Stable Symmetric Series, Differential Operators, and Jack Deformations

What the math gives to ML

The paper develops stable algebraic bases for permutation-class multiplication and connects them to differential operators and Jack-polynomial deformations. The most directly transferable construction in the supplied material is the exact Wick-matching expansion of Gaussian matrix polynomial features: an expectation of trace products of AZBZ^t becomes a finite sum of products of power-sum invariants of A and B, with integer matching-count coefficients. This gives a structured polynomial interaction layer whose parameters are tied by combinatorial matching counts rather than learned independently. A practical first implementation is a low-degree invariant module for set, graph, or token representations, using precomputed matching coefficients for n=2, 3, or 4 and comparing it with an unconstrained polynomial interaction baseline.