A symmetric polynomial on 
variables 
, ..., 
(also called a totally symmetric polynomial) is a polynomial
that is unchanged by any permutation of its variables.
In other words, the symmetric polynomials satisfy
 |
(1)
|
where 
and 
is an arbitrary permutation of the indices 1, 2,
..., 
.
For fixed 
and coefficients in a field, the symmetric polynomials
form an infinite-dimensional algebra generated freely by the 
elementary symmetric
polynomials. These generators are algebraically
independent; they are not a vector-space basis. They occur, with alternating
signs, as the coefficients of the monic polynomial
whose roots are 
,
..., 
.
There are four common homogeneous bases for the symmetric polynomials over 
, indexed by appropriate partitions 
(Dumitriu et al. 2007). Letting
be the length of 
,
the elementary symmetric polynomials 
, complete
homogeneous symmetric polynomials 
, and power-sum functions 
are defined for 
by
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
where 
,
..., 
are nonnegative integers, and for 
by
 |
(5)
|
where 
is one of 
,
or 
.
In addition, the monomial symmetric polynomials 
are defined as
 |
(6)
|
where 
is the set of permutations
giving distinct terms in the sum and 
is padded with zeros to length 
.
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 a symmetric polynomial, 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)
|
 |  |  |
(22)
|
so the first few values are
 |  |  |
(23)
|
 |  |  |
(24)
|
 |  |  |
(25)
|
 |  |  |
(26)
|
In general, 
can be computed from the determinant
 |
(27)
|
(Littlewood 1958, Cadogan 1971). In particular,
 |  |  |
(28)
|
 |  |  |
(29)
|
 |  |  |
(30)
|
 |  |  |
(31)
|
(Schroeppel 1972), as can be verified by plugging in and multiplying through.
