Duality and bicrystals on infinite binary matrices

The set of finite binary matrices a given size is known to carry type $A$ bicrystal structure. We first review this classical construction, explain how it yields short proof the equality between Kostka polynomials and one-dimensional sums together with natural generalisation $2M-X$ Pitman transform. Next, we show that, once relevant formalism on families infinite introduced, particular case much more general phenomenon. Each such family proved be endowed Kac–Moody tricrystal structures defined from root systems. Moreover, give an explicit decomposition these multicrystals, reminiscent characters yielding Cauchy identities.