In algebra, an alternating polynomial is a
polynomial
In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
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
symmetric
Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
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
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.
T ...
(a
-
graded algebra
In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that . The index set is usually the set of nonnegative integers or the set of integers, ...
), 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 the
Vandermonde polynomial in ''n'' variables as generator.
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.
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.
[Rather, it only rearranges the other terms: for , switching and changes to , and exchanges with , but does not change their sign.]
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
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 polynomials are defined in this way, as an alternating polynomial divided by the Vandermonde polynomial.
Ring structure
Thus, denoting the ring of symmetric polynomials by Λ
''n'', the ring of symmetric and alternating polynomials is