A combinatorial model for the fermionic diagonal coinvariant ring

TL;DR

This paper develops a combinatorial basis for the fermionic diagonal coinvariant ring, extending previous work to the entire ring and linking it to noncrossing partitions and symmetric group actions.

Contribution

It introduces a new combinatorial basis for the entire fermionic diagonal coinvariant ring, generalizing prior results limited to maximal degree components.

Findings

Basis indexed by noncrossing partitions

Extended combinatorial interpretation to the whole ring

Links to symmetric group actions and noncrossing partitions

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 Rhoades defined and studied the fermionic diagonal coinvariant ring $FDR_n$ obtained from $\wedge \{ \Theta_n, \Xi_n \}$ by modding out by the $\mathfrak{S}_n$-invariants with vanishing constant term. In joint work with Rhoades we gave a basis for the maximal degree components of this ring where the action of $\mathfrak{S}_n$ could be interpreted combinatorially via noncrossing set partitions. This paper will do similarly for the entire ring, although the combinatorial interpretation will be limited to the action of $\mathfrak{S}_{n-1} \subset \mathfrak{S}_n$. The basis will be indexed by a certain class of noncrossing partitions.