Loewy Truncations and Lifting Obstructions for Commuting Nilpotent Matrix Pairs

Two pairs of commuting nilpotent n×n complex matrices are simultaneously similar if and only if the associated modules over R = C[x,y] are isomorphic. We compare such modules along their Loewy truncations M/m^r M, where m = (x,y). The obstruction to lifting an isomorphism of truncations from level r to level r+1 is a class in an Ext^1 group, and we compute it from the defects of an arbitrary linear lift. For a single Jordan block J_n, every commuting nilpotent matrix is p(J_n) for a polynomial p without constant term, and the truncations of the module are given by the Taylor polynomials of p. The obstruction at level r vanishes for one, and hence for every, isomorphism at that level if and only if p and q have the same Taylor coefficient of order r, so two such pairs first differ at the lowest order at which the Taylor coefficients of p and q differ. We compare this with the pencil rank criterion of Derksen, Klep, Makam and Volčič, describe the family of ideals with Hilbert function (1,2,1), where the argument no longer reduces to a scalar, and propose an application to historical phonology. In it, an unconditioned sound change acts as a linear map on the segment inventory, a chain shift ending in deletion acts as J_n, and a second change that commutes with it acts as a power of J_n. The level of the first obstruction is the position in the chain at which two varieties first differ, and the difference of their exponents counts the segments of a standard form that cannot be recovered from the corresponding dialect form.