Symmetric polynomial
Encyclopedia
In mathematics
, a symmetric polynomial is a polynomial
P(X1, X2, …, Xn) in n variables, such that if any of the variables are interchanged, one obtains the same polynomial. Formally, P is a symmetric polynomial, if for any permutation
σ of the subscripts 1, 2, ..., n one has P(Xσ(1), Xσ(2), …, Xσ(n)) = P(X1, X2, …, Xn).
Symmetric polynomials arise naturally in the study of the relation between the roots of a polynomial in one variable and its coefficients, since the coefficients can be given by polynomial expressions in the roots, and all roots play a similar role in this setting. From this point of view the elementary symmetric polynomial
s are the most fundamental symmetric polynomials. A theorem states that any symmetric polynomial can be expressed in terms of elementary symmetric polynomials, which implies that every symmetric polynomial expression
in the roots of a monic polynomial can alternatively be given as a polynomial expression in the coefficients of the polynomial.
Symmetric polynomials also form an interesting structure by themselves, independently of any relation to the roots of a polynomial. In this context other collections of specific symmetric polynomials, such as complete homogeneous
, power sum
, and Schur polynomial
s play important roles alongside the elementary ones. The resulting structures, and in particular the ring of symmetric functions, are of great importance in combinatorics
and in representation theory
.
and in three variables X1, X2, X3 one has for instance
There are many ways to make specific symmetric polynomials in any number of variables, see the various types below. An example of a somewhat different flavor is
where first a polynomial is constructed that changes sign under every exchange of variables, and taking the square renders it completely symmetric (if the variables represent the roots of a monic polynomial, this polynomial gives its discriminant
).
On the other hand the polynomial in two variables
is not symmetric, since if one exchanges
and 
one gets a different polynomial, 
. Similarly in three variables
has only symmetry under cyclic permutations of the three variables, which is not sufficient to be a symmetric polynomial.
polynomials of degree n having n roots in a given field
. These n roots determine the polynomial, and when they are considered as independent variables, the coefficients of the polynomial are symmetric polynomial functions of the roots. Moreover the fundamental theorem of symmetric polynomials implies that a polynomial function f of the n roots can be expressed as (another) polynomial function of the coefficients of the polynomial determined by the roots if and only if
f is given by a symmetric polynomial.
This yields the approach to solving polynomial equations in terms of inverting this map, "breaking" the symmetry – given the coefficients of the polynomial (the elementary symmetric polynomial
s in the roots), how can one recover the roots?
This leads to studying solutions of polynomials in terms of the permutation group
of the roots, originally in the form of Lagrange resolvents, later developed in Galois theory
.
with coefficients ai in some field k. There exist n roots x1,…,xn of P in some possibly larger field (for instance if k is the field of real number
s, the roots will exist in the field of complex number
s); some of the roots might be equal, but the fact that one has all roots is expressed by the relation
By comparison of the coefficients one finds that
These are in fact just instances of Viète's formulas. They show that all coefficients of the polynomial are given in terms of the roots by a symmetric polynomial expression
: although for a given polynomial P there may be qualitative differences between the roots (like lying in the base field k or not, being simple or multiple roots), none of this affects the way the roots occur in these expressions.
Now one may change the point of view, by taking the roots rather than the coefficients as basic parameters for describing P, and considering them as indeterminates rather than as constants in an appropriate field; the coefficients ai then become just the particular symmetric polynomials given by the above equations. Those polynomials, without the sign
, are known as the elementary symmetric polynomial
s in x1,…,xn. A basic fact, known as the fundamental theorem of symmetric polynomials states that any symmetric polynomial in n variables can be given by a polynomial expression in terms of these elementary symmetric polynomials. It follows that any symmetric polynomial expression in the roots of a monic polynomial can be expressed as a polynomial in the coefficients of the polynomial, and in particular that its value lies in the base field k that contains those coefficients. Thus, when working only with such symmetric polynomial expressions in the roots, it is unnecessary to know anything particular about those roots, or to compute in any larger field than k in which those roots may lie. In fact the values of the roots themselves become rather irrelevant, and the necessary relations between coefficients and symmetric polynomial expressions can be found by computations in terms of symmetric polynomials only. An example of such relations are Newton's identities
, which express the sum of any fixed power of the roots in terms of the elementary symmetric polynomials.
