Séminaire Lotharingien de Combinatoire, 86B.22 (2022), 12 pp.
Jesse Kim and Brendon Rhoades
Set Partitions, Fermions, and Skein Relations
Abstract.
The second author defined an action of the symmetric group
Sn<(sub> on the vector space spanned by noncrossing partitions of
{1, ... , n} by introducing
new skein relations which resolve local crossings in set partitions.
On the other hand, the second author and Jongwon Kim defined and studied
the {\em fermionic diagonal coinvariant
ring} FDRn which has a definition analogous to the Garsia-Haiman diagonal coinvariant ring DRn,
but with fermionic (anticommuting) variables. We prove that set partition skein relations arises naturally
in the context of FDRn. This clarifies and sharpens results on the skein action and gives an
Sn-equivariant way to resolve an arbitrary set partition into a linear combination of
noncrossing partitions.
Received: November 25, 2021.
Accepted: March 4, 2022.
Final version: April 1, 2022.
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