A web basis of invariant polynomials from noncrossing partitions

Patrias, Rebecca; Pechenik, Oliver; Striker, Jessica

doi:10.1016/j.aim.2022.108603

Cited by 4 publications

(2 citation statements)

References 47 publications

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“…Their proof proceeded via direct calculation of N (n, k, q) and sizes of fixed point sets; the skein action allowed for an alternate proof utilizing Springer's theorem on regular elements [7,9]. The skein action has since been found within coinvariant rings and coordinate rings of certain partial flag varieties [3,5], strengthening the claim that it is an action worth studying in its own right.…”

Section: → +mentioning

confidence: 97%

An Embedding of the Skein Action on Set Partitions into the Skein Action on Matchings.

Kim

2024

Electron. J. Combin.

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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.

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Section: → +mentioning

confidence: 97%

An Embedding of the Skein Action on Set Partitions into the Skein Action on Matchings.

Kim

2024

Electron. J. Combin.

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“…The action of S n on that basis could be understood in terms of Skein-like relations described by Rhoades [Rho17]. Patrias, Pechenik, and Striker [PPS22] independently discovered an alternate algebraic/geometric model for the irreducible submodule of this action generated by singleton-free noncrossing set partitions as the coordinate ring of a certain algebraic variety. They associated to each partition a polynomial in this coordinate ring defined in terms of matrix minors, and showed that these polynomials satisfied the Skein relations described in [Rho17].…”

Section: Maximal Bidegrees Cyclic Sieving and Further Directionsmentioning

confidence: 99%

A combinatorial basis for the fermionic diagonal coinvariant ring

Kim¹

2023

Combinatorial Theory

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Let Θ n = (θ 1 , . . . , θ n ) and Ξ n = (ξ 1 , . . . , ξ n ) be two lists of n variables and consider the diagonal action of S n on the exterior algebra ∧{Θ n , Ξ n } generated by these variables. Jongwon Kim and Rhoades defined and studied the fermionic diagonal coinvariant ring FDR n obtained from ∧{Θ n , Ξ n } by modding out by the S n -invariants with vanishing constant term. The author and Rhoades gave a basis for the maximal degree components of this ring where the action of S n could be interpreted combinatorially via noncrossing set partitions. This paper will do similarly for the entire ring, although the combinatorial interpretation will be limited to the action of S n−1 ⊂ S n . The basis will be indexed by a certain class of noncrossing partitions.

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Tableau evacuation and webs

Patrias,

Pechenik

2023

Proc. Amer. Math. Soc. Ser. B

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Webs are certain planar diagrams embedded in disks. They index and describe bases of tensor products of representations of s l 2 \mathfrak {sl}_2 and s l 3 \mathfrak {sl}_3 . There are explicit bijections between webs and certain rectangular tableaux. Work of Petersen–Pylyavskyy–Rhoades (2009) and Russell (2013) shows that these bijections relate web rotation to tableau promotion. We describe the analogous relation between web reflection and tableau evacuation.

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A web basis of invariant polynomials from noncrossing partitions

Cited by 4 publications

References 47 publications

An Embedding of the Skein Action on Set Partitions into the Skein Action on Matchings.

An Embedding of the Skein Action on Set Partitions into the Skein Action on Matchings.

A combinatorial basis for the fermionic diagonal coinvariant ring

Tableau evacuation and webs

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