For example, for n = 2, the relevant elementary symmetric polynomials are e1(X1, X2) = X1+X2, and e2(X1, X2) = X1X2. The first polynomial in the list of examples above can then be written as
(for a proof that this is always possible see the fundamental theorem of symmetric polynomials).
Powers and products of elementary symmetric polynomials work out to rather complicated expressions. If one seeks basic additive building blocks for symmetric polynomials, a more natural choice is to take those symmetric polynomials that contain only one type of monomial, with only those copies required to obtain symmetry. Any monomial
in X1, …, Xn can be written as X1α1…Xnαn where the exponents αi are natural numbers (possibly zero); writing α = (α1,…,αn) this can be abbreviated to Xα. The monomial symmetric polynomial mα(X1, …, Xn) is defined as the sum of all monomials xβ where β ranges over all distinct permutations of (α1,…,αn). For instance one has
,
Clearly mα = mβ when β is a permutation of α, so one usually considers only those mα for which α1 ≥ α2 ≥ … ≥ αn, in other words for which α is a partition
.
These monomial symmetric polynomials form a vector space basis: every symmetric polynomial P can be written as a linear combination
of the monomial symmetric polynomials; to do this it suffices to separate the different types of monomials occurring in P. In particular if P has integer coefficients, then so will the linear combination.
The elementary symmetric polynomials are particular cases of monomial symmetric polynomials: for 0 ≤ k ≤ n one has
where α is the partition of k into k parts 1 (followed by n − k zeros).
All symmetric polynomials can be obtained from the first n power sum symmetric polynomials by additions and multiplications, possibly involving rational coefficients. More precisely,
In particular, the remaining power sum polynomials pk(X1, …, Xn) for k > n can be so expressed in the first n power sum polynomials; for example
In contrast to the situation for the elementary and complete homogeneous polynomials, a symmetric polynomial in n variables with integral coefficients need not be a polynomial function with integral coefficients of the power sum symmetric polynomials.
For an example, for n = 2, the symmetric polynomial
has the expression
Using three variables one gets a different expression
The corresponding expression was valid for two variables as well (it suffices to set X3 to zero), but since it involves p3, it could not be used to illustrate the statement for n = 2. The example shows that whether or not the expression for a given monomial symmetric polynomial in terms of the first n power sum polynomials involves rational coefficients may depend on n. But rational coefficients are always needed to express elementary symmetric polynomials (except the constant ones, and e1 which coincides with the first power sum) in terms of power sum polynomials. The Newton identities provide an explicit method to do this; it involves division by integers up to n, which explains the rational coefficients. Because of these divisions, the mentioned statement fails in general when coefficients are taken in a field of finite characteristic
; however it is valid with coefficients in any ring containing the rational numbers.
s of degree k in the variables X1, …, Xn. For instance
The polynomial hk(X1, …, Xn) is also the sum of all distinct monomial symmetric polynomials of degree k in X1, …, Xn, for instance for the given example
All symmetric polynomials in these variables can be built up from complete homogeneous ones: any symmetric polynomial in X1, …, Xn can be obtained from the complete homogeneous symmetric polynomials h1(X1, …, Xn), …, hn(X1, …, Xn) via multiplications and additions. More precisely:
For example, for
, the relevant complete homogeneous symmetric polynomials are h1(X1,X2) = X1+X2), and h2(X1,X2) = X12+X1X2+X22. The first polynomial in the list of examples above can then be written as
Like in the case of power sums, the given statement applies in particular to the complete homogeneous symmetric polynomials beyond hn(X1, …, Xn), allowing them to be expressed in the ones up to that point; again the resulting identities become invalid when the number of variables is increased.
An important aspect of complete homogeneous symmetric polynomials is their relation to elementary symmetric polynomials, which can be given as the identities
, for all k > 0, and any number of variables n.
