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A polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s0917001.png" />, with coefficients in a field or a commutative associative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s0917002.png" /> with a unit, which is a [[Symmetric function|symmetric function]] in its variables, that is, is invariant under all permutations of the variables:
+
A polynomial $f$ with coefficients in a field or a commutative associative ring $K$ with a unit, which is a [[Symmetric function|symmetric function]] in its variables, that is, is invariant under all permutations of the variables:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s0917003.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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\[
 +
\label{symm}
 +
f(x_1,\ldots,x_n) = f(\pi(x_1),\ldots,\pi(x_n)).
 +
\]
  
The symmetric polynomials form the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s0917004.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s0917005.png" />.
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The symmetric polynomials form the algebra $S(x_1,\ldots,x_n)$ over $K$.
  
 
The most important examples of symmetric polynomials are the elementary symmetric polynomials
 
The most important examples of symmetric polynomials are the elementary symmetric polynomials
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s0917006.png" /></td> </tr></table>
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\[
 +
s_k(x_1,\ldots,x_n) = \sum_{1 \leq i_1 < \ldots < i_k \leq n} x_{i_1} \ldots x_{i_k}
 +
\]
  
 
and the power sums
 
and the power sums
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s0917007.png" /></td> </tr></table>
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\[
 +
p_k(x_1,\ldots,x_n) = x_1^k + \ldots + x_n^k.
 +
\]
  
 
The latter can be expressed in terms of elementary symmetric polynomials by recurrence formulas, called Newton's formulas:
 
The latter can be expressed in terms of elementary symmetric polynomials by recurrence formulas, called Newton's formulas:
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The fundamental theorem on symmetric polynomials: Every symmetric polynomial is a polynomial in the elementary symmetric polynomials, and this representation is unique. In other words, the elementary symmetric polynomials are a set of free generators for the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170016.png" />. If the field has characteristic 0, then the polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170017.png" /> also form a set of free generators of this algebra.
 
The fundamental theorem on symmetric polynomials: Every symmetric polynomial is a polynomial in the elementary symmetric polynomials, and this representation is unique. In other words, the elementary symmetric polynomials are a set of free generators for the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170016.png" />. If the field has characteristic 0, then the polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170017.png" /> also form a set of free generators of this algebra.
  
A skew-symmetric, or alternating, polynomial is a polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170018.png" /> satisfying the relation (*) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170019.png" /> is even and the relation
+
A skew-symmetric, or alternating, polynomial is a polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170018.png" /> satisfying the relation (1) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170019.png" /> is even and the relation
  
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170020.png" /></td> </tr></table>
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170020.png" /></td> </tr></table>

Revision as of 10:10, 3 April 2015

A polynomial $f$ with coefficients in a field or a commutative associative ring $K$ with a unit, which is a symmetric function in its variables, that is, is invariant under all permutations of the variables:

\[ \label{symm} f(x_1,\ldots,x_n) = f(\pi(x_1),\ldots,\pi(x_n)). \]

The symmetric polynomials form the algebra $S(x_1,\ldots,x_n)$ over $K$.

The most important examples of symmetric polynomials are the elementary symmetric polynomials

\[ s_k(x_1,\ldots,x_n) = \sum_{1 \leq i_1 < \ldots < i_k \leq n} x_{i_1} \ldots x_{i_k} \]

and the power sums

\[ p_k(x_1,\ldots,x_n) = x_1^k + \ldots + x_n^k. \]

The latter can be expressed in terms of elementary symmetric polynomials by recurrence formulas, called Newton's formulas:

For the elementary symmetric polynomials () of the roots of an arbitrary polynomial in one variable with leading coefficient 1, , one has (see Viète theorem).

The fundamental theorem on symmetric polynomials: Every symmetric polynomial is a polynomial in the elementary symmetric polynomials, and this representation is unique. In other words, the elementary symmetric polynomials are a set of free generators for the algebra . If the field has characteristic 0, then the polynomials also form a set of free generators of this algebra.

A skew-symmetric, or alternating, polynomial is a polynomial satisfying the relation (1) if is even and the relation

if is odd. Any skew-symmetric polynomial can be written in the form , where is a symmetric polynomial and

This representation is not unique, in view of the relation .

References

[1] A.G. Kurosh, "Higher algebra" , MIR (1972) (Translated from Russian)
[2] A.I. Kostrikin, "Introduction to algebra" , Springer (1982) (Translated from Russian)
[3] A.P. Mishina, I.V. Proskuryakov, "Higher algebra. Linear algebra, polynomials, general algebra" , Pergamon (1965) (Translated from Russian)


Comments

Another important set of symmetric polynomials, which appear in the representations of the symmetric group, are the Schur polynomials (-functions) . These are defined for any partition , and include as special cases the above functions, e.g. , (see, e.g., [a4], Chapt. VI).

In general, the discriminant of the polynomial with roots is defined as , and satisfies

with .

See Discriminant.

Let be the alternating group, consisting of the even permutations. The ring of polynomials of polynomials over a field obviously contains the elementary symmetric functions and . If is not of characteristic , the ring of polynomials is generated by and , and the ideal of relations is generated by . The condition is also necessary for the statement that every skew-symmetric polynomial is of the form with symmetric. More precisely, what is needed for this is that implies for .

References

[a1] N. Jacobson, "Basic algebra" , 1 , Freeman (1974)
[a2] A.G. Kurosh, "An introduction to algebra" , MIR (1971) (Translated from Russian)
[a3] B.L. van der Waerden, "Algebra" , 1 , Springer (1967) (Translated from German)
[a4] D.E. Littlewood, "The theory of group characters and matrix representations of groups" , Clarendon Press (1950)
[a5] V. Poénaru, "Singularités en présence de symmétrie" , Springer (1976) pp. 14ff
[a6] P.M. Cohn, "Algebra" , 1 , Wiley (1982) pp. 181
How to Cite This Entry:
Symmetric polynomial. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Symmetric_polynomial&oldid=36386
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article