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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. 

Let $$V_1, V_2, V_3, \dots $$ be a sequence of $$\mathbb {Q}$$-vector spaces where $$V_n$$ carries an action of $$\mathfrak{S}_n$$. Representation stability and multiplicity stability are two related notions of when the sequence $$V_n$$ has a limit. An important source of stability phenomena arises when $$V_n$$ is the $$d^{th}$$ homology group (for fixed $$d$$) of the configuration space of $$n$$ distinct points in some fixed topological space $$X$$. We replace these configuration spaces with moduli spaces of tuples $$(W_1, \dots, W_n)$$ of subspaces of a fixed complex vector space $$\mathbb {C}^N$$ such that $$W_1 + \cdots + W_n = \mathbb {C}^N$$. These include the varieties of spanning line configurations which are tied to the Delta Conjecture of symmetric function theory. 

Abstract The superspace ring $$\Omega _n$$ is a rank n polynomial ring tensored with a rank n exterior algebra. Using an extension of the Vandermonde determinant to $$\Omega _n$$ , the authors previously defined a family of doubly graded quotients $${\mathbb {W}}_{n,k}$$ of $$\Omega _n$$ , which carry an action of the symmetric group $${\mathfrak {S}}_n$$ and satisfy a bigraded version of Poincaré Duality. In this paper, we examine the duality modules $${\mathbb {W}}_{n,k}$$ in greater detail. We describe a monomial basis of $${\mathbb {W}}_{n,k}$$ and give combinatorial formulas for its bigraded Hilbert and Frobenius series. These formulas involve new combinatorial objects called ordered set superpartitions . These are ordered set partitions $$(B_1 \mid \cdots \mid B_k)$$ of $$\{1,\dots ,n\}$$ in which the nonminimal elements of any block $$B_i$$ may be barred or unbarred.