Since e0(X1, …, Xn) and h0(X1, …, Xn) are both equal to 1, one can isolate either the first or the last terms of these summations; the former gives a set of equations that allows to recursively express the successive complete homogeneous symmetric polynomials in terms of the elementary symmetric polynomials, and the latter gives a set of equations that allows doing the inverse. This implicitly shows that any symmetric polynomial can be expressed in terms of the hk(X1, …, Xn) with 1 ≤ k ≤ n: one first expresses the symmetric polynomial in terms of the elementary symmetric polynomials, and then expresses those in terms of the mentioned complete homogeneous ones.
. They are however not as easy to describe as the other kinds of special symmetric polynomials; see the main article for details.
, representation theory
, and Galois theory
. They are also important in combinatorics
, where they are mostly studied through the ring of symmetric functions, which avoids having to carry around a fixed number of variables all the time.
These are all products of the Vandermonde polynomial and a symmetric polynomial, and form a quadratic extension of the ring of symmetric polynomials: the Vandermonde polynomial is a square root of the discriminant.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a symmetric polynomial is a polynomial
Polynomial ring
In mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more variables with coefficients in another ring. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of...
P(X1, X2, …, Xn) in n variables, such that if any of the variables are interchanged, one obtains the same polynomial. Formally, P is a symmetric polynomial, if for any permutation
Permutation
In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order...
σ of the subscripts 1, 2, ..., n one has P(Xσ(1), Xσ(2), …, Xσ(n)) = P(X1, X2, …, Xn).
Symmetric polynomials arise naturally in the study of the relation between the roots of a polynomial in one variable and its coefficients, since the coefficients can be given by polynomial expressions in the roots, and all roots play a similar role in this setting. From this point of view the elementary symmetric polynomial
Elementary symmetric polynomial
In mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial P can be expressed as a polynomial in elementary symmetric polynomials: P can be given by an...
s are the most fundamental symmetric polynomials. A theorem states that any symmetric polynomial can be expressed in terms of elementary symmetric polynomials, which implies that every symmetric polynomial expression
Polynomial expression
In mathematics, and in particular in the field of algebra, a polynomial expression in one or more given entities E1, E2, ..., is any meaningful expression constructed from copies of those entities together with constants, using the operations of addition and multiplication...
in the roots of a monic polynomial can alternatively be given as a polynomial expression in the coefficients of the polynomial.
Symmetric polynomials also form an interesting structure by themselves, independently of any relation to the roots of a polynomial. In this context other collections of specific symmetric polynomials, such as complete homogeneous
Complete homogeneous symmetric polynomial
In mathematics, specifically in algebraic combinatorics and commutative algebra, the complete homogeneous symmetric polynomials are a specific kind of symmetric polynomials...
, power sum
Power sum symmetric polynomial
In mathematics, specifically in commutative algebra, the power sum symmetric polynomials are a type of basic building block for symmetric polynomials, in the sense that every symmetric polynomial with rational coefficients can be expressed as a sum and difference of products of power sum symmetric...
, and Schur polynomial
Schur polynomial
In mathematics, Schur polynomials, named after Issai Schur, are certain symmetric polynomials in n variables, indexed by partitions, that generalize the elementary symmetric polynomials and the complete homogeneous symmetric polynomials. In representation theory they are the characters of...
s play important roles alongside the elementary ones. The resulting structures, and in particular the ring of symmetric functions, are of great importance in combinatorics
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
and in representation theory
Representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
.
Examples
In two variables X1, X2 one has symmetric polynomials like
and in three variables X1, X2, X3 one has for instance
There are many ways to make specific symmetric polynomials in any number of variables, see the various types below. An example of a somewhat different flavor is
where first a polynomial is constructed that changes sign under every exchange of variables, and taking the square renders it completely symmetric (if the variables represent the roots of a monic polynomial, this polynomial gives its discriminant
Discriminant
In algebra, the discriminant of a polynomial is an expression which gives information about the nature of the polynomial's roots. For example, the discriminant of the quadratic polynomialax^2+bx+c\,is\Delta = \,b^2-4ac....
