Posets for Specht ideals of essential real reflection groups

TL;DR

This paper extends the combinatorial study of Specht ideals to all essential real reflection groups, including types D and dihedral groups, completing the understanding across these infinite families.

Contribution

It introduces the theory of Specht ideals for type D and dihedral groups, filling gaps in the combinatorial understanding of these ideals for all essential real reflection groups.

Findings

Completed the combinatorial analysis of Specht ideals for type D groups.

Extended Specht ideal theory to dihedral groups.

Unified the understanding of Specht ideals across all infinite families.

Abstract

Specht ideals are symmetric ideals in the polynomial ring generated by Specht polynomials associated with group representations. These ideals were previously studied for reflection groups of types $A$ and $B$, where their inclusion relations and their varieties reflect rich combinatorial structures. In this paper, we extend this theory to type $D$ and the dihedral groups. Our results complete the combinatorial study of Specht ideals across all infinite families of essential real reflection groups.