A skein action of the symmetric group on noncrossing partitions

2015/01/20 by Brendon Rhoades, Rhoades, Brendon · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.1501.04680

openalex publication_date 2015/01/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce and study a new action of the symmetric group \mathfrakS_n on the vector space spanned by noncrossing partitions of \1, 2, …, n\ in which the adjacent transpositions (i, i+1) ∈ \mathfrakS_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 \mathfrakS_n-equivariant way.

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