The sign character of the triagonal fermionic coinvariant ring

TL;DR

This paper determines the multiplicity of the sign character in certain fermionic coinvariant rings, proving a conjecture and providing explicit formulas for specific cases, advancing understanding of their algebraic structure.

Contribution

It proves a conjecture about the sign character multiplicity in $R_n^{(0,3)}$ and provides explicit formulas for double hook characters and methods for other cases.

Findings

Proves that the multiplicity of the sign character in $R_n^{(0,3)}$ is $n^2 - n + 1$.

Provides an explicit formula for double hook characters in $R_n^{(0,2)}$.

Discusses methods for calculating the sign character in $R_n^{(0,4)}$ and refines a conjecture involving Fibonacci numbers.

Abstract

We determine the trigraded multiplicity of the sign character of the triagonal fermionic coinvariant ring $R_n^{(0,3)}$. As a corollary, this proves a conjecture of Bergeron (2020) that the multiplicity of the sign character of $R_n^{(0,3)}$ is $n^2-n+1$. We also give an explicit formula for double hook characters in the diagonal fermionic coinvariant ring $R_n^{(0,2)}$, and discuss methods towards calculating the sign character of $R_n^{(0,4)}$. Finally, we give a multigraded refinement of a conjecture of Bergeron (2020) that the multiplicity of the sign character of the $(1,3)$-bosonic-fermionic coinvariant ring $R_n^{(1,3)}$ is $\frac{1}{2}F_{3n}$, where $F_n$ is a Fibonacci number.