Alternating polynomial
Encyclopedia
In algebra, an alternating polynomial is a polynomial
such that if one switches any two of the variables, the polynomial changes sign:
Equivalently, if one permutes the variables, the polynomial changes in value by the sign of the permutation:
More generally, a polynomial
is said to be alternating in 
if it changes sign if one switches any two of the 
, leaving the 
fixed.
and alternating polynomials (in the same variables
) behave thus:
This is exactly the addition table for parity, with "symmetric" corresponding to "even" and "alternating" corresponding to "odd". Thus, the direct sum of the spaces of symmetric and alternating polynomials forms a superalgebra
(a
-graded algebra
), where the symmetric polynomials are the even part, and the alternating polynomials are the odd part.
This grading is unrelated to the grading of polynomials by degree
.
In particular, alternating polynomials form a module
over the algebra of symmetric polynomials (the odd part of a superalgebra is a module over the even part); in fact it is a free module of rank 1, with as generator the Vandermonde polynomial in n variables.
If the characteristic of the coefficient ring
is 2, there is no difference between the two concepts: the alternating polynomials are precisely the symmetric polynomials.
This is clearly alternating, as switching two variables changes the sign of one term and does not change the others.
The alternating polynomials are exactly the Vandermonde polynomial times a symmetric polynomial:
where 
is symmetric.
This is because:
Conversely, the ratio of two alternating polynomials is a symmetric function, possibly rational (not necessarily a polynomial), though the ratio of an alternating polynomial over the Vandermonde polynomial is a polynomial.
Schur polynomial
s are defined in this way, as an alternating polynomial divided by the Vandermonde polynomial.
, or more precisely 
, where 
is a symmetric polynomial, the discriminant
.
That is, the ring of symmetric and alternating polynomials is a quadratic extension of the ring of symmetric polynomials, where one has adjoined a square root of the discriminant.
Alternatively, it is:
If 2 is not invertible, the situation is somewhat different, and one must use a different polynomial
, and obtains a different relation; see Romagny.
, the symmetric and alternating polynomials are subrepresentations of the action of the symmetric group
on n letters on the polynomial ring in n variables. (Formally, the symmetric group acts on n letters, and thus acts on derived objects, particularly free object
s on n letters, such as the ring of polynomials.)
The symmetric group has two 1-dimensional representations: the trivial representation and the sign representation. The symmetric polynomials are the trivial representation, and the alternating polynomials are the sign representation. Formally, the scalar span of any symmetric (resp., alternating) polynomial is a trivial (resp., sign) representation of the symmetric group, and multiplying the polynomials tensors the representations.
In characteristic 2, these are not distinct representations, and the analysis is more complicated.
If
, there are also other subrepresentations of the action of the symmetric group on the ring of polynomials, as discussed in representation theory of the symmetric group
.
): the ring of symmetric polynomials in n variables can be obtained from the ring of symmetric polynomials in arbitrarily many variables by evaluating all variables above
to zero: symmetric polynomials are thus stable or compatibly defined. However, this is not the case for alternating polynomials, in particular the Vandermonde polynomial.
, and its square (the discriminant) corresponds to the top Pontryagin class
. This is formalized in the splitting principle
, which connects characteristic classes to polynomials.
From the point of view of stable homotopy theory
, the fact that the Euler class
is an unstable class corresponds to the fact that alternating polynomials (and the Vandermonde polynomial in particular) are unstable.
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
such that if one switches any two of the variables, the polynomial changes sign:Equivalently, if one permutes the variables, the polynomial changes in value by the sign of the permutation:
More generally, a polynomial
is said to be alternating in if it changes sign if one switches any two of the , leaving the fixed.Relation to symmetric polynomials
Products of symmetricSymmetric polynomial
In mathematics, a symmetric polynomial is a polynomial P in n variables, such that if any of the variables are interchanged, one obtains the same polynomial...
and alternating polynomials (in the same variables
) behave thus:
- the product of two symmetric polynomials is symmetric,
- the product of a symmetric polynomial and an alternating polynomial is alternating, and
- the product of two alternating polynomials is symmetric.
This is exactly the addition table for parity, with "symmetric" corresponding to "even" and "alternating" corresponding to "odd". Thus, the direct sum of the spaces of symmetric and alternating polynomials forms a superalgebra
Superalgebra
In mathematics and theoretical physics, a superalgebra is a Z2-graded algebra. That is, it is an algebra over a commutative ring or field with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading....
(a
-graded algebraGraded algebra
In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field with an extra piece of structure, known as a gradation ....
), where the symmetric polynomials are the even part, and the alternating polynomials are the odd part.
This grading is unrelated to the grading of polynomials by degree
Degree of a polynomial
The degree of a polynomial represents the highest degree of a polynominal's terms , should the polynomial be expressed in canonical form . The degree of an individual term is the sum of the exponents acting on the term's variables...
.
In particular, alternating polynomials form a module
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
over the algebra of symmetric polynomials (the odd part of a superalgebra is a module over the even part); in fact it is a free module of rank 1, with as generator the Vandermonde polynomial in n variables.
