Ring of symmetric functions: Difference between revisions

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=== Generating functions ===

The first definition of &Lambda;_''R'' as a subring of ''R''[<span/>[''X''₁,''X''₂,…]] allows expression the [[generating function]]s of several sequences of symmetric functions to be elegantly expressed. Contrary to the relations mentioned earlier, which are internal to &Lambda;_''R'', these expressions involve operations taking place in ''R''<nowiki>[[<span/nowiki>[''X''₁,''X''₂,…;''t''<nowiki>]]</nowiki> but outside its subring &Lambda;_''R''<nowiki>[[<span/nowiki>[''t''<nowiki>]]</nowiki>, so they are meaningful only if symmetric functions are viewed as formal power series in indeterminates ''X''_''i''. We shall write "(''X'')" after the symmetric functions to stress this interpretation.

The generating function for the elementary symmetric functions is