Crystals from 5-vertex ice models

Crystal structures for symmetric Grothendieck polynomials

“…Note that GT patterns, and hence 5-vertex configurations, have a natural crystal structure coming from the bijection with semistandard tableaux. On 5-vertex configurations, this was explicitly described in [EV17]. This crystal structure is a "coarse" version of the crystals on semistandard set-valued tableaux obtained by grouping together multiple terms.…”

“…Using the single variable symmetric Grothendieck polynomial, we add a marking to the GT pattern, and thus can write a G λ as a sum over these marked GT patterns. On the other hand, since GT patterns are naturally in bijection with semistandard tableaux, we can impose a natural crystal structure on the configurations of the 5-vertex model, as was described explicitly in [EV17]. This gives a "coarse" crystal structure of B(λ) that can compute G λ in analogy to the Tokuyama formulas for Whittaker functions (see, e.g., [BBF11, BBC + 12]).…”