  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.
