Open Access Publications from the University of California

## Published Web Location

https://doi.org/10.5070/C63160415
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. The author and Rhoades 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.

Mathematics Subject Classifications: 05E10, 05E18, 20C30

Keywords: Skein relation, coinvariant algebra, noncrossing set partition, cyclic sieving