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A fundamental result in the study of graph homomorphisms is Lov\'asz'stheorem that two graphs are isomorphic if and only if they admit the samenumber of homomorphisms from every graph. A line of work extending Lov\'asz'sresult to more general types of graphs was recently capped by Cai and Govorov,who showed that it holds for graphs with vertex and edge weights from anarbitrary field of characteristic 0. In this work, we generalize from graphhomomorphism -- a special case of #CSP with a single binary function -- togeneral #CSP by showing that two sets $\mathcal{F}$ and $\mathcal{G}$ ofarbitrary constraint functions are isomorphic if and only if the partitionfunction of any #CSP instance is unchanged when we replace the functions in$\mathcal{F}$ with those in $\mathcal{G}$. We give two very different proofs ofthis result. First, we demonstrate the power of the simple Vandermondeinterpolation technique of Cai and Govorov by extending it to general #CSP.Second, we give a proof using the intertwiners of the automorphism group of aconstraint function set, a concept from the representation theory of compactgroups. This proof is a generalization of a classical version of the recentproof of the Lov\'asz-type result by Man\v{c}inska and Roberson relatingquantum isomorphism and homomorphisms from planar graphs.

Equality on all #CSP Instances Yields Constraint Function Isomorphism via Interpolation and Intertwiners