Set partitions, fermions, and skein relations

Abstract

Let Θn = (θ1, . . . , θn) and Ξn = (ξ1, . . . , ξn) be two lists of n variables and consider the diagonal action of Sn on the exterior algebra ∧{Θn, Ξn} generated by these variables. Jongwon Kim and the second author defined and studied the fermionic diagonal coinvariant ring F DRn obtained from ∧{Θn, Ξn} by modding out by the Sn-invariants with vanishing constant term. On the other hand, the second author described an action of Sn on the vector space with basis given by noncrossing set partitions of {1, . . . , n} using a novel family of skein relations which resolve crossings in set partitions. We give an isomorphism between a natural Catalan-dimensional submodule of F DRn 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 Sn-equivariant way to resolve crossings in set partitions. We use fermions to clarify, sharpen, and extend the theory of set partition crossing resolution.

A combinatorial model for the fermionic diagonal coinvariant ring

“…Such a proof would complete an alternative representation theoretic proof of Thiel's result. In joint work with Rhoades [6] we developed a similar combinatorial model for the maximal bidegree components of F DR n , with a basis indexed by all noncrossing set partitions. The action of S n on that basis could be understood in terms of skein-like relations described by Rhoades [11].…”

“…They remark that in the case when i + j = n − 1, the above shows that the dimension of (F DR n ) n−k,k−1 is given by the Narayana number Nar(n, k). Narayana numbers count noncrossing set partitions of [n] into k blocks, and in joint work with Rhoades [6] we gave a combinatorial basis of (F DR n ) n−k,k−1 indexed by set partitions for which the S n -action was given by a skein action on noncrossing partitions first described by Rhoades in [11].…”

Quasisymmetric harmonics of the exterior algebra

“…The symmetric function expressions and representation theoretic interpretation was extended further to include the quotient of two sets of commuting and two sets of anticommuting variables in [8] to what is known as the Theta Conjecture. At present, this also remains an open conjecture, but progress has been made on some special cases [20,21,28,29].…”

A proof of the fermionic Theta coinvariant conjecture

“…The ring DR n is a bigraded S n -module; Haiman used algebraic geometry to calculate its isomorphism type [10]. In recent years, researchers in algebraic combinatorics studied variants of DR n involving mixtures of commuting and anticommuting variables [1,2,3,9,11,12,13,14,16,17,18,19]. Drawing terminology from supersymmetry, we will refer to commuting variables as bosonic and anticommuting variables as fermionic.…”