).
On the other hand the polynomial in two variables
is not symmetric, since if one exchanges
and one gets a different polynomial, . Similarly in three variables
has only symmetry under cyclic permutations of the three variables, which is not sufficient to be a symmetric polynomial.
Galois theory
One context in which symmetric polynomial functions occur is in the study of monic univariateUnivariate
In mathematics, univariate refers to an expression, equation, function or polynomial of only one variable. Objects of any of these types but involving more than one variable may be called multivariate...
polynomials of degree n having n roots in a given field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
. These n roots determine the polynomial, and when they are considered as independent variables, the coefficients of the polynomial are symmetric polynomial functions of the roots. Moreover the fundamental theorem of symmetric polynomials implies that a polynomial function f of the n roots can be expressed as (another) polynomial function of the coefficients of the polynomial determined by the roots if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
f is given by a symmetric polynomial.
This yields the approach to solving polynomial equations in terms of inverting this map, "breaking" the symmetry – given the coefficients of the polynomial (the elementary symmetric polynomial
Elementary symmetric polynomial
In mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial P can be expressed as a polynomial in elementary symmetric polynomials: P can be given by an...
s in the roots), how can one recover the roots?
This leads to studying solutions of polynomials in terms of the permutation group
Permutation group
In mathematics, a permutation group is a group G whose elements are permutations of a given set M, and whose group operation is the composition of permutations in G ; the relationship is often written as...
of the roots, originally in the form of Lagrange resolvents, later developed in Galois theory
Galois theory
In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...
.
Relation with the roots of a monic univariate polynomial
Consider a monic polynomial in t of degree nwith coefficients ai in some field k. There exist n roots x1,…,xn of P in some possibly larger field (for instance if k is the field of real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s, the roots will exist in the field of complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s); some of the roots might be equal, but the fact that one has all roots is expressed by the relation
By comparison of the coefficients one finds that
These are in fact just instances of Viète's formulas. They show that all coefficients of the polynomial are given in terms of the roots by a symmetric polynomial expression
Polynomial expression
In mathematics, and in particular in the field of algebra, a polynomial expression in one or more given entities E1, E2, ..., is any meaningful expression constructed from copies of those entities together with constants, using the operations of addition and multiplication...
: although for a given polynomial P there may be qualitative differences between the roots (like lying in the base field k or not, being simple or multiple roots), none of this affects the way the roots occur in these expressions.
Now one may change the point of view, by taking the roots rather than the coefficients as basic parameters for describing P, and considering them as indeterminates rather than as constants in an appropriate field; the coefficients ai then become just the particular symmetric polynomials given by the above equations. Those polynomials, without the sign
, are known as the elementary symmetric polynomialElementary symmetric polynomial
In mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial P can be expressed as a polynomial in elementary symmetric polynomials: P can be given by an...
s in x1,…,xn. A basic fact, known as the fundamental theorem of symmetric polynomials states that any symmetric polynomial in n variables can be given by a polynomial expression in terms of these elementary symmetric polynomials. It follows that any symmetric polynomial expression in the roots of a monic polynomial can be expressed as a polynomial in the coefficients of the polynomial, and in particular that its value lies in the base field k that contains those coefficients. Thus, when working only with such symmetric polynomial expressions in the roots, it is unnecessary to know anything particular about those roots, or to compute in any larger field than k in which those roots may lie. In fact the values of the roots themselves become rather irrelevant, and the necessary relations between coefficients and symmetric polynomial expressions can be found by computations in terms of symmetric polynomials only. An example of such relations are Newton's identities
Newton's identities
In mathematics, Newton's identities, also known as the Newton–Girard formulae, give relations between two types of symmetric polynomials, namely between power sums and elementary symmetric polynomials...
, which express the sum of any fixed power of the roots in terms of the elementary symmetric polynomials.
