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Symmetric Polynomial


A symmetric polynomial on n variables x_1, ..., x_n (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

 f(y_1,y_2,...,y_n)=f(x_1,x_2,...,x_n),
(1)

where y_i=x_(pi(i)) and pi being an arbitrary permutation of the indices 1, 2, ..., n.

For fixed n, the set of all symmetric polynomials in n variables forms an algebra of dimension n. The coefficients of a univariate polynomial f(x) of degree n are algebraically independent symmetric polynomials in the roots of f, 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 lambda (Dumitriu et al. 2004). Letting l be the length of lambda, the elementary functions e_lambda, complete homogeneous functions h_lambda, and power-sum functions p_lambda are defined for l=1 by

e_(lambda_1)=sum_(j_1<j_2<...<j_(lambda_1))x_(j_1)...x_(j_(lambda_1))
(2)
h_(lambda_1)=sum_(m_1+...+m_n=lambda_1)product_(j=1)^(n)x^(m_j)
(3)
p_(lambda_1)=sum_(j=1)^(n)x^(lambda_1),
(4)

and for l>1 by

 s_lambda=product_(i=1)^ls_(lambda_i)
(5)

where s is one of e, h or p. In addition, the monomial functions m_lambda are defined as

 m_lambda=sum_(sigma in S_lambda)x_(sigma(1))^(lambda_1)x_(sigma(2))^(lambda_2)...x_(sigma(m))^(lambda_m),
(6)

where S_lambda is the set of permutations giving distinct terms in the sum and lambda 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 Pi_k(x_1,...,x_n) (sometimes denoted sigma_k or e_lambda) on n variables {x_1,...,x_n} are defined by

Pi_1(x_1,...,x_n)=sum_(1<=i<=n)x_i
(7)
Pi_2(x_1,...,x_n)=sum_(1<=i<j<=n)x_ix_j
(8)
Pi_3(x_1,...,x_n)=sum_(1<=i<j<k<=n)x_ix_jx_k
(9)
Pi_4(x_1,...,x_n)=sum_(1<=i<j<k<l<=n)x_ix_jx_kx_l
(10)
|
(11)
Pi_n(x_1,...,x_n)=product_(1<=i<=n)x_i.
(12)

The kth elementary symmetric polynomial is implemented in the Wolfram Language as SymmetricPolynomial[k, {x1, ..., xn}]. SymmetricReduction[f, {x1, ..., xn}] gives a pair of polynomials {p,q} in x_1, ..., x_n where p is the symmetric part and q is the remainder.

Alternatively, Pi_j(x_1,...,x_n) can be defined as the coefficient of x^(n-j) in the generating function

 product_(1<=i<=n)(x+x_i).
(13)

For example, on four variables x_1, ..., x_4, the elementary symmetric polynomials are

Pi_1(x_1,x_2,x_3,x_4)=x_1+x_2+x_3+x_4
(14)
Pi_2(x_1,x_2,x_3,x_4)=x_1x_2+x_1x_3+x_1x_4+x_2x_3+x_2x_4+x_3x_4
(15)
Pi_3(x_1,x_2,x_3,x_4)=x_1x_2x_3+x_1x_2x_4+x_1x_3x_4+x_2x_3x_4
(16)
Pi_4(x_1,x_2,x_3,x_4)=x_1x_2x_3x_4.
(17)

The power sum S_p(x_1,...,x_n) is defined by

 S_p(x_1,...,x_n)=sum_(k=1)^nx_k^p.
(18)

The relationship between S_p and Pi_1, ..., Pi_p is given by the so-called Newton-Girard formulas. The related function s_p(Pi_1,...,Pi_n) with arguments given by the elementary symmetric polynomials (not x_n) is defined by

s_p(Pi_1,...,Pi_n)=(-1)^(p-1)S_p(x_1,...,x_n)
(19)
=(-1)^(p-1)sum_(k=1)^(n)x_k^p.
(20)

It turns out that s_p(Pi_1,...,Pi_n) is given by the coefficients of the generating function

 ln(1+Pi_1t+Pi_2t^2+Pi_3t^3+...)=sum_(k=1)^infty(s_k)/kt^k 
=Pi_1t+1/2(-Pi_1^2+2Pi_2)t^2+1/3(Pi_1^3-3Pi_1Pi_2+3Pi_3)t^3+...,
(21)

so the first few values are

s_1=Pi_1
(22)
s_2=-Pi_1^2+2Pi_2
(23)
s_3=Pi_1^3-3Pi_1Pi_2+3Pi_3
(24)
s_4=-Pi_1^4+4Pi_1^2Pi_2-2Pi_2^2-4Pi_1Pi_3+4Pi_4.
(25)

In general, s_p can be computed from the determinant

 s_p=(-1)^(p-1)|Pi_1 1 0 0 ... 0; 2Pi_2 Pi_1 1 0 ... 0; 3Pi_3 Pi_2 Pi_1 1 ... 0; 4Pi_4 Pi_3 Pi_2 Pi_1 ... 0; | | | | ... 1; pPi_p Pi_(p-1) Pi_(p-2) Pi_(p-3) ... Pi_1|
(26)

(Littlewood 1958, Cadogan 1971). In particular,

S_1(x_1,...,x_n)=sum_(k=1)^(n)x_k=Pi_1
(27)
S_2(x_1,...,x_n)=Pi_1^2-2Pi_2
(28)
S_3(x_1,...,x_n)=Pi_1^3-3Pi_1Pi_2+3Pi_3
(29)
S_4(x_1,...,x_n)=Pi_1^4-4Pi_1^2Pi_2+2Pi_2^2+4Pi_1Pi_3-4Pi_4
(30)

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


See also

Fundamental Theorem of Symmetric Functions, Jack Polynomial, Zonal Polynomial, Multivariate Hermite Polynomial, Multivariate Jacobi Polynomial, Multivariate Laguerre Polynomial, Multivariate Orthogonal Polynomials, Newton-Girard Formulas, Orthogonal Polynomials, Power Sum, Symmetric Function, Vieta's Formulas

Portions of this entry contributed by David Terr

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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.

Referenced on Wolfram|Alpha

Symmetric Polynomial

Cite this as:

Terr, David and Weisstein, Eric W. "Symmetric Polynomial." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SymmetricPolynomial.html

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