Symmetric Polynomial

A symmetric polynomial on  variables , ...,  (also called a totally symmetric polynomial) is a function that is unchanged by any permutation of its variables. In other words, the symmetric polynomials satisfy

(1)

where  and  being an arbitrary permutation of the indices 1, 2, ..., .

For fixed , the set of all symmetric polynomials in  variables forms an algebra of dimension . The coefficients of a univariate polynomial  of degree  are algebraically independent symmetric polynomials in the roots of , and thus form a basis for the set of all such symmetric polynomials.

There are four common homogeneous bases for the symmetric polynomials, each of which is indexed by a partition (Dumitriu et al. 2004). Letting  be the length of , the elementary functions , complete homogeneous functions , and power-sum functions  are defined for  by

			(2)
			(3)
			(4)

and for  by

(5)

where  is one of ,  or . In addition, the monomial functions  are defined as

(6)

where  is the set of permutations giving distinct terms in the sum and  is considered to be infinite.

As several different abbreviations and conventions are in common use, care must be taken when determining which symmetric polynomial is in use.

The elementary symmetric polynomials  (sometimes denoted  or ) on  variables  are defined by

			(7)
			(8)
			(9)
			(10)
			(11)
			(12)

The th elementary symmetric polynomial is implemented in the Wolfram Language as SymmetricPolynomial[k, x1, ..., xn]. SymmetricReduction[f, x1, ..., xn] gives a pair of polynomials  in , ...,  where  is the symmetric part and  is the remainder.

Alternatively,  can be defined as the coefficient of  in the generating function

(13)

For example, on four variables , ..., , the elementary symmetric polynomials are

			(14)
			(15)
			(16)
			(17)

The power sum  is defined by

(18)

The relationship between  and , ...,  is given by the so-called Newton-Girard formulas. The related function  with arguments given by the elementary symmetric polynomials (not ) is defined by

			(19)
			(20)

It turns out that  is given by the coefficients of the generating function

(21)

so the first few values are

			(22)
			(23)
			(24)
			(25)

In general,  can be computed from the determinant

(26)

(Littlewood 1958, Cadogan 1971). In particular,

			(27)
			(28)
			(29)
			(30)

(Schroeppel 1972), as can be verified by plugging in and multiplying through.

REFERENCES:

Borwein, P. and Erdélyi, T. Polynomials and Polynomial Inequalities. New York: Springer-Verlag, p. 5, 1995.

Cadogan, C. C. "The Möbius Function and Connected Graphs." J. Combin. Th. B 11, 193-200, 1971.

Dumitriu, I.; Edelman, A.; and Shuman, G. "MOPS: Multivariate Orthogonal Polynomials (Symbolically)." Preprint. March 26, 2004.

Littlewood, J. E. A University Algebra, 2nd ed. London: Heinemann, 1958.

Schroeppel, R. Item 6 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 4, Feb. 1972. http://www.inwap.com/pdp10/hbaker/hakmem/geometry.html#item6.

Séroul, R. "Newton-Girard Formulas." §10.12 in Programming for Mathematicians. Berlin: Springer-Verlag, pp. 278-279, 2000.