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In mathematics, the term "symmetric function" can mean two different things. A symmetric function of n variables is one whose value at any n-tuple of arguments is the same as its value at any permutation of that n-tuple. While this notion can apply to any type of function whose n arguments live in the same set, it is most often used for polynomial functions, in which case these are the functions given by symmetric polynomials. There is hardly any systematic theory of symmetric non-polynomial functions of n variables, which are therefore not considered in this article.## Symmetric polynomials

## Symmetric functions of the roots of a univariate polynomial

The right hand side of the relation for a_{n−d}, without the sign (−1)^{d}, is the elementary symmetric polynomial of degree d of the roots.## 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 leads to
_{k}(X_{1},…,X_{n}) = 0 whenever n < k. One would like to write this as an identity 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 polynomials. In that ring there are elements e_{k} for all integers k ≥ 1, and an arbitrary element can be given by a polynomial expression in them.
### Definitions

#### As a ring of formal power series

#### As an algebraic limit

### Defining individual symmetric functions

_{n} denotes the ring of symmetric polynomials in n indeterminates), and also in (Stanley, 1999)_{n} (decreasing the number of indeterminates is obtained by setting some of them to zero, so that the coefficients of any monomial in the remaining indeterminates is unchanged), and their degree should remain bounded. (An example of a family of symmetric polynomials that fails both conditions is $Pi\_\{i=1\}^nX\_i$; the family $Pi\_\{i=1\}^n(X\_i+1)$ fails only the second condition.) Any symmetric polynomial in n indeterminates can be used to construct a compatible family of symmetric polynomials, using the morphisms ρ_{i} for i < n to decrease the number of indeterminates, and φ_{i} for i ≥ n to increase the number of indeterminates (which amounts to adding all monomials in new indeterminates obtained by symmetry from monomials already present).### A principle relating symmetric polynomials and symmetric functions

## Properties of the ring of symmetric functions

### Identities

### Structural properties of Λ_{R}

Important properties of Λ_{R} include the following.### Generating functions

_{k>0} p_{k}(X)t^{k−1}, and its expressions therefore lack a factor t with respect to those given here). The two final expressions, involving the formal derivatives of the generating functions E(t) and H(t), imply Newton's identities and their variants for the complete homogeneous symmetric functions. These expressions are sometimes written as
## See also

## References

One context in which symmetric polynomial functions occur is in the study of monic univariate polynomials of degree n having n roots in a given field. These n roots determine the polynomial, and when they are considered as independent variables, the coefficients of the polynomial are symmetric polynomial functions of the roots. Moreover the fundamental theorem of symmetric polynomials implies that a polynomial function f of the n roots can be expressed as (another) polynomial function of the coefficents of the polynomial determined by the roots if and only if f is given by a symmetric polynomial.

However, in algebra and in particular in algebraic combinatorics, the term "symmetric function" is often used instead to refer to elements of the ring of symmetric functions, where that ring 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 groups.

The study of symmetric functions is based on that of symmetric polynomials. In a polynomial ring in some finite set of indeterminates, there is an action by ring automorphisms of the symmetric group on (the indices of) the indeterminates (simultaneaously substituting each of them for another according to the permutation used). The invariants for this action form the subring of symmetric polynomials. If the indeterminates are X_{1},…,X_{n}, then examples of such symmetric polynomials are

- $X\_1+X\_2+cdots+X\_n,$

- $X\_1^3+X\_2^3+cdots+X\_n^3,$

and

- $X\_1X\_2cdots\; X\_n.$

A somewhat more complicated example is
X_{1}^{3}X_{2}X_{3} +X_{1}X_{2}^{3}X_{3} +X_{1}X_{2}X_{3}^{3} +X_{1}^{3}X_{2}X_{4} +X_{1}X_{2}^{3}X_{4} +X_{1}X_{2}X_{4}^{3} +…
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.

If P is a monic polynomial in t of degree n with n roots x_{1},…,x_{n}, that is

- $P=t^n+a\_\{n-1\}t^\{n-1\}+cdots+a\_2t^t+a\_1t+a\_0=(t-x\_1)(t-x\_2)cdots(t-x\_n),$

then there are relations expressing the coefficients a_{n−d} as symmetric functions of the roots, in fact as homogeneous polynomials of degree d:

- $begin\{align\}$

Using these relations, any expression involving only the coefficients a_{1},…,a_{n} can be rewritten as a symmetric function of the roots x_{1},…,x_{n}; in particular this holds for polynomial expressions. The fundamental theorem of symmetric polynomials implies that, less obviously, the converse also holds: any symmetric polynomial expression in the roots corresponds to a (unique) polynomial expression in the coefficients.

