You may be interested in Yuval Filmus's survey paper Spectral Methods for Intersection Problems (Friedgut's Research Program in Extremal Combinatorics) (and/or his thesis which seems to have exactly the same title), which purports to explain how a number of combinatorial results along the lines of the Erdős-Ko-Rado theorem can be proved using representation theory. (I'm not sure one should call such results "identities", however: they're more along the lines of maximal combinatorics.)
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