Geodesic
An embedding of the skein action on set partitions into the skein action on matchings
Jesse Kim1
1UCSD
The electronic journal of combinatorics, Tome 31 (2024) no. 1
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Rhoades defined a skein action of the symmetric group on noncrossing set partitions which generalized an action of the symmetric group on matchings. The $\mathfrak{S}_n$-action on matchings is made possible via the Ptolemy relation, while the action on set partitions is defined in terms of a set of skein relations that generalize the Ptolemy relation. The skein action on noncrossing set partitions has seen applications to coinvariant theory and coordinate rings of partial flag varieties. In this paper, we will show how Rhoades' $\mathfrak{S}_n$-module can be embedded into the $\mathfrak{S}_n$-module generated by matchings, thereby explaining how Rhoades' generalized skein relations all arise from the Ptolemy relation.
DOI : 10.37236/11578
Classification : 05E18, 05A18, 05E10, 20B30
Mots-clés : cyclic sieving, noncrossing partition, promotion, rotation, skein relation, symmetric group

Jesse Kim  1

1 UCSD 
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Jesse Kim. An embedding of the skein action on set partitions into the skein action on matchings. The electronic journal of combinatorics, Tome 31 (2024) no. 1. doi: 10.37236/11578

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