- $p\_3(X\_1,ldots,X\_n)=e\_1(X\_1,ldots,X\_n)^3-3e\_2(X\_1,ldots,X\_n)e\_1(X\_1,ldots,X\_n)+3e\_3(X\_1,ldots,X\_n),$

A ring of symmetric polynomials can be defined over any commutative ring R, and will be denoted Λ_{R}; the basic case is for R = Z. The ring Λ_{R} 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).

The easiest (though somewhat heavy) construction starts with the ring of formal power series RX_{1},X_{2},… over R in infinitely many indeterminates; one defines Λ_{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 > 1 in order to be symmetric. Unlike the whole power series ring, the subring Λ_{R} is graded by the total degree of monomials: due to condition 2, every element of Λ_{R} is a finite sum of homogeneous elements of Λ_{R} (which are themselves infinite sums of terms of equal degree). For every k ≥ 0, the element e_{k} ∈ Λ_{R} is defined as the formal sum of all products of k distinct indeterminates, which is clearly homogeneous of degree k.

Another construction of Λ_{R} takes somewhat longer to describe, but better indicates the relationship with the rings R[X_{1},…,X_{n}]^{Sn} of symmetric polynomials in n indeterminates. For every n there is a surjective ring homomorphism ρ_{n} from the analoguous ring R[X_{1},…,X_{n+1}]^{Sn+1} with one more indeterminate onto R[X_{1},…,X_{n}]^{Sn}, defined by setting the last indeterminate X_{n+1} to 0. Although ρ_{n} has a non-trivial kernel, the nonzero elements of that kernel have degree at least $n+1$ (they are multiples of X_{1}X_{2}…X_{n+1}). This means that the restriction of ρ_{n} to elements of degree at most n is a bijective linear map, and ρ_{n}(e_{k}(X_{1},…,X_{n+1})) = e_{k}(X_{1},…,X_{n}) for all k ≤ n. The inverse of this restriction can be extended uniquely to a ring homomorphism φ_{n} from R[X_{1},…,X_{n}]^{Sn} to R[X_{1},…,X_{n+1}]^{Sn+1}, as follows for instance from the fundamental theorem of symmetric polynomials. Since the images φ_{n}(e_{k}(X_{1},…,X_{n})) = e_{k}(X_{1},…,X_{n+1}) for k = 1,…,n are still algebraically independent over R, the homomorphism φ_{n} is injective and can be viewed as a (somewhat unusual) inclusion of rings. The ring Λ_{R} is then the "union" (direct limit) of all these rings subject to these inclusions. Since all φ_{n} are compatible with the grading by total degree of the rings involved, Λ_{R} obtains the structure of a graded ring.

This construction differs slightly from the one in (Macdonald, 1979). That construction only uses the surjective morphisms ρ_{n} without mentioning the injective morphisms φ_{n}: it constructs the homogeneous components of Λ_{R} separately, and equips their direct sum with a ring structure using the ρ_{n}. It is also observed that the result can be described as an inverse limit in the category of graded rings. That description however somewhat obscures an important property typical for a direct limit of injective morphisms, namely that every individual element (symmetric function) is already faithfully represented in some object used in the limit construction, here a ring R[X_{1},…,X_{d}]^{Sd}. It suffices to take for d the degree of the symmetric function, since the part in degree d is of that ring is mapped isomorphically to rings with more indeterminates by φ_{n} for all n ≥ d. This implies that for studying relations between individual elements, there is no fundamental difference between symmetric polynomials and symmetric functions.

It should be noted that the name "symmetric function" for elements of Λ_{R} is a misnomer: in neither construction the elements are functions, and in fact, unlike symmetric polynomials, no function of independent variables can be associated to such elements (for instance e_{1} would be the sum of all infinitely many variables, which is not defined unless restrictions are imposed on the variables). However the name is traditional and well established; it can be found both in (Macdonald, 1979), which says (footnote on p.12)

The elements of Λ (unlike those of Λ(here Λ_{n}) are no longer polynomials: they are formal infinite sums of monomials. We have therefore reverted to the older terminology of symmetric functions.

