The sign character of the triagonal fermionic coinvariant ring
John Lentfer
TL;DR
This work determines the trigraded multiplicity of the sign character in the triagonal fermionic coinvariant ring $R_n^{(0,3)}$ by realizing its structure through triagonal fermionic harmonics $T_n$ and proving that the sign component of the multigraded Frobenius character equals $s_{(n-1)}(u,v,w)+s_{(n-2,1,1)}(u,v,w)$, which evaluates to $n^2-n+1$ at $(1,1,1)$. The authors construct two explicit highest-weight vectors in $T_n$ to show the upper bound for the sign multiplicity is tight, thereby proving the main theorem. They also derive explicit formulas for double-hook characters of the diagonal fermionic coinvariant ring $R_n^{(0,2)}$, using Rosas’ Kronecker-coefficient formulas, and discuss methods and partial results for the four-fermion case $R_n^{(0,4)}$. In addition, they examine the sign character in $R_n^{(1,3)}$, providing a conjectured $q$-refined formula and proving a combinatorial identity linking to the Fibonacci numbers, which supports Bergeron’s conjecture under the Theta framework. Altogether, the paper advances understanding of sign characters across fermionic-bosonic coinvariants, linking representation theory, combinatorics, and conjectural Theta/Fibonacci phenomena.
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 dimension 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 dimension 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.