Set Partitions, Fermions, and Skein Relations.

This article focuses on establishing a deep connection between two modules over the symmetric group \(\mathfrak{S}_n\): a combinatorial module defined via skein relations resolving crossings in noncrossing set partitions, and an algebraic module arising from the fermionic diagonal coinvariant ring \(FDR_n\). The authors construct explicit fermionic elements \(F_\pi\) and \(f_\pi\) in an exterior algebra that correspond to set partitions \(\pi\), proving that these fermions satisfy the skein relations and form bases for submodules of \(FDR_n\). They demonstrate an isomorphism between the skein representation on noncrossing partitions and a natural Catalan-dimensional submodule \(\overline{FDR_n}\) of \(FDR_n\), thereby providing an algebraic model for the skein action. The paper also develops a crossing resolution map \(p\) projecting arbitrary set partitions onto linear combinations of noncrossing partitions, characterized by equivariance under a sign-twisted permutation action, and explores related quadratic ideals encoding these relations. Finally, the authors discuss extensions of their framework to other reflection groups and multidiagonal fermionic coinvariant rings, posing open problems for further combinatorial and algebraic interpretations.