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

A web basis of a representation of $\mathfrak{S}_n$ is a basis of the representation for which the action of $\mathfrak{S}_n$ can be understood through combinatorial rules called skein relations. In this thesis, we study web bases for two families of irreducible $\mathfrak{S}_n$ representations, indexed by the partitions $(d^2, 1^{n-2d})$ and $(d^3, 1^{n-3d})$. The first was introduced by Rhoades and is indexed by noncrossing set partitions of $n$. We use it to give a model for the top degree component of the fermionic diagonal coinvariant ring, and introduce another similar basis to model the entire fermionic diagonal coinvariant ring. We also give an embedding of the noncrossing set partition representation into an induction product of the Temperley-Lieb representation with a sign representation, thereby providing alternate proofs that the skein relations which define the noncrossing set partition representation are in fact well defined. The second web basis is new, and simultaneously generalizes the $SL_3$ web basis of Kuperberg and the noncrossing set partition web basis. To define it and show it gives a basis, we draw on the combinatorics of Plabic graphs, jellyfish invariants, and weblike subgraphs.