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Crystal invariant theory II: Pseudo-energies

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 third-named 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 single-row geometric crystals, and a new derivation of Kirillov and Berenstein's piecewise-linear formula for cocharge. The formulas manifest the symmetries possessed by these functions.