Foulkes characters for complex reflection groups

Abstract

Foulkes discovered a marvelous set of characters for the symmetric group by summing Specht modules of certain ribbon shapes according to height. These characters have many remarkable properties and have been the subject of many investigations, including a recent one [2] by Diaconis and Fulman, which established some new formulas, a conjecture of Isaacs, and a connection with Eulerian idempotents. We widen our consideration to complex reflection groups and find ourselves equipped from the start with a simple formula for (generalized) Foulkes characters which explains and extends these properties. In particular, it gives a factorization of the Foulkes character table which explains Diaconis and Fulman’s formula for the determinant, their link to Eulerian idempotents, and their formula for the inverse. We present a natural extension of a conjecture of Isaacs, and then use properties of Foulkes characters which resemble those of supercharacters to establish the result. We also discover a remarkable refinement of Diaconis and Fulman’s determinantal formula by considering Smith normal forms. Classic type A Foulkes characters have connections with adding random numbers, shuffling cards, the Veronese embedding, and combinatorial Hopf algebras [2, 7]. Our formula brings Orlik–Solomon coexponents from [12] the cohomology theory of [10] complements into the picture with the geometry of the Milnor fiber complex [8], and it gives rise to a curious classification at the end of the paper. The paper is structured as follows. Section 1 introduces Foulkes characters for Shephard and Coxeter groups. Key properties are quickly gathered, including our main formula. In Section 2, properties of type A Foulkes characters are explained and extended from the symmetric group to the infinite family of wreath products. In Section 3, Isaacs’ type A conjecture is sharpened for the Coxeter–Shephard–Koster family. Diaconis and Fulman’s type A determinantal formula is also extended here. Lastly, we determine exactly when the Foulkes characters are a basis for the space of class functions .g/ that depend only on the dimension of the fixed space of g.