Open questions for operators related to rectangular Catalan combinatorics

We formulate many open questions, most ot them new, regarding Schur positivity, Schur inclusion, $e$-positivity, and $e$-inclusion of interesting symmetric functions arising from operators in the elliptic Hall algebra, and give supporting evidence for why one should expect such behavior. This ties in with many recent advances in the study of Rectangular Catalan Combinatorics and other subjects pertaining to Algebraic Geometry, Representation Theory, and the Homology of Torus Knots.

Keywords

Catalan combinatorics, Schur positivity, $(m,n)$-Dyck paths, elliptic Hall algebra

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This work was supported by NSERC.

Received 30 March 2016

Published 17 July 2017