Set Partitions, Fermions, and Skein Relations

Abstract

Abstract Let $\Theta _n = (\theta _1, \dots , \theta _n)$ and $\Xi _n = (\xi _1, \dots , \xi _n)$ be two lists of $n$ variables, and consider the diagonal action of ${{\mathfrak {S}}}_n$ on the exterior algebra $\wedge \{ \Theta _n, \Xi _n \}$ generated by these variables. Jongwon Kim and the 2nd author defined and studied the fermionic diagonal coinvariant ring$FDR_n$ obtained from $\wedge \{ \Theta _n, \Xi _n \}$ by modding out by the ideal generated by the ${{\mathfrak {S}}}_n$-invariants with vanishing constant term. On the other hand, the 2nd author described an action of ${{\mathfrak {S}}}_n$ on the vector space with basis given by noncrossing set partitions of $\{1,\dots ,n\}$ using a novel family of skein relations that resolve crossings in set partitions. We give an isomorphism between a natural Catalan-dimensional submodule of $FDR_n$ and the skein representation. To do this, we show that set partition skein relations arise naturally in the context of exterior algebras. Our approach yields an ${{\mathfrak {S}}}_n$-equivariant way to resolve crossings in set partitions. We use fermions to clarify, sharpen, and extend the theory of set partition crossing resolution.