Hu, Jun Wang, Jianpan
Published in
Science in China Series A: Mathematics

This is a continuation of our previous work. We classify all the simple ℋq(Dn)-modules via an automorphismh defined on the set { λ | Dλ ≠ 0}. Whenfn(q) ≠ 0, this yields a classification of all the simple ℋq(Dn)- modules for arbitrary n. In general ( i. e., q arbitrary), if λ(1) = λ(2),wegivea necessary and sufficient condition ( in terms of some po...

Liu, Ricky Ini
Published in
Journal of Algebraic Combinatorics

We introduce a Littlewood-Richardson rule based on an algorithmic deformation of skew Young diagrams and present a bijection with the classical rule. The result is a direct combinatorial interpretation and proof of the geometric rule presented by Coskun (2000). We also present a corollary regarding the Specht modules of the intermediate diagrams.

Biggs, Norman
Published in
Journal of Algebraic Combinatorics

The identity linking the Tutte polynomial with the Potts model on a graph implies the existence of a decomposition resembling that previously obtained for the chromatic polynomial. Specifically, let {Gn} be a family of bracelets in which the base graph has b vertices. It is shown here (Theorems 3 and 4) that the Tutte polynomial of Gn can be writte...

Ryom-Hansen, Steen
Published in
Journal of Algebraic Combinatorics

We consider the algebra ℰn(u) introduced by Aicardi and Juyumaya as an abstraction of the Yokonuma–Hecke algebra. We construct a tensor space representation for ℰn(u) and show that this is faithful. We use it to give a basis of ℰn(u) and to classify its irreducible representations.

Peccati, Giovanni Pycke, Jean-Renaud

It is proved that each Hoeffding space associated with a random permutation (or, equivalently, with extractions without replacement from a finite population) carries an irreducible representation of the symmetric group, equivalent to a two-block Specht module.

Weber, Christian

Ordinary representation theory of the symmetric groups is quite well understood, but there are still many open questions concerning modular representation theory of the symmetric groups. About cohomology of S_n-modules, there is little known as well. This thesis has its starting point where the latter two fields meet. Possibilities of making statem...

Fayers, Matthew Lyle, Sinéad
Published in
Journal of Algebraic Combinatorics

The reducible Specht modules for the Hecke algebra \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {H}_{\mathbb{F},q}(\mathfrak{S}_{n})$\end{document} have been...

Billey, Sara Pawlowski, Brendan

Generalizing the notion of a vexillary permutation, we introduce a filtration of $S_{\infty}$ by the number of Edelman-Greene tableaux of a permutation, and show that each filtration level is characterized by avoiding a finite set of patterns. In doing so, we show that if $w$ is a permutation containing $v$ as a pattern, then there is an injection ...

Donkin, Stephen Santana, Ana Paula Yudin, Ivan
Published in
Journal of Algebra

We continue the development of the homological theory of quantum general linear groups previously considered by the first author. The development is used to transfer information to the representation theory of quantised Schur algebras. The acyclicity of induction from some rank-one modules for quantised Borel–Schur subalgebras is deduced. This is u...

Morse, Jennifer Schilling, Anne

We apply ideas from crystal theory to affine Schubert calculus and flag Gromov-Witten invariants. By defining operators on certain decompositions of elements in the type-$A$ affine Weyl group, we produce a crystal reflecting the internal structure of Specht modules associated to permutation diagrams. We show how this crystal framework can be applie...