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