To define a symmetric function one must either indicate directly a power series as in the first construction, or give a symmetric polynomial in n indeterminates for every natural number n in a way compatible with the second construction. An expression in an unspecified number of indeterminates may do both, for instance

- $e\_2=sum\_\{i\}x\_ix\_j,\; math>$

The following are fundamental examples of symmetric functions.

- The monomial symmetric functions m
_{α}, determined by monomial X^{α}(where α = (α_{1},α_{2},…) is a sequence of natural numbers); m_{α}is the sum of all monomials obtained by symmetry from X^{α}. For a formal definition, consider such sequences to be infinite by appending zeroes (which does not alter the monomial), and define the relation "~" between such sequences that expresses that one is a permutation of the other; then

- $m\_alpha=sumnolimits\_\{betasimalpha\}X^beta.$

- This symmetric function corresponds to the monomial symmetric polynomial m
_{α}(X_{1},…,X_{n}) for any n large enough to have the monomial X^{α}. The distinct monomial symmetric functions are parametrized by the integer partitions (each m_{α}has a unique representative monomial X^{λ}with the parts λ_{i}in weakly decreasing order). Since any symmetric function containing any of the monomials of some m_{α}must contain all of them with the same coefficient, each symmetric function can be written as an R-linear combination of monomial symmetric functions, and the distinct monomial symmetric functions form a basis of Λ_{R}as R-module.

- The elementary symmetric functions e
_{k}, for any natural number k; one has e_{k}= m_{α}where $X^alpha=Pi\_\{i=1\}^kX\_i$. As a power series, this is the sum of all distinct products of k distinct indeterminates. This symmetric function corresponds to the elementary symmetric polynomial e_{k}(X_{1},…,X_{n}) for any n ≥ k. - The power sum symmetric functions p
_{k}, for any positive integer k; one has p_{k}= m_{(k)}, the monomial symmetric function for the monomial X_{1}^{k}. This symmetric function corresponds to the power sum symmetric polynomial p_{k}(X_{1},…,X_{n}) = X_{1}^{k}+…+X_{n}^{k}for any n ≥ 1. - The complete homogeneous symmetric functions h
_{k}, for any natural number k; h_{k}is the sum of all monomial symmetric functions m_{α}where α is a partition of k. As a power series, this is the sum of all monomials of degree k, which is what motivates its name. This symmetric function corresponds to the complete homogeneous symmetric polynomial h_{k}(X_{1},…,X_{n}) for any n ≥ k. - The Schur functions s
_{λ}for any partition λ, which corresponds to the Schur polynomial s_{λ}(X_{1},…,X_{n}) for any n large enough to have the monomial X^{λ}.

There is no power sum symmetric function p_{0}: although it is possible (and in some contexts natural) to define $p\_0(X\_1,ldots,X\_n)=Sigma\_\{i=1\}^nX\_i^0=n$ as a symmetric polynomial in n variables, these values are not compatible with the morphisms ρ_{n}. The "discriminant" $textstyle(prod\_\{i\}(x\_i-x\_j))^2\; math>\; is\; another\; example\; of\; an\; expression\; giving\; a\; symmetric\; polynomial\; for\; all$n, but not defining any symmetric function. The expressions defining Schur polynomials as a quotient of alternating polynomials are somewhat similar to that for the discriminant, but the polynomials s_{λ}(X_{1},…,X_{n}) turn out to be compatible for varying n, and therefore do define a symmetric function.

For any symmetric function P, the corresponding symmetric polynomials in n indeterminates for any natural number n may be designated by P(X_{1},…,X_{n}). The second definition of the ring of symmetric functions implies the following fundamental principle:

- If P and Q are symmetric functions of degree d, then one has the identity $P=Q$ of symmetric functions if and only one has the identity P(X
_{1},…,X_{d}) = Q(X_{1},…,X_{d}) of symmetric polynomials in d indeterminates. In this case one has in fact P(X_{1},…,X_{n}) = Q(X_{1},…,X_{n}) for any number n of indeterminates.

This is because one can always reduce the number of variables by substituting zero for some variables, and one can increase the number of variables by applying the homomorphisms φ_{n}; the definition of those homomorphisms assures that φ_{n}(P(X_{1},…,X_{n})) = P(X_{1},…,X_{n+1}) (and similarly for Q) whenever n ≥ d. See a proof of Newton's identities for an effective application of this principle.

