Combinatorial Theory

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