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I am not a native speaker of English and moreover belong to the ethnic group that is known to mess up the articles, but I certainly don't feel that the sentence "Behind these and several other attractive theories stands one immutable source" necessarily implies that the theory of symmetric functions is "THE" core of those theories. Many mathematical theories are connected to symmetric functions and their various structures, most notably the plethysm, and indeed experts do develop their own language and notation for those structures, sometimes reinventing the bicycle, and that, in my opinion was what Rota wanted to express, in his usual provocative way.