Elementary symmetric polynomial

In mathematics, specifically in commutative algebra, theelementary symmetric polynomials are one type of ba-sic building block for symmetric polynomials, in the sensethat any symmetric polynomial can be expressed as apolynomial in elementary symmetric polynomials. Thatis, any symmetric polynomial P is given by an expressioninvolving only additions and multiplication of constantsand elementary symmetric polynomials. There is one el-ementary symmetric polynomial of degree d in n vari-ables for each nonnegative integer d n, and it is formedby adding together all distinct products of d distinct vari-ables.

1 Definition

The elementary symmetric polynomials inn variablesX1,, Xn, written ek(X1, , Xn) for k = 0, 1, ..., n, aredefined by

e0(X1, X2, . . . , Xn) = 1,

e1(X1, X2, . . . , Xn) =

1jn Xj ,

e2(X1, X2, . . . , Xn) =

1j

2 4 THE FUNDAMENTAL THEOREM OF SYMMETRIC POLYNOMIALS

That is, when we substitute numerical values for the vari-ables X1, X2, . . . , Xn , we obtain the monic univariatepolynomial (with variable ) whose roots are the valuessubstituted for X1, X2, . . . , Xn and whose coefficientsare the elementary symmetric polynomials.The characteristic polynomial of a linear operator is anexample of this. The roots are the eigenvalues of theoperator. When we substitute these eigenvalues into theelementary symmetric polynomials, we obtain the coef-ficients of the characteristic polynomial, which are nu-merical invariants of the operator. This fact is usefulin linear algebra and its applications and generalizations,like tensor algebra and disciplines which extensively em-ploy tensor fields, such as differential geometry.The set of elementary symmetric polynomials in n vari-ables generates the ring of symmetric polynomials in nvariables. More specifically, the ring of symmetric poly-nomials with integer coefficients equals the integral poly-nomial ring Z[e1(X1, . . . , Xn), . . . , en(X1, . . . , Xn)].(See below for a more general statement and proof.) Thisfact is one of the foundations of invariant theory. Forother systems of symmetric polynomials with a simi-lar property see power sum symmetric polynomials andcomplete homogeneous symmetric polynomials.

4 The fundamental theorem ofsymmetric polynomials

For any commutative ringA denote the ring of symmetricpolynomials in the variablesX1, . . . , Xn with coefficientsin A by A[X1, . . . , Xn]Sn .

A[X1, . . . , Xn]Sn is a polynomial ring in

the n elementary symmetric polynomialsek(X1, . . . , Xn) for k = 1, ..., n.

(Note that e0 is not among these polynomials; since e0 =1 , it cannot be member of any set of algebraically inde-pendent elements.)This means that every symmetric polynomialP (X1, . . . , Xn) A[X1, . . . , Xn]Sn has a uniquerepresentation

P (X1, . . . , Xn) = Q(e1(X1, . . . , Xn), . . . , en(X1, . . . , Xn))

for some polynomial Q A[Y1, . . . , Yn] . Another wayof saying the same thing is that A[X1, . . . , Xn]Sn is iso-morphic to the polynomial ring A[Y1, . . . , Yn] throughan isomorphism that sends Yk to ek(X1, . . . , Xn) fork = 1, . . . , n .

4.1 Proof sketch

The theoremmay be proved for symmetric homogeneouspolynomials by a double mathematical induction with re-spect to the number of variables n and, for fixed n, withrespect to the degree of the homogeneous polynomial.The general case then follows by splitting an arbitrarysymmetric polynomial into its homogeneous components(which are again symmetric).In the case n = 1 the result is obvious because every poly-nomial in one variable is automatically symmetric.Assume now that the theorem has been proved for allpolynomials form < n variables and all symmetric poly-nomials in n variables with degree < d. Every homo-geneous symmetric polynomial P in A[X1, . . . , Xn]Sncan be decomposed as a sum of homogeneous symmetricpolynomials

P (X1, . . . , Xn) = Placunary(X1, . . . , Xn)+X1 XnQ(X1, . . . , Xn).

Here the lacunary part Placunary is defined as the sumof all monomials in P which contain only a proper sub-set of the n variables X1, ..., Xn, i.e., where at least onevariable Xj is missing.Because P is symmetric, the lacunary part is determinedby its terms containing only the variables X1, ..., Xn,i.e., which do not contain Xn. These are precisely theterms that survive the operation of setting Xn to 0, sotheir sum equals P (X1, . . . , Xn1, 0) , which is a sym-metric polynomial in the variables X1, ..., Xn that weshall denote by P (X1, . . . , Xn1) . By the inductive as-sumption, this polynomial can be written as

P (X1, . . . , Xn1) = Q(1,n1, . . . , n1,n1)

for some Q . Here the doubly indexed j,n1 denote theelementary symmetric polynomials in n1 variables.Consider now the polynomial

R(X1, . . . , Xn) := Q(1,n, . . . , n1,n) .

Then R(X1, . . . , Xn) is a symmetric polynomial in X1,..., Xn, of the same degree as Placunary , which satisfies

R(X1, . . . , Xn1, 0) = Q(1,n1, . . . , n1,n1) = P (X1, . . . , Xn1, 0)

(the first equality holds because setting Xn to 0 in j,ngives j,n1 , for all j < n ), in other words, the lacu-nary part of R coincides with that of the original poly-nomial P. Therefore the difference PR has no lacunarypart, and is therefore divisible by the product X1 Xnof all variables, which equals the elementary symmetric

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3

polynomial n,n . Then writing P R = n,n Q , thequotient Q is a homogeneous symmetric polynomial ofdegree less than d (in fact degree at most d n) which bythe inductive assumption can be expressed as a polyno-mial in the elementary symmetric functions. Combiningthe representations for PR and R one finds a polynomialrepresentation for P.The uniqueness of the representation can be proved in-ductively in a similar way. (It is equivalent to the factthat the n polynomials e1, . . . , en are algebraically inde-pendent over the ring A.) The fact that the polynomialrepresentation is unique implies that A[X1, . . . , Xn]Snis isomorphic to A[Y1, . . . , Yn] .

