Ring of symmetric functions: Difference between revisions

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# The set of monomial symmetric functions parametrized by partitions form a basis of Λ_''R'' as graded ''R''-[[module (mathematics)|module]], those parametrized by partitions of ''d'' being homogeneous of degree ''d''; the same is true for the set of Schur functions (also parametrized by partitions).

# Λ_''R'' is [[isomorphic]] as a graded ''R''-algebra to a polynomial ring ''R''[''Y''₁,''Y''₂,…] in infinitely many variables, where ''Y''_''i'' is given degree ''i'' for all ''i'' > 0, one isomorphism being the one that sends ''Y''_''i'' to ''e''_''i'' ∈ Λ_''R'' for every ''i''.

# There is an [[Involution (mathematics)|involutaryinvolutory]] [[automorphism]] ω of Λ_''R'' that interchanges the elementary symmetric functions ''e''_''i'' and the complete homogeneous symmetric function ''h''_''i'' for all ''i''. It also sends each power sum symmetric function ''p''_''i'' to (−1)^''i''−1 ''p''_''i'', and it permutes the Schur functions among each other, interchanging ''s''_λ and ''s''_λ^t where λ^t is the transpose partition of λ.

Property 2 is the essence of the [[fundamental theorem of symmetric polynomials]]. It immediately implies some other properties: