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arXiv · 1811.05440

Cyclic quasi-symmetric functions

Ron M. Adin·Ira M. Gessel·Victor Reiner·Yuval Roichman
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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 $s_λ$ 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.

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Ron M. Adin, Ira M. Gessel, Victor Reiner, Yuval Roichman. 2020-05-25. Cyclic quasi-symmetric functions. https://arxiv.org/abs/1811.05440

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