Ring of symmetric functions

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

${\displaystyle p_3(X_1,\dots ,X_n)=e_1(X_1,\dots ,X_n{)}^3-3e_2(X_1,\dots ,X_n)e_1(X_1,\dots ,X_n)+3e_3(X_1,\dots ,X_n),}$

where the ${\displaystyle e_i}$ denote elementary symmetric polynomials; this formula is valid for all natural numbers n, and the only notable dependency on it is that e_k(X₁,...,X_n) = 0 whenever n < k. One would like to write this as an identity

${\displaystyle p_3=e_1^3-3e_2{e}_1+3e_3}$

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 e_k for all integers k ≥ 1, and any element of the ring can be given by a polynomial expression in the elements e_k.

Definitions

A ring of symmetric functions 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).

As a ring of formal power series

The easiest (though somewhat heavy) construction starts with the ring of formal power series ${\displaystyle R[[X_1,X_2,...]]}$ over R in infinitely (countably) many indeterminates; the elements of this power series ring are formal infinite sums of terms, each of which consists of a coefficient from R multiplied by a monomial, where each monomial is a product of finitely many finite powers of indeterminates. One defines Λ_R as its subring consisting of those power series S that satisfy

1. S is invariant under any permutation of the indeterminates, and
2. 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 X₁ 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.

As an algebraic limit

Another construction of Λ_R takes somewhat longer to describe, but better indicates the relationship with the rings R[X₁,...,X_n]^S_n of symmetric polynomials in n indeterminates. For every n there is a surjective ring homomorphism ρ_n from the analogous ring R[X₁,...,X_n+1]^S_n+1 with one more indeterminate onto R[X₁,...,X_n]^S_n, 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 ${\displaystyle n+1}$ (they are multiples of X₁X₂...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(X₁,...,X_n+1)) = e_k(X₁,...,X_n) for all k ≤ n. The inverse of this restriction can be extended uniquely to a ring homomorphism φ_n from R[X₁,...,X_n]^S_n to R[X₁,...,X_n+1]^S_n+1, as follows for instance from the fundamental theorem of symmetric polynomials. Since the images φ_n(e_k(X₁,...,X_n)) = e_k(X₁,...,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; applying φ_n to a polynomial amounts to adding all monomials containing the new indeterminate obtained by symmetry from monomials already present. 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[X₁,...,X_d]^S_d. It suffices to take for d the degree of the symmetric function, since the part in degree d 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.

Defining individual symmetric functions

The name "symmetric function" for elements of Λ_R is a misnomer: in neither construction are the elements functions, and in fact, unlike symmetric polynomials, no function of independent variables can be associated to such elements (for instance e₁ 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_n) are no longer polynomials: they are formal infinite sums of monomials. We have therefore reverted to the older terminology of symmetric functions.

(here Λ_n denotes the ring of symmetric polynomials in n indeterminates), and also in (Stanley, 1999).

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

${\displaystyle e_2=\sum _{i<j}{X}_i{X}_j}$

can be taken as the definition of an elementary symmetric function if the number of indeterminates is infinite, or as the definition of an elementary symmetric polynomial in any finite number of indeterminates. Symmetric polynomials for the same symmetric function should be compatible with the homomorphisms ρ_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 ${\displaystyle \prod _{i=1}^n{X}_i}$; the family ${\displaystyle \prod _{i=1}^n(X_i+1)}$ fails only the second condition.) Any symmetric polynomial in n indeterminates can be used to construct a compatible family of symmetric polynomials, using the homomorphisms 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).

The following are fundamental examples of symmetric functions.

• The monomial symmetric functions m_α. Suppose α = (α₁,α₂,...) is a sequence of non-negative integers, only finitely many of which are non-zero. Then we can consider the monomial defined by α: X^α = X₁^α₁X₂^α₂X₃^α₃.... Then m_α is the symmetric function determined by X^α, i.e. the sum of all monomials obtained from X^α by symmetry. For a formal definition, define β ~ α to mean that the sequence β is a permutation of the sequence α and set

${\displaystyle m_{\alpha }={\sum }_{\beta \sim \alpha }{X}^{\beta }.}$

This symmetric function corresponds to the monomial symmetric polynomial m_α(X₁,...,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 therefore form a basis of Λ_R as an R-module.

• The elementary symmetric functions e_k, for any natural number k; one has e_k = m_α where ${\displaystyle X^{\alpha }=\prod _{i=1}^k{X}_i}$. 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(X₁,...,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 X₁^k. This symmetric function corresponds to the power sum symmetric polynomial p_k(X₁,...,X_n) = X₁^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(X₁,...,X_n) for any n ≥ k.
• The Schur functions s_λ for any partition λ, which corresponds to the Schur polynomial s_λ(X₁,...,X_n) for any n large enough to have the monomial X^λ.

