Journal of Combinatorics

Volume 12 (2021)

Number 2

Some conjectures on the Schur expansion of Jack polynomials

Pages: 215 – 233

DOI: https://dx.doi.org/10.4310/JOC.2021.v12.n2.a2

Authors

Per Alexandersson (KTH Royal Institute of Technology, Stockholm, Sweden)

James Haglund (Department of Mathematics, University of Pennsylvania, Philadelphia, Penn., U.S.A.)

George Wang (Department of Mathematics, University of Pennsylvania, Philadelphia, Penn., U.S.A.)

Abstract

We present positivity conjectures for the Schur expansion of Jack symmetric functions in two bases given by binomial coefficients. Partial results suggest that there are rich combinatorics to be found in these bases, including Eulerian numbers, Stirling numbers, quasi-Yamanouchi tableaux, and rook boards. These results also lead to further conjectures about the fundamental quasisymmetric expansions of these bases, which we prove for special cases.

Keywords

Jack polynomials, Schur polynomials, quasi-Yamanouchi tableaux, Eulerian numbers, Stirling numbers, Rook polynomials

2010 Mathematics Subject Classification

Primary 05E05. Secondary 05A15.

Full Text (PDF format)

Per Alexandersson was partially supported by the Knut and Alice Wallenberg Foundation, James Haglund was partially supported by NSF grant DMS-1600670, and George Wang was partially supported by the NSF Graduate Research Fellowship, DGE-1321851.

Received 11 November 2018

Accepted 7 May 2020

Published 16 July 2021