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Title: Set Partitions, Fermions, and Skein Relations
Abstract Let $$\Theta _n = (\theta _1, \dots , \theta _n)$$ and $$\Xi _n = (\xi _1, \dots , \xi _n)$$ be two lists of $$n$$ variables, and consider the diagonal action of $${{\mathfrak {S}}}_n$$ on the exterior algebra $$\wedge \{ \Theta _n, \Xi _n \}$$ generated by these variables. Jongwon Kim and the 2nd author defined and studied the fermionic diagonal coinvariant ring$$FDR_n$$ obtained from $$\wedge \{ \Theta _n, \Xi _n \}$$ by modding out by the ideal generated by the $${{\mathfrak {S}}}_n$$-invariants with vanishing constant term. On the other hand, the 2nd author described an action of $${{\mathfrak {S}}}_n$$ on the vector space with basis given by noncrossing set partitions of $$\{1,\dots ,n\}$$ using a novel family of skein relations that resolve crossings in set partitions. We give an isomorphism between a natural Catalan-dimensional submodule of $$FDR_n$$ and the skein representation. To do this, we show that set partition skein relations arise naturally in the context of exterior algebras. Our approach yields an $${{\mathfrak {S}}}_n$$-equivariant way to resolve crossings in set partitions. We use fermions to clarify, sharpen, and extend the theory of set partition crossing resolution.  more » « less
Award ID(s):
1953781
PAR ID:
10367053
Author(s) / Creator(s):
;
Publisher / Repository:
Oxford University Press
Date Published:
Journal Name:
International Mathematics Research Notices
Volume:
2023
Issue:
11
ISSN:
1073-7928
Page Range / eLocation ID:
p. 9427-9480
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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