Cyclic quasi-symmetric functions

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

Research output: Contribution to journalArticlepeer-review

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

Bibliographical note

Funding Information:
VR was partially supported by NSF grant DMS-1601961. Acknowledgements

Funding Information:
IMG was partially supported by grant no. 427060 from the Simons Foundation.

Funding Information:
RMA and YR were partially supported by an MIT-Israel MISTI grant and by the Israel Science Foundation, grant no. 1970/18.

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

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