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arXiv · 2501.09920

The sign character of the triagonal fermionic coinvariant ring

John Lentfer
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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.

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John Lentfer. 2025-01-17. The sign character of the triagonal fermionic coinvariant ring. https://doi.org/10.37236/13818

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