The poset of Specht ideals for hyperoctahedral groups

TL;DR

This paper explores the structure of Specht ideals related to hyperoctahedral groups, establishing a poset framework that connects combinatorics with algebraic properties, and extends classical symmetric group results.

Contribution

It introduces a bidominance order on bipartitions to describe Specht ideal inclusions for hyperoctahedral groups, advancing understanding of their algebraic and combinatorial structure.

Findings

Defined a bidominance order on bipartitions

Described the poset of Specht ideal inclusions

Studied algebraic consequences for B_n-invariant ideals

Abstract

Specht polynomials classically realize the irreducible representations of the symmetric group. The ideals defined by these polynomials provide a strong connection with the combinatorics of Young tableaux and have been intensively studied by several authors. We initiate similar investigations for the ideals defined by the Specht polynomials associated to the hyperoctahedral group $B_n$. We introduce a bidominance order on bipartitions which describes the poset of inclusions of these ideals and study algebraic consequences on general $B_n$-invariant ideals and varieties, which can lead to computational simplifications.