A skein action of the symmetric group on noncrossing partitions

Abstract

We introduce and study a new action of the symmetric group $${\mathfrak {S}}_n$$Sn on the vector space spanned by noncrossing partitions of $$\{1, 2,\ldots , n\}$${1,2,ź,n} in which the adjacent transpositions $$(i, i+1) \in {\mathfrak {S}}_n$$(i,i+1)źSn act on noncrossing partitions by means of skein relations. We characterize the isomorphism type of the resulting module and use it to obtain new representation-theoretic proofs of cyclic sieving results due to Reiner---Stanton---White and Pechenik for the action of rotation on various classes of noncrossing partitions and the action of K-promotion on two-row rectangular increasing tableaux. Our skein relations generalize the Kauffman bracket (or Ptolemy relation) and can be used to resolve any set partition as a linear combination of noncrossing partitions in a $${\mathfrak {S}}_n$$Sn-equivariant way.