Alternating polynomial

In algebra, an alternating polynomial is a polynomial f ( x 1 , … , x n ) {displaystyle f(x_{1},dots ,x_{n})} such that if one switches any two of the variables, the polynomial changes sign: In algebra, an alternating polynomial is a polynomial f ( x 1 , … , x n ) {displaystyle f(x_{1},dots ,x_{n})} 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 f ( x 1 , … , x n , y 1 , … , y t ) {displaystyle f(x_{1},dots ,x_{n},y_{1},dots ,y_{t})} is said to be alternating in x 1 , … , x n {displaystyle x_{1},dots ,x_{n}} if it changes sign if one switches any two of the x i {displaystyle x_{i}} , leaving the y j {displaystyle y_{j}} fixed. Products of symmetric and alternating polynomials (in the same variables x 1 , … , x n {displaystyle x_{1},dots ,x_{n}} ) 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 Z 2 {displaystyle mathbf {Z} _{2}} -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. 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.