The ring of symmetric functions is a convenient tool for writing identities between symmetric polynomials that are independent of the number of indeterminates: in Λ_{R} there is no such number, yet by the above principle any identity in Λ_{R} automatically gives identities the rings of symmetric polynomials over R in any number of indeterminates. Some fundamental identities are

- $sum\_\{i=0\}^k(-1)^ie\_ih\_\{k-i\}=0=sum\_\{i=0\}^k(-1)^ih\_ie\_\{k-i\}quadmbox\{for\; all\; \}k>0,$

- $ke\_k=sum\_\{i=1\}^k(-1)^\{i-1\}p\_ie\_\{k-i\}quadmbox\{for\; all\; \}kgeq0,$

- $kh\_k=sum\_\{i=1\}^kp\_ih\_\{k-i\}quadmbox\{for\; all\; \}kgeq0.$

- The set of monomial symmetric functions parametrized by partitions form a basis of Λ
_{R}as graded R-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 a involutary 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_{λt}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:

- The subring of Λ
_{R}generated by its elements of degree at most n is isomorphic to the ring of symmetric polynomials over R in n variables; - The Hilbert–Poincaré series of Λ
_{R}is $textstyleprod\_\{i=1\}^inftyfrac1\{1-t^i\}$, the generating function of the integer partitions (this also follows from property 1); - For every n > 0, the R-module formed by the homogeneous part of Λ
_{R}of degree n, modulo its intersection with the subring generated by its elements of degree strictly less than n, is free of rank 1, and (the image of) e_{n}is a generator of this R-module; - For every family of symmetric functions (f
_{i})_{i>0}in which f_{i}is homogeneous of degree i and gives a generator of the free R-module of the previous point (for all i), there is an alternative isomorphism of graded R-algebras from R[Y_{1},Y_{2},…] as above to Λ_{R}that sends Y_{i}to f_{i}; in other words, the family (f_{i})_{i>0}forms a set of free polynomial generators of Λ_{R}.

This final point applies in particular to the family (h_{i})_{i>0} of complete homogeneous symmetric functions.
If R contains the field Q of rational numbers, it applies also to the family (p_{i})_{i>0} of power sum symmetric functions. This explains why the first n elements of each of these families define sets of symmetric polynomials in n variables that are free polynomial generators of that ring of symmetric polynomials.

The fact that the complete homogeneous symmetric functions form a set of free polynomial generators of Λ_{R} already shows the existence of an automorphism ω sending the elementary symmetric functions to the complete homogeneous ones, as mentioned in property 3. The fact that ω is an involution of Λ_{R} follows from the symmetry between elementary and complete homogeneous symmetric functions expressed by the first set of relations given above.

The first definition of Λ_{R} as a subring of RX_{1},X_{2},… allows expression the generating functions of several sequences of symmetric functions to be elegantly expressed. Contrary to the relations mentioned earlier, which are internal to Λ_{R}, these expressions involve operations taking place in RX_{1},X_{2},…;t but outside its subring Λ_{R}t, 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

- $E(t)=sum\_\{kgeq0\}e\_k(X)t^k=prod\_\{i=1\}^infty(1+X\_it).$

- $H(t)=sum\_\{kgeq0\}h\_k(X)t^k=prod\_\{i=1\}^inftyleft(sum\_\{kgeq0\}(X\_it)^kright)=prod\_\{i=1\}^inftyfrac1\{1-X\_it\}.$

- $P(t)=sum\_\{k>0\}p\_k(X)t^k=sum\_\{k>0\}sum\_\{i=1\}^infty(X\_it)^k=sum\_\{i=1\}^inftyfrac\{X\_it\}\{1-X\_it\}=frac\{tE\text{'}(-t)\}\{E(-t)\}=frac\{tH\text{'}(t)\}\{H(t)\}$

- $P(t)=-tfrac\; d\{dt\}log(E(-t))=\; tfrac\; d\{dt\}log(H(t)),$

- Macdonald, I. G. Symmetric functions and Hall polynomials. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 1979. viii+180 pp. ISBN 0-19-853530-9
- Macdonald, I. G. Symmetric functions and Hall polynomials. Second edition. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1995. x+475 pp. ISBN 0-19-853489-2
- Stanley, Richard P. Enumerative Combinatorics, Vol. 2, Cambridge University Press, 1999. ISBN 0-521-56069-1 (hardback) ISBN 0-521-78987-7 (paperback).

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Last updated on Wednesday August 13, 2008 at 09:17:21 PDT (GMT -0700)

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