Special kinds of symmetric polynomials
There are a few types of symmetric polynomials in the variables X1, X2, …, Xn that are fundamental.Elementary symmetric polynomials
For each nonnegative integer k, the elementary symmetric polynomial ek(X1, …, Xn) is the sum of all distinct products of k distinct variables (some authors denote it by σk instead). For k = 0 there is only the empty product so e0(X1, …, Xn) = 1, while for k > n, no products at all can be formed, so ek(X1, X2, …, Xn) = 0 in these cases. The remaining n elementary symmetric polynomials are building blocks for all symmetric polynomials in these variables: as mentioned above, any symmetric polynomial in the variables considered can be obtained from these elementary symmetric polynomials using multiplications and additions only. In fact one has the following more detailed facts:- any symmetric polynomial P in X1, …, Xn can be written as a polynomial expressionPolynomial expressionIn mathematics, and in particular in the field of algebra, a polynomial expression in one or more given entities E1, E2, ..., is any meaningful expression constructed from copies of those entities together with constants, using the operations of addition and multiplication...
in the polynomials ek(X1, …, Xn) with 1 ≤ k ≤ n; - this expression is unique up to equivalence of polynomial expressions;
- if P has integral coefficients, then the polynomial expression also has integral coefficients.
For example, for n = 2, the relevant elementary symmetric polynomials are e1(X1, X2) = X1+X2, and e2(X1, X2) = X1X2. The first polynomial in the list of examples above can then be written as
(for a proof that this is always possible see the fundamental theorem of symmetric polynomials).
Monomial symmetric polynomials
Powers and products of elementary symmetric polynomials work out to rather complicated expressions. If one seeks basic additive building blocks for symmetric polynomials, a more natural choice is to take those symmetric polynomials that contain only one type of monomial, with only those copies required to obtain symmetry. Any monomial
Monomial
In mathematics, in the context of polynomials, the word monomial can have one of two different meanings:*The first is a product of powers of variables, or formally any value obtained by finitely many multiplications of a variable. If only a single variable x is considered, this means that any...
in X1, …, Xn can be written as X1α1…Xnαn where the exponents αi are natural numbers (possibly zero); writing α = (α1,…,αn) this can be abbreviated to Xα. The monomial symmetric polynomial mα(X1, …, Xn) is defined as the sum of all monomials xβ where β ranges over all distinct permutations of (α1,…,αn). For instance one has
,Clearly mα = mβ when β is a permutation of α, so one usually considers only those mα for which α1 ≥ α2 ≥ … ≥ αn, in other words for which α is a partition
Partition (number theory)
In number theory and combinatorics, a partition of a positive integer n, also called an integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered to be the same partition; if order matters then the sum becomes a...
.
These monomial symmetric polynomials form a vector space basis: every symmetric polynomial P can be written as a linear combination
Linear combination
In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results...
of the monomial symmetric polynomials; to do this it suffices to separate the different types of monomials occurring in P. In particular if P has integer coefficients, then so will the linear combination.
The elementary symmetric polynomials are particular cases of monomial symmetric polynomials: for 0 ≤ k ≤ n one has
where α is the partition of k into k parts 1 (followed by n − k zeros).Power-sum symmetric polynomials
For each integer k ≥ 1, the monomial symmetric polynomial m(k,0,…,0)(X1, …, Xn) is of special interest, and called the power sum symmetric polynomial pk(X1, …, Xn), soAll symmetric polynomials can be obtained from the first n power sum symmetric polynomials by additions and multiplications, possibly involving rational coefficients. More precisely,
- Any symmetric polynomial in X1, …, Xn can be expressed as a polynomial expression with rational coefficients in the power sum symmetric polynomials p1(X1, …, Xn), …, pn(X1, …, Xn).
In particular, the remaining power sum polynomials pk(X1, …, Xn) for k > n can be so expressed in the first n power sum polynomials; for example
In contrast to the situation for the elementary and complete homogeneous polynomials, a symmetric polynomial in n variables with integral coefficients need not be a polynomial function with integral coefficients of the power sum symmetric polynomials.