4.2 An alternative proof

The following proof is also inductive, but does not involveother polynomials than those symmetric in X1,...,Xn, andalso leads to a fairly direct procedure to effectively writea symmetric polynomial as a polynomial in the elemen-tary symmetric ones. Assume the symmetric polynomialto be homogeneous of degree d; different homogeneouscomponents can be decomposed separately. Order themonomials in the variables Xi lexicographically, wherethe individual variables are ordered X1 > > Xn, inother words the dominant term of a polynomial is onewith the highest occurring power of X1, and among thosethe one with the highest power of X2, etc. Furthermoreparametrize all products of elementary symmetric poly-nomials that have degree d (they are in fact homogeneous)as follows by partitions of d. Order the individual elemen-tary symmetric polynomials ei(X1,,Xn) in the productso that those with larger indices i come first, then build foreach such factor a column of i boxes, and arrange thosecolumns from left to right to form a Young diagram con-taining d boxes in all. The shape of this diagram is a par-tition of d, and each partition of d arises for exactly oneproduct of elementary symmetric polynomials, which weshall denote by e (X1,,Xn) (the t is present only be-cause traditionally this product is associated to the trans-pose partition of ). The essential ingredient of the proofis the following simple property, which uses multi-indexnotation for monomials in the variables Xi.Lemma. The leading term of e (X1,,Xn) is X.

Proof. The leading term of the product is theproduct of the leading terms of each factor(this is true whenever one uses a monomial or-der, like the lexicographic order used here),and the leading term of the factor ei(X1,,Xn)is clearly X1X2Xi. To count the occur-rences of the individual variables in the result-ing monomial, fill the column of the Young di-agram corresponding to the factor concernedwith the numbers 1,i of the variables, thenall boxes in the first row contain 1, those in the

second row 2, and so forth, which means theleading term is X.

Now one proves by induction on the leading monomial inlexicographic order, that any nonzero homogeneous sym-metric polynomial P of degree d can be written as poly-nomial in the elementary symmetric polynomials. SinceP is symmetric, its leading monomial has weakly decreas-ing exponents, so it is someX with a partition of d. Letthe coefficient of this term be c, then P ce (X1,,Xn)is either zero or a symmetric polynomial with a strictlysmaller leading monomial. Writing this difference in-ductively as a polynomial in the elementary symmetricpolynomials, and adding back ce (X1,,Xn) to it, oneobtains the sought for polynomial expression for P.The fact that this expression is unique, or equivalently thatall the products (monomials) e (X1,,Xn) of elemen-tary symmetric polynomials are linearly independent, isalso easily proved. The lemma shows that all these prod-ucts have different leading monomials, and this suffices:if a nontrivial linear combination of the e (X1,,Xn)were zero, one focusses on the contribution in the lin-ear combination with nonzero coefficient and with (aspolynomial in the variables Xi) the largest leading mono-mial; the leading term of this contribution cannot be can-celled by any other contribution of the linear combina-tion, which gives a contradiction.

5 See also Symmetric polynomial

Complete homogeneous symmetric polynomial

Schur polynomial

Newtons identities

MacMahon Master theorem

Symmetric function

Representation theory

6 References Macdonald, I.G. (1995), Symmetric Functions andHall Polynomials, second ed. Oxford: ClarendonPress. ISBN 0-19-850450-0 (paperback, 1998).

Richard P. Stanley (1999), Enumerative Combina-torics, Vol. 2. Cambridge: Cambridge UniversityPress. ISBN 0-521-56069-1

https://en.wikipedia.org/wiki/Algebraically_independenthttps://en.wikipedia.org/wiki/Algebraically_independenthttps://en.wikipedia.org/wiki/Monomialhttps://en.wikipedia.org/wiki/Lexicographic_orderhttps://en.wikipedia.org/wiki/Integer_partitionhttps://en.wikipedia.org/wiki/Young_diagramhttps://en.wikipedia.org/wiki/Monomial#Notationhttps://en.wikipedia.org/wiki/Monomial#Notationhttps://en.wikipedia.org/wiki/Monomial_orderhttps://en.wikipedia.org/wiki/Monomial_orderhttps://en.wikipedia.org/wiki/Symmetric_polynomialhttps://en.wikipedia.org/wiki/Complete_homogeneous_symmetric_polynomialhttps://en.wikipedia.org/wiki/Schur_polynomialhttps://en.wikipedia.org/wiki/Newton%2527s_identitieshttps://en.wikipedia.org/wiki/MacMahon_Master_theoremhttps://en.wikipedia.org/wiki/Symmetric_functionhttps://en.wikipedia.org/wiki/Representation_theoryhttps://en.wikipedia.org/wiki/I._G._Macdonaldhttps://en.wikipedia.org/wiki/Special:BookSources/0198504500https://en.wikipedia.org/wiki/Richard_P._Stanleyhttps://en.wikipedia.org/wiki/Special:BookSources/0521560691

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DefinitionExamplesPropertiesThe fundamental theorem of symmetric polynomials Proof sketch An alternative proof

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