If the characteristic of the coefficient ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
is 2, there is no difference between the two concepts: the alternating polynomials are precisely the symmetric polynomials.
Vandermonde polynomial
The basic alternating polynomial is the Vandermonde polynomial:This is clearly alternating, as switching two variables changes the sign of one term and does not change the others.
The alternating polynomials are exactly the Vandermonde polynomial times a symmetric polynomial:
where is symmetric.This is because:
-
is a factor of every alternating polynomial: 
is a factor of every alternating polynomial, as if 
, the polynomial is zero (since switching them does not change the polynomial, we get
- so
is a factor), and thus 
is a factor.
- an alternating polynomial times a symmetric polynomial is an alternating polynomial; thus all multiples of
are alternating polynomials
- an alternating polynomial times a symmetric polynomial is an alternating polynomial; thus all multiples of
Conversely, the ratio of two alternating polynomials is a symmetric function, possibly rational (not necessarily a polynomial), though the ratio of an alternating polynomial over the Vandermonde polynomial is a polynomial.
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 are defined in this way, as an alternating polynomial divided by the Vandermonde polynomial.
Ring structure
Thus, denoting the ring of symmetric polynomials by Λn, the ring of symmetric and alternating polynomials is, or more precisely , where is a symmetric polynomial, the discriminantDiscriminant
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....
.
That is, the ring of symmetric and alternating polynomials is a quadratic extension of the ring of symmetric polynomials, where one has adjoined a square root of the discriminant.
Alternatively, it is:
If 2 is not invertible, the situation is somewhat different, and one must use a different polynomial
, and obtains a different relation; see Romagny.Representation theory
From the perspective of 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...
, the symmetric and alternating polynomials are subrepresentations of the action of the symmetric group
Representation theory of the symmetric group
In mathematics, the representation theory of the symmetric group is a particular case of the representation theory of finite groups, for which a concrete and detailed theory can be obtained. This has a large area of potential applications, from symmetric function theory to problems of quantum...
on n letters on the polynomial ring in n variables. (Formally, the symmetric group acts on n letters, and thus acts on derived objects, particularly free object
Free object
In mathematics, the idea of a free object is one of the basic concepts of abstract algebra. It is a part of universal algebra, in the sense that it relates to all types of algebraic structure . It also has a formulation in terms of category theory, although this is in yet more abstract terms....
s on n letters, such as the ring of polynomials.)
The symmetric group has two 1-dimensional representations: the trivial representation and the sign representation. The symmetric polynomials are the trivial representation, and the alternating polynomials are the sign representation. Formally, the scalar span of any symmetric (resp., alternating) polynomial is a trivial (resp., sign) representation of the symmetric group, and multiplying the polynomials tensors the representations.
In characteristic 2, these are not distinct representations, and the analysis is more complicated.
If
, there are also other subrepresentations of the action of the symmetric group on the ring of polynomials, as discussed in representation theory of the symmetric groupRepresentation theory of the symmetric group
In mathematics, the representation theory of the symmetric group is a particular case of the representation theory of finite groups, for which a concrete and detailed theory can be obtained. This has a large area of potential applications, from symmetric function theory to problems of quantum...
.
Unstable
Alternating polynomials are an unstable phenomenon (in the language of stable homotopy theoryStable homotopy theory
In mathematics, stable homotopy theory is that part of homotopy theory concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor...
): the ring of symmetric polynomials in n variables can be obtained from the ring of symmetric polynomials in arbitrarily many variables by evaluating all variables above
to zero: symmetric polynomials are thus stable or compatibly defined. However, this is not the case for alternating polynomials, in particular the Vandermonde polynomial.Characteristic classes
In characteristic classes, the Vandermonde polynomial corresponds to the Euler classEuler class
In mathematics, specifically in algebraic topology, the Euler class, named after Leonhard Euler, is a characteristic class of oriented, real vector bundles. Like other characteristic classes, it measures how "twisted" the vector bundle is...
, and its square (the discriminant) corresponds to the top Pontryagin class
Pontryagin class
In mathematics, the Pontryagin classes are certain characteristic classes. The Pontryagin class lies in cohomology groups with degree a multiple of four...
. This is formalized in the splitting principle
Splitting principle
In mathematics, the splitting principle is a technique used to reduce questions about vector bundles to the case of line bundles.In the theory of vector bundles, one often wishes to simplify computations, say of Chern classes. Often computations are well understood for line bundles and for direct...
, which connects characteristic classes to polynomials.
From the point of view of stable homotopy theory
Stable homotopy theory
In mathematics, stable homotopy theory is that part of homotopy theory concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor...
, the fact that the Euler class
Euler class
In mathematics, specifically in algebraic topology, the Euler class, named after Leonhard Euler, is a characteristic class of oriented, real vector bundles. Like other characteristic classes, it measures how "twisted" the vector bundle is...
is an unstable class corresponds to the fact that alternating polynomials (and the Vandermonde polynomial in particular) are unstable.