We define permutation modules and Young modules for the Brauer algebra B k (r, δ), and show that if the characteristic of the field k is neither 2 nor 3 then every permutation module is a sum of Young modules, respecting an ordering condition similar to that for symmetric groups. Moreover, we determine precisely in which cases cell module filtration multiplicities are well-defined, as done by Hemmer and Nakano for symmetric groups.

No abstract

We study the permutation module arising from the action of the symmetric group S 2n on the conjugacy class of fixed-point-free involutions, defined over an arbitrary field. The indecomposable direct summands of these modules are shown to possess filtrations by Specht modules and also filtrations by dual Specht modules. We see that these provide counterexamples to a conjecture by Hemmer. Twisted permutation modules are also considered, as is an application to the Brauer algebra.

Abstract. Given natural numbers m and n, we define a deflation map from the characters of the symmetric group S mn to the characters of S n . This map is obtained by first restricting a character of S mn to the wreath product S m o S n , and then taking the sum of the irreducible constituents of the restricted character on which the base group S m S m acts trivially. We prove a combinatorial formula which gives the values of the images of the irreducible characters of S mn under this map. We also prove an analogous result for more general deflation maps in which the base group is not required to act trivially. These results generalize the Murnaghan-Nakayama rule and special cases of the LittlewoodRichardson rule. As a corollary we obtain a new combinatorial formula for the character multiplicities that are the subject of the long-standing Foulkes' Conjecture. Using this formula we verify Foulkes' Conjecture in some new cases.

This paper proves a combinatorial rule giving all maximal and minimal partitions λ such that the Schur function sλ appears in a plethysm of two arbitrary Schur functions. Determining the decomposition of these plethysms has been identified by Stanley as a key open problem in algebraic combinatorics. As corollaries we prove three conjectures of Agaoka on the partitions labelling the lexicographically greatest and least Schur functions appearing in an arbitrary plethysm. We also show that the multiplicity of the Schur function labelled by the lexicographically least constituent may be arbitrarily large. The proof is carried out in the symmetric group and gives an explicit non‐zero homomorphism corresponding to each maximal or minimal partition.