Young modules and filtration multiplicities for Brauer algebras

Abstract

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.

The Brauer algebra and the symplectic Schur algebra

“…Following [21] (see also [27]), we define the Specht (or cell) module S(λ) and twisted Specht (or cell) module S(λ) for the Brauer algebra by Following Hartmann and Paget [27], we define the permutation module M(λ) and the twisted permutation module M(λ) for the Brauer algebra by…”

“…The same must hold for J i , since it is a sum of images of I i . By the proof of [27,Prop. 3.3] the irreducible module D(µ), µ p-regular, is killed by I i if and only if |µ| > t i .…”

The blocks of the Brauer algebra in characteristic zero

“…A necessary and sufficient condition for semisimplicity (which holds over arbitrary fields) was given recently by Rui [Rui05]. The study of Young and permutation modules for these algebras has been started in [HP06].…”

Dominant Dimension and Almost Relatively True Versions of Schur’s Theorem

“…for any cellular algebra). The first such analogue followed quickly on the heels of Hemmer and Nakano's article: Hartmann and Paget [28] were able to show that cell modules of Brauer algebras form a standard system, provided the parameter δ is not zero (or the parameter r is odd) and the characteristic of the ground field k is neither two nor three. Brauer algebras occur in various different contexts, ranging from knot theory to statistical mechanics.…”

“…Other instances of Schur-Weyl dualities have been established for various classes of diagram algebras such as Brauer algebras, in [28] and more generally in [29]. [33]), or more generally of Schur algebras of diagram algebras (including BMW algebras and partition algebras, see [29]).…”