Crystal invariant theory II: Pseudoenergies
Abstract
The geometric crystal operators and geometric $R$matrices (or geometric Weyl group actions) give commuting actions on the field of rational functions in $mn$ variables. We study the invariants of various combinations of these actions, which we view as "crystal analogues" of the invariants of $S_m$, ${\rm SL}_m$, $S_n \times S_m$, ${\rm SL}_n \times \, S_m$, and ${\rm SL}_n \times {\rm SL}_m$ acting on the polynomial ring in an $m \times n$ matrix of variables. The polynomial invariants of the $S_m$action generated by the ${\rm GL}_m$geometric $R$matrices were described by Lam and the thirdnamed author as the ring of loop symmetric functions. In a previous paper of the authors, the polynomial invariants of the ${\rm GL}_m$geometric crystal operators were described as a subring of the ring of loop symmetric functions. In this paper, we give conjectural generating sets for the fields of rational invariants in the remaining cases, and we give formulas expressing a large class of loop symmetric functions in terms of these conjectural generators. Our results include new positive formulas for the central charge and energy function of a product of singlerow geometric crystals, and a new derivation of Kirillov and Berenstein's piecewiselinear formula for cocharge. The formulas manifest the symmetries possessed by these functions.
 Publication:

arXiv eprints
 Pub Date:
 May 2022
 DOI:
 10.48550/arXiv.2205.12681
 arXiv:
 arXiv:2205.12681
 Bibcode:
 2022arXiv220512681B
 Keywords:

 Mathematics  Quantum Algebra;
 Mathematics  Combinatorics;
 Mathematics  Representation Theory
 EPrint:
 51 pages, 12 figures