Consider a variety $X$ in the space of matrices, stable under the action of a product of general linear groups by row and column operations. How does its coordinate ring decompose as a direct sum of irreducible representations? We argue that this question is effectively studied by imposing a crystal graph structure on the standard monomials of the defining ideal of $X$ (with respect to some term order). For the standard monomials of "bicrystalline" ideals, we obtain such a crystal structure from the crystal graph on monomials introduced by Danilov–Koshevoi and van Leeuwen. This yields an explicit combinatorial rule we call "filtered RSK" for their irreducible representation multiplicities. In this talk, we will explain our rule and show that Schubert determinantal ideals (among others) are bicrystalline. Based on joint work with Abigail Price and Alexander Yong, https://arxiv.org/abs/2403.09938.