A combinatorial basis for the fermionic diagonal coinvariant ring

Abstract

Let Θ n = (θ 1 , . . . , θ n ) and Ξ n = (ξ 1 , . . . , ξ n ) be two lists of n variables and consider the diagonal action of S n on the exterior algebra ∧{Θ n , Ξ n } generated by these variables. Jongwon Kim and Rhoades defined and studied the fermionic diagonal coinvariant ring FDR n obtained from ∧{Θ n , Ξ n } by modding out by the 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 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 S n−1 ⊂ S n . The basis will be indexed by a certain class of noncrossing partitions.