Rosa Orellana

The row-strict 0-Hecke action on standard immaculate skew tableaux was studied by the present authors, who showed that it gives rise to a bounded poset, called the \emph{skew immaculate Hecke poset}, and consequently to a cyclic 0-Hecke module. It was further shown that the subposet of skew standard extended immaculate tableaux always has a unique maximal element, but may have multiple minimal elements. In this paper we focus on these minimal elements, completely classifying them for a family of skew shapes that we call \emph{lobsters}. Moreover, we prove that when the skew shape is connected, the skew extended Hecke poset does have a unique minimal element, thereby showing that the associated 0-Hecke module is cyclic for both the row-strict and the dual immaculate actions.

The immaculate Hecke poset was introduced and investigated by Niese, Sundaram, van Willigenburg, Vega and Wang, who established the full poset structure, and determined modules for the 0-Hecke algebra action on immaculate and row-strict immaculate tableaux. In this paper, we extend their results by introducing the skew immaculate Hecke poset. We investigate the poset structure, and construct modules for the 0-Hecke algebra action on skew immaculate and skew row-strict immaculate tableaux, thus showing that the skew immaculate Hecke poset captures representation-theoretic information analogous to the immaculate Hecke poset. We also describe branching rules for the resulting skew modules.

The immaculate Hecke poset was introduced and investigated by Niese, Sundaram, van Willigenburg, Vega and Wang, who established the full poset structure, and determined modules for the 0-Hecke algebra action on immaculate and row-strict immaculate tableaux. In this paper, we extend their results by introducing the skew immaculate Hecke poset. We investigate the poset structure, and construct modules for the 0-Hecke algebra action on skew immaculate and skew row-strict immaculate tableaux, thus showing that the skew immaculate Hecke poset captures representation-theoretic information analogous to the immaculate Hecke poset. We also describe branching rules for the resulting skew modules.

We study the lattice of submonoids of the uniform block permutation monoid containing the symmetric group (which is its group of units). We prove that this lattice is distributive under union and intersection by relating the submonoids containing the symmetric group to downsets in a new partial order on integer partitions. Furthermore, we show that the sizes of the $\mathscr{J}$-classes of the uniform block permutation monoid are sums of squares of dimensions of irreducible modules of the monoid algebra.

The quasi-partition algebras were introduced by Daugherty and the first author as centralizers of the symmetric group. In this article, we give a more general definition of these algebras and give a construction of their simple modules. In addition, we introduce two new algebras, we give linear bases and show that for specializations of their parameters, these new algebras are isomorphic to centralizer algebras. We provide a generalized Bratteli diagram that illustrates how the representation theory of the three algebras discussed in this paper are related. Moreover, we give combinatorial formulas for the dimensions of the simple modules of these algebras.

We study the representation theory of the uniform block permutation algebra in the context of the representation theory of factorizable inverse monoids. The uniform block permutation algebra is a subalgebra of the partition algebra and is also known as the party algebra. We compute its characters and provide a Frobenius characteristic map to symmetric functions. This reveals connections of the characters of the uniform block permutation algebra and plethysms of Schur functions.