There is no power sum symmetric function p₀: although it is possible (and in some contexts natural) to define ${\displaystyle p_0(X_1,\dots ,X_n)=\sum _{i=1}^n{X}_i^0=n}$ as a symmetric polynomial in n variables, these values are not compatible with the morphisms ρ_n. The "discriminant" ${\displaystyle (\prod _{i<j}(X_i-X_j){)}^2}$ 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(X₁,...,X_n) turn out to be compatible for varying n, and therefore do define a symmetric function.

A principle relating symmetric polynomials and symmetric functions

For any symmetric function P, the corresponding symmetric polynomials in n indeterminates for any natural number n may be designated by P(X₁,...,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 ${\displaystyle P=Q}$ of symmetric functions if and only if one has the identity P(X₁,...,X_d) = Q(X₁,...,X_d) of symmetric polynomials in d indeterminates. In this case one has in fact P(X₁,...,X_n) = Q(X₁,...,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(X₁,...,X_n)) = P(X₁,...,X_n+1) (and similarly for Q) whenever n ≥ d. See a proof of Newton's identities for an effective application of this principle.

Properties of the ring of symmetric functions

Identities

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

${\displaystyle \sum _{i=0}^k(-1{)}^i{e}_i{h}_{k-i}=0=\sum _{i=0}^k(-1{)}^i{h}_i{e}_{k-i}\text{for all }k>0,}$

which shows a symmetry between elementary and complete homogeneous symmetric functions; these relations are explained under complete homogeneous symmetric polynomial.

${\displaystyle k{e}_k=\sum _{i=1}^k(-1{)}^{i-1}{p}_i{e}_{k-i}\text{for all }k\ge 0,}$

the Newton identities, which also have a variant for complete homogeneous symmetric functions:

${\displaystyle k{h}_k=\sum _{i=1}^k{p}_i{h}_{k-i}\text{for all }k\ge 0.}$

Structural properties of ΛR

Important properties of Λ_R include the following.

1. The set of monomial symmetric functions parametrized by partitions form a basis of Λ_R as a 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).
2. Λ_R is isomorphic as a graded R-algebra to a polynomial ring R[Y₁,Y₂, ...] 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.
3. There is an involutory 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⁻¹p_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:

• 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 ${\displaystyle \prod _{i=1}^{\mathrm{\infty }}\frac{1}{1-t^i}}$, 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[Y₁,Y₂, ...] 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 ${\displaystyle \mathbb{Q}}$ 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 ring of symmetric functions Λ_Z is the Exp ring of the integers Z. It is also a lambda-ring in a natural fashion; in fact it is the universal lambda-ring in one generator.

Generating functions

The first definition of Λ_R as a subring of ${\displaystyle R[[X_1,X_2,...]]}$ allows 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 R[[X₁,X₂,...;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

${\displaystyle E(t)=\sum _{k\ge 0}{e}_k(X)t^k=\prod _{i=1}^{\mathrm{\infty }}(1+X_{i}t).}$

Similarly one has for complete homogeneous symmetric functions

${\displaystyle H(t)=\sum _{k\ge 0}{h}_k(X)t^k=\prod _{i=1}^{\mathrm{\infty }}\left(\sum _{k\ge 0}(X_{i}t{)}^k\right)=\prod _{i=1}^{\mathrm{\infty }}\frac{1}{1-X_{i}t}.}$

The obvious fact that ${\displaystyle E(-t)H(t)=1=E(t)H(-t)}$ explains the symmetry between elementary and complete homogeneous symmetric functions. The generating function for the power sum symmetric functions can be expressed as

${\displaystyle P(t)=\sum _{k>0}{p}_k(X)t^k=\sum _{k>0}\sum _{i=1}^{\mathrm{\infty }}(X_{i}t{)}^k=\sum _{i=1}^{\mathrm{\infty }}\frac{X_{i}t}{1-X_{i}t}=\frac{t{E}^{\prime }(-t)}{E(-t)}=\frac{t{H}^{\prime }(t)}{H(t)}}$

((Macdonald, 1979) defines P(t) as Σ_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

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

which amounts to the same, but requires that R contain the rational numbers, so that the logarithm of power series with constant term 1 is defined (by ${\displaystyle \log (1-tS)=-\sum _{i>0}\frac{1}{i}(tS{)}^i}$).

Specializations

Let ${\displaystyle \mathrm{\Lambda }}$ be the ring of symmetric functions and ${\displaystyle R}$ a commutative algebra with unit element. An algebra homomorphism ${\displaystyle \phi :\mathrm{\Lambda }\to R,f\mapsto f(\phi )}$ is called a specialization.

Example:

• Given some real numbers ${\displaystyle a_1,\dots ,a_k}$ and ${\displaystyle f(x_1,x_2,\dots ,)\in \mathrm{\Lambda }}$, then the substitution ${\displaystyle x_1=a_1,\dots ,x_k=a_k}$ and ${\displaystyle x_j=0,\mathrm{\forall }j>k}$ is a specialization.
• Let ${\displaystyle f\in \mathrm{\Lambda }}$, then ${\displaystyle \mathrm{ps}(f):=f(1,q,q^2,q^3,\dots )}$ is called principal specialization.