Cyclic quasi-symmetric functions

Ron M. Adin, Ira M. Gessel, Victor Reiner, Yuval Roichman

Research output: Contribution to journalArticlepeer-review

12 Scopus citations

Abstract

The ring of cyclic quasi-symmetric functions and its non-Escher subring are introduced in this paper. A natural basis consists of fundamental cyclic quasi-symmetric functions; for the non-Escher subring they arise as toric P-partition enumerators, for toric posets P with a total cyclic order. The associated structure constants are determined by cyclic shuffles of permutations. We then prove the following positivity phenomenon: for every non-hook shape λ, the coefficients in the expansion of the Schur function sλ in terms of fundamental cyclic quasi-symmetric functions are nonnegative. The proof relies on the existence of a cyclic descent map on the standard Young tableaux (SYT) of shape λ. The theory has applications to the enumeration of cyclic shuffles and SYT by cyclic descents.

Original languageEnglish
Pages (from-to)437-500
Number of pages64
JournalIsrael Journal of Mathematics
Volume243
Issue number1
DOIs
StatePublished - Jun 2021

Bibliographical note

Publisher Copyright:
© 2021, The Hebrew University of Jerusalem.

Funding

VR was partially supported by NSF grant DMS-1601961. Acknowledgements IMG was partially supported by grant no. 427060 from the Simons Foundation. RMA and YR were partially supported by an MIT-Israel MISTI grant and by the Israel Science Foundation, grant no. 1970/18.

FundersFunder number
MIT-Israel MISTI
National Science FoundationDMS-1601961
Simons Foundation
Israel Science Foundation1970/18

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