A representation-theoretic interpretation of positroid classes

Pawlowski, Brendan

doi:10.1016/j.aim.2023.109178

Cited by 2 publications

References 24 publications

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On Cyclic Descents for Tableaux

Adin

Reiner

Roichman

2018

International Mathematics Research Notices

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The notion of descent set, for permutations as well as for standard Young tableaux (SYT), is classical. Cellini introduced a natural notion of cyclic descent set for permutations, and Rhoades introduced such a notion for SYT -but only for rectangular shapes. In this work we define cyclic extensions of descent sets in a general context, and prove existence and essential uniqueness for SYT of almost all shapes. The proof applies nonnegativity properties of Postnikov's toric Schur polynomials, providing a new interpretation of certain Gromov-Witten invariants. Date: November 22, 2017. Proof of Lemma 2.2(i). The nonnegativity property (a) is obvious, with m(∅) = m([n]) = 0 following from the non-Escher axiom, while properties (b) and (c) are consequences of the equivariance and extension axioms in Definition 2.1. Proof of Lemma 2.2(ii). Let (m(J)) J⊆[n] be integers satisfying conditions (a),(b),(c) above. For each J ⊆ [n − 1] choose, arbitrarily, a subset T 0 J of T J := Des −1 (J) of size #T 0 J = m(J), and denote T 1 J := T J \ T 0 J ; by condition (c), #T 1 J = m(J ⊔ {n}). By construction, T = J⊆[n−1](T 0 J ⊔ T 1 J ).Define cDes : T −→ 2 [n] by

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On Cyclic Descents for Tableaux

Adin

Reiner

Roichman

2018

International Mathematics Research Notices

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Quasiparabolic sets and Stanley symmetric functions for affine fixed-point-free involutions

Zhang

2019

Preprint

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We introduce and study affine analogues of the fixed-point-free (FPF) involution Stanley symmetric functions of Hamaker, Marberg, and Pawlowski. Our methods use the theory of quasiparabolic sets introduced by Rains and Vazirani, and we prove that the subset of FPF-involutions is a quasiparabolic set for the affine symmetric group under conjugation. Using properties of quasiparabolic sets, we prove a transition formula for the affine FPF involution Stanley symmetric functions, analogous to Lascoux and Schützenberger's transition formula for Schubert polynomials. Our results suggest several conjectures and open problems.

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A representation-theoretic interpretation of positroid classes

Cited by 2 publications

References 24 publications

On Cyclic Descents for Tableaux

On Cyclic Descents for Tableaux

Quasiparabolic sets and Stanley symmetric functions for affine fixed-point-free involutions

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