The Littlewood-Richardson rule for wreath products with symmetric groups and the quiver of the categoryF≀FI_n

Abstract

We give a new proof for the Littlewood-Richardson rule for the wreath product F ≀Sn where F is a finite group. Our proof does not use symmetric functions but more elementary representation theoretic tools. We also derive a branching rule for inducing the natural embedding of F ≀ Sn to F ≀ Sn+1. We then apply the generalized Littlewood-Richardson rule for computing the ordinary quiver of the category F ≀ FIn where FIn is the category of all injective functions between subsets of an n-element set. Frobenius reciprocity, answering one of these questions essentially answers the other one. Moreover, since both induction and restriction are additive, it is enough to consider the case where U or V are irreducible representations. Even considering this reduction, the question, in general, is very difficult. If G = S n the answer is known for certain natural choices of H and these solutions are often called branching rules. The most classical case is where H = S n−1 viewed as the subgroup of all permutations that fix n. An important generalization is the Littlewood-Richardson rule which gives the answer for the case H = S k × S n−k .Let F and G be finite groups such that G acts on the left of a finite set X. We denote by F ≀ X G the wreath product of F and G. The representation theory of F ≀ X G is a well-studied subject (see [3] and [6, Chapter 4]) and the case G = S n with the natural action on {1, . . . , n} is of special importance. Finding generalizations for the branching rules is a natural question. The "classical" branching rule for inducing from F ≀ S n to F ≀ S n+1 was found by Pushkarev [10]. In this paper we generalize the Littlewood-Richardson rule to the group F ≀ S n . After the present paper was already circulating, we became aware of the paper [5] by Ingram, Jing and Stitzinger, where the same result was obtained using symmetric functions. However, our approach is different. We use only elementary representation theoretic tools and base our proof on the explicit description of the irreducible representations of F ≀ S n . In Section §5 we use the generalized Littlewood-Richardson rule to retrieve Pushkarev's result.Then we turn to give an application to the representation theory of a natural family of categories. Denote by FI the category of finite sets and injective functions. The representation theory of FI is currently under active research which was initiated in [4]. There is also research on the representation theory of the wreath product F ≀ FI (see [8,11,13]). We will be interested here in the finite version of this category. We denote by FI n the category of all subsets of {1, . . . , n} and injective functions. In Section §6 we will give a description of the ordinary quiver of the algebra of F ≀ FI n . The case where F is the trivial group was originally done by [2] and a simple proof was later given in [9]. For the general case, we imitate the method of [9] but where they use usual branching rule for S n we will use the generalization for F ≀ S n .

On the Dowling and Rhodes lattices and wreath products

“…The results in Theorem 2.1 have been used, along with appropriate branching rules for the groups G ≀ S n to study the representation theory of the monoid G ≀ P T n in [16,17]. This has applications to the representation theory of the group G ≀ S n via natural semigroup theoretic representations like those discussed in this section.…”

Polynomial relations between operators on chains of representation rings

“…The categorification of the Heisenberg algebra continues to be a topic of active interest [3] [4] [5] [6], and wreath products have naturally arisen when studying Heisenberg algebra structures [7] [8]. Littlewood-Richardson type rules have been found for wreath products [9]. Nowadays, the branching graph perspective for symmetric group representation theory in which Induction and Restriction operators play the central role has been incredibly successful and has now become part of our everyday thinking ( [10] [11] [12]).…”