For an example, for n = 2, the symmetric polynomial
has the expression
Using three variables one gets a different expression
The corresponding expression was valid for two variables as well (it suffices to set X3 to zero), but since it involves p3, it could not be used to illustrate the statement for n = 2. The example shows that whether or not the expression for a given monomial symmetric polynomial in terms of the first n power sum polynomials involves rational coefficients may depend on n. But rational coefficients are always needed to express elementary symmetric polynomials (except the constant ones, and e1 which coincides with the first power sum) in terms of power sum polynomials. The Newton identities provide an explicit method to do this; it involves division by integers up to n, which explains the rational coefficients. Because of these divisions, the mentioned statement fails in general when coefficients are taken in a field of finite characteristic
Characteristic (algebra)
In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...
; however it is valid with coefficients in any ring containing the rational numbers.
Complete homogeneous symmetric polynomials
For each nonnegative integer k, the complete homogeneous symmetric polynomial hk(X1, …, Xn) is the sum of all distinct monomialMonomial
In mathematics, in the context of polynomials, the word monomial can have one of two different meanings:*The first is a product of powers of variables, or formally any value obtained by finitely many multiplications of a variable. If only a single variable x is considered, this means that any...
s of degree k in the variables X1, …, Xn. For instance
The polynomial hk(X1, …, Xn) is also the sum of all distinct monomial symmetric polynomials of degree k in X1, …, Xn, for instance for the given example
All symmetric polynomials in these variables can be built up from complete homogeneous ones: any symmetric polynomial in X1, …, Xn can be obtained from the complete homogeneous symmetric polynomials h1(X1, …, Xn), …, hn(X1, …, Xn) via multiplications and additions. More precisely:
- Any symmetric polynomial P in X1, …, Xn can be written as a polynomial expression in the polynomials hk(X1, …, Xn) with 1 ≤ k ≤ n.
- If P has integral coefficients, then the polynomial expression also has integral coefficients.
For example, for
, the relevant complete homogeneous symmetric polynomials are h1(X1,X2) = X1+X2), and h2(X1,X2) = X12+X1X2+X22. The first polynomial in the list of examples above can then be written asLike in the case of power sums, the given statement applies in particular to the complete homogeneous symmetric polynomials beyond hn(X1, …, Xn), allowing them to be expressed in the ones up to that point; again the resulting identities become invalid when the number of variables is increased.
An important aspect of complete homogeneous symmetric polynomials is their relation to elementary symmetric polynomials, which can be given as the identities
, for all k > 0, and any number of variables n.Since e0(X1, …, Xn) and h0(X1, …, Xn) are both equal to 1, one can isolate either the first or the last terms of these summations; the former gives a set of equations that allows to recursively express the successive complete homogeneous symmetric polynomials in terms of the elementary symmetric polynomials, and the latter gives a set of equations that allows doing the inverse. This implicitly shows that any symmetric polynomial can be expressed in terms of the hk(X1, …, Xn) with 1 ≤ k ≤ n: one first expresses the symmetric polynomial in terms of the elementary symmetric polynomials, and then expresses those in terms of the mentioned complete homogeneous ones.
Schur polynomials
Another class of symmetric polynomials is that of the Schur polynomials, which are of fundamental importance in the applications of symmetric polynomials to representation theoryRepresentation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
. They are however not as easy to describe as the other kinds of special symmetric polynomials; see the main article for details.
Symmetric polynomials in algebra
Symmetric polynomials are important to linear algebraLinear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
, representation theory
Representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
, and Galois theory
Galois theory
In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...
. They are also important in combinatorics
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
, where they are mostly studied through the ring of symmetric functions, which avoids having to carry around a fixed number of variables all the time.
Alternating polynomials
Analogous to symmetric polynomials are alternating polynomials: polynomials that, rather than being invariant under permutation of the entries, change according to the sign of the permutation.These are all products of the Vandermonde polynomial and a symmetric polynomial, and form a quadratic extension of the ring of symmetric polynomials: the Vandermonde polynomial is a square root of the discriminant.