On Cyclic Descents for Tableaux

Ron M. Adin; Victor Reiner; Yuval Roichman

doi:10.1093/imrn/rny280

On Cyclic Descents for Tableaux

Ron M. Adin
, Victor Reiner
, Yuval Roichman

Department of Mathematics

University of Minnesota Twin Cities

Research output: Contribution to journal › Article › peer-review

10 Scopus citations

Abstract

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.

Original language	English
Pages (from-to)	10231-10276
Number of pages	46
Journal	International Mathematics Research Notices
Volume	2020
Issue number	24
DOIs	https://doi.org/10.1093/imrn/rny280
State	Published - 1 Dec 2020

Bibliographical note

Publisher Copyright:
© 2018 The Author(s).

Funding

This work was partially supported by a MIT-Israel International Science and Technology Initiatives grant [to R.M.A. and Y.R.] and a National Science Foundation grant [DMS-1601961 to V.R.].

Funders	Funder number
National Science Foundation	DMS-1601961

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AB - 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.

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On Cyclic Descents for Tableaux

Abstract

Bibliographical note

Funding

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