A skein action of the symmetric group on noncrossing partitions |
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Authors: | Brendon Rhoades |
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Institution: | 1.Department of Mathematics,University of California, San Diego,La Jolla,USA |
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Abstract: | We introduce and study a new action of the symmetric group \({\mathfrak {S}}_n\) on the vector space spanned by noncrossing partitions of \(\{1, 2,\ldots , n\}\) in which the adjacent transpositions \((i, i+1) \in {\mathfrak {S}}_n\) 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 \({\mathfrak {S}}_n\)-equivariant way. |
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