group algebra.…”

“…Note that each entry in T i is congruent to i+1 (mod 3). The entries partition of T is θ(T ) = (14,14,10,9,9,8,7,6,5,3,3). Let us compute the image of T , φ λ (T ) = (T, ∆).…”

“…The standard Young tableau T is drawn in Figure 2. Now, Des(T ) = {3, 7, 9} and f (T ) = (10,10,9,8,8,7,6,5,4,2,2). It follows that (θ(T ) − f (T )) n i=1 = (4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1), and so ∆ = (0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0).…”

“…It is worth mentioning, that in a very recent preprint concerning wreath products, Baumann and Hohlweg [6], following Solomon's descent algebra approach [27], define some special characters of G(r, n) as images, through a "generalized Solomon homomorphism", of the elements of a basis of the Mantaci-Reutenaur algebra [21]. Nevertheless, they don't provide any module having them as characters.…”

Colored Posets and Colored Quasisymmetric Functions

“…Just as there is a Hopf algebra of permutations, there is a colored permutation Hopf algebra. See [3] and [4].…”