  Matrix Schubert varieties (Fulton '92) carry natural actions of Levi groups.
Their coordinate rings are thereby Levi-representations; what is a
combinatorial counting rule for the multiplicities of their irreducibles? When
the Levi group is a torus, (Knutson-Miller '04) answers the question. We
present a general solution, a common refinement of the multigraded Hilbert
series, the Cauchy identity, and the Littlewood-Richardson rule. Our result
applies to any ``bicrystalline'' algebraic variety; we define these using the
operators of (Kashiwara '95) and of (Danilov-Koshevoi '05, van Leeuwen '06).
The proof introduces a ``filtered'' generalization of the
Robinson-Schensted-Knuth correspondence.
