In algebra and in particular in algebraic combinatorics, the ring of symmetric functions is a specific limit of the rings of symmetric polynomials in n indeterminates, as n goes to infinity. This ring serves as universal structure in which relations between symmetric polynomials can be expressed in a way independent of the number n of indeterminates (but its elements are neither polynomials nor functions). Among other things, this ring plays an important role in the representation theory of the symmetric group. The ring of symmetric functions can be given a coproduct and a bilinear form making it into a positive selfadjoint graded Hopf algebra that is both commutative and cocommutative.
Symmetric polynomials
The study of symmetric functions is based on that of symmetric polynomials. In a polynomial ring in some finite set of indeterminates, a polynomial is called symmetric if it stays the same whenever the indeterminates are permuted in any way. More formally, there is an action by ring automorphisms of the symmetric group Sn on the polynomial ring in n indeterminates, where a permutation acts on a polynomial by simultaneously substituting each of the indeterminates for another according to the permutation used. The invariants for this action form the subring of symmetric polynomials. If the indeterminates are X1, ..., Xn, then examples of such symmetric polynomials are
X 1 + X 2 + ⋯ + X n , {\displaystyle X_{1}+X_{2}+\cdots +X_{n},\,}
X 1 3 + X 2 3 + ⋯ + X n 3 , {\displaystyle X_{1}^{3}+X_{2}^{3}+\cdots +X_{n}^{3},\,}
and
X 1 X 2 ⋯ X n . {\displaystyle X_{1}X_{2}\cdots X_{n}.\,}
A somewhat more complicated example is X13X2X3 + X1X23X3 + X1X2X33 + X13X2X4 + X1X23X4 + X1X2X43 + ... where the summation goes on to include all products of the third power of some variable and two other variables. There are many specific kinds of symmetric polynomials, such as elementary symmetric polynomials, power sum symmetric polynomials, monomial symmetric polynomials, complete homogeneous symmetric polynomials, and Schur polynomials.
The ring of symmetric functions Most relations between symmetric polynomials do not depend on the number n of indeterminates, other than that some polynomials in the relation might require n to be large enough in order to be defined. For instance the Newton's identity for the third power sum polynomial p3 leads to
p 3 ( X 1 , … , X n ) = e 1 ( X 1 , … , X n ) 3 − 3 e 2 ( X 1 , … , X n ) e 1 ( X 1 , … , X n ) + 3 e 3 ( X 1 , … , X n ) , {\displaystyle p_{3}(X_{1},\ldots ,X_{n})=e_{1}(X_{1},\ldots ,X_{n})^{3}-3e_{2}(X_{1},\ldots ,X_{n})e_{1}(X_{1},\ldots ,X_{n})+3e_{3}(X_{1},\ldots ,X_{n}),}
where the e i {\displaystyle e_{i}} denote elementary symmetric polynomials; this formula is valid for all natural numbers n, and the only notable dependency on it is that ek(X1,...,Xn) = 0 whenever n < k. One would like to write this as an identity
p 3 = e 1 3 − 3 e 2 e 1 + 3 e 3 {\displaystyle p_{3}=e_{1}^{3}-3e_{2}e_{1}+3e_{3}}
that does not depend on n at all, and this can be done in the ring of symmetric functions. In that ring there are nonzero elements ek for all integers k ≥ 1, and any element of the ring can be given by a polynomial expression in the elements ek.
Definitions A ring of symmetric functions can be defined over any commutative ring R, and will be denoted ΛR; the basic case is for R = Z. The ring ΛR is in fact a graded R-algebra. There are two main constructions for it; the first one given below can be found in (Stanley, 1999), and the second is essentially the one given in (Macdonald, 1979).
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