Ring of symmetric functions: Difference between revisions

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In [[algebra]] and in particular in [[algebraic combinatorics]], the '''ring of symmetric functions''' is a specific limit of the rings of [[symmetric polynomial]]s in ''n'' indeterminates, as ''n'' goes to infinity. This [[ring (mathematics)|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 algebra|graded]] [[Hopf algebra]] that is both commutative and cocommutative.

== Symmetric polynomials ==

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{{main | Symmetric polynomial }}

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 [[Group action (mathematics)|action]] by [[ring homomorphism|ring automorphism]]s of the [[symmetric group]] ''S_n'' 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 [[Invariant (mathematics)#Unchanged under group action|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

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

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where the <math>e_i</math> denote elementary symmetric polynomials; this formula is valid for all natural numbers ''n'', and the only notable dependency on it is that ''e''_''k''(''X''₁,...,''X''_''n'')&nbsp;=&nbsp;0 whenever ''n''&nbsp;<&nbsp;''k''. One would like to write this as an identity

:<math>p_3=e_1^3-3e_2 e_1 + 3e_3</math>

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''&nbsp;≥&nbsp;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''&nbsp;=&nbsp;'''Z'''. The ring Λ_''R'' is in fact a [[graded ring|graded]] ''R''-[[Algebra over a ring|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 ====

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

Another construction of Λ_''R'' takes somewhat longer to describe, but better indicates the relationship with the rings ''R''[''X''₁,...,''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''[''X''₁,...,''X''_''n''+1]^{'''S'''_''n''+1} with one more indeterminate onto ''R''[''X''₁,...,''X''_''n'']^{'''S'''_''n''}, defined by setting the last indeterminate ''X''_''n''+1 to&nbsp;0. Although ''ρ''_''n'' has a non-trivial kernel, the nonzero elements of that kernel have degree at least <math>n+1</math> (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))&nbsp;=&nbsp;''e''_''k''(''X''₁,...,''X''_''n'') for all ''k''&nbsp;≤&nbsp;''n''. The inverse of this restriction can be extended uniquely to a ring homomorphism ''φ''_''n'' from ''R''[''X''₁,...,''X''_''n'']^{'''S'''_''n''} to ''R''[''X''₁,...,''X''_''n''+1]^{'''S'''_''n''+1}, as follows for instance from the [[fundamental theorem of symmetric polynomials]]. Since the images ''φ''_''n''(''e''_''k''(''X''₁,...,''X''_''n''))&nbsp;=&nbsp;''e''_''k''(''X''₁,...,''X''_''n''+1) for ''k''&nbsp;=&nbsp;1,...,''n'' are still [[algebraically independent]] over&nbsp;''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''[''X''₁,...,''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''&nbsp;≥&nbsp;''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 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.&nbsp;12)

<blockquote>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.</blockquote>

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

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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

:<math>e_2=\sum_{i<j}X_iX_j\,</math>

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 morphisms ''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 <math>\textstyle\prod_{i=1}^nX_i</math>; the family <math>\textstyle\prod_{i=1}^n(X_i+1)</math> 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''&nbsp;<&nbsp;''n'' to decrease the number of indeterminates, and ''φ''_''i'' for ''i''&nbsp;≥&nbsp;''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.

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* 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 <math>\textstyle p_0(X_1,\ldots,X_n)=\sum_{i=1}^nX_i^0=n</math> as a symmetric ''polynomial'' in ''n'' variables, these values are not compatible with the morphisms ''ρ''_''n''. The "discriminant" <math>\textstyle(\prod_{i<j}(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 polynomial]]s 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.

=== A principle relating symmetric polynomials and symmetric functions ===

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:If ''P'' and ''Q'' are symmetric functions of degree ''d'', then one has the identity <math>P=Q</math> of symmetric functions if and only one has the identity ''P''(''X''₁,...,''X''_''d'')&nbsp;=&nbsp;''Q''(''X''₁,...,''X''_''d'') of symmetric polynomials in ''d'' indeterminates. In this case one has in fact ''P''(''X''₁,...,''X''_''n'')&nbsp;=&nbsp;''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''))&nbsp;=&nbsp;''P''(''X''₁,...,''X''_''n''+1) (and similarly for ''Q'') whenever ''n''&nbsp;≥&nbsp;''d''. See [[Newton's identities#Derivation of the identities|a proof of Newton's identities]] for an effective application of this principle.

== Properties of the ring of symmetric functions ==