Representation stability in the intrinsic hyperplane arrangements associated to irreducible representations of the symmetric-groups

TL;DR

This paper investigates the stability properties of hyperplane arrangements derived from irreducible symmetric group representations, combining theoretical proofs with computational experiments to understand their topological and combinatorial behaviors.

Contribution

It introduces the concept of representation stability for intrinsic hyperplane arrangements associated with symmetric group representations, providing new theoretical results and empirical analysis.

Findings

Proves stability theorems for hyperplane complements

Shows diverse behaviors in associated compliment spaces

Provides simulation data on flats enumeration

Abstract

Some of the most classically relevant Hyperplane arrangements are the Braid Arrangements $B_n$ and their associated compliment spaces $\mathcal{F}_n$. In their recent work, Tsilevich, Vershik, and Yuzvinsky construct what they refer to as the intrinsic hyperplane arrangement within any irreducible representation of the symmetric group that generalize the classical braid arrangements. Through examples it is also shown that the associated compliment spaces to these intrinsic arrangements display behaviors far removed from $\mathcal{F}_n$. In this work we study the intrinsic hyperplane arrangements of irreducible representations of the symmetric group from the perspective of representation stability. This work is both theoretical, proving representation stability theorems for hyperplane complements, as well as statistical, examining the outputs of a number of simulations designed to enumerate flats.