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.
