Partition algebras

Abstract

IntroductionA centerpiece of representation theory is the Schur-Weyl duality, which says that, (a) the general linear group GL n (C) and the symmetric group S k both act on tensor spacewith dim(V ) = n, (b) these two actions commute and (c) each action generates the full centralizer of the other, so that (d) as a (GL n (C), S k )-bimodule, the tensor space has a multiplicity free decomposition,

On the decomposition matrix of the partition algebra in positive characteristic

“…First notice that if δ < 0 then P K n (δ) is always semisimple, as we can never have a δ-pair µ ֒→ δ λ. Combining this with [6,Theorem 3.27] we see that P K r (δ) is non-semisimple if and only if 0 ≤ δ < 2n − 1.…”

“…It was shown in [6] that these elements generate P R n (δ). Notice that multiplication in P R n (δ) cannot increase the number of propagating blocks.…”

Cellular Bases for Algebras with a Jones Basic Construction

“…allowed for the simultaneous analysis of the whole tower of algebras (6.11) using the Jones basic construction, by Martin [25] and Halverson and Ram [17]. Cellularity of the partition algebras was proved in [6,41,42].…”

Stability Patterns in Representation Theory

“…Generators of A n . Generators for the partition algebra are given in [HR,Theorem 1.11(d)]. We explain the action of these generators on T d n (see [HR,Section 3]).…”