Ring of symmetric functions: Difference between revisions

Revision as of 08:02, 18 April 2014

Revision as of 12:31, 5 May 2014

Line 13:

: <math>X_1^3+X_2^3+\cdots+X_n^3, \, </math>

and

:<math>X_1X_2\cdots X_n. \, </math>

Line 32:

==== As a ring of formal power series ====

The easiest (though somewhat heavy) construction starts with the ring of [[Formal_power_seriesFormal power series#Power_series_in_several_variablesPower series in several variables|formal power series]] ''R''[[''X''₁,''X''₂,…]] over ''R'' in infinitely many indeterminates; one defines &Lambda;_''R'' as its subring consisting of power series ''S'' that satisfy

#''S'' is invariant under any permutation of the indeterminates, and

#the degrees of the monomials occurring in ''S'' are bounded.

Note that because of the second condition, power series are used here only to allow infinitely many terms of a fixed degree, rather than to sum terms of all possible degrees. Allowing this is necessary because an element that contains for instance a term ''X''₁ should also contain a term ''X''_''i'' for every ''i''&nbsp;&gt;&nbsp;1 in order to be symmetric. Unlike the whole power series ring, the subring &Lambda;_''R'' is graded by the total degree of monomials: due to condition&nbsp;2, every element of &Lambda;_''R'' is a finite sum of [[Homogeneous_polynomialHomogeneous polynomial|homogeneous]] elements of &Lambda;_''R'' (which are themselves infinite sums of terms of equal degree). For every ''k''&nbsp;&ge;&nbsp;0, the element ''e''_''k''&nbsp;&isin;&nbsp;&Lambda;_''R'' is defined as the formal sum of all products of ''k'' distinct indeterminates, which is clearly homogeneous of degree ''k''.

==== As an algebraic limit ====

Line 73:

== Properties of the ring of symmetric functions ==

=== Identities ===