Algebra/Chapter 10/Symmetric Polynomials/Permutations and Symmetry

Definition 1 A permutation is a bijective function from a set to itself. Let be a finite set. The function is called a permutation if and only if it is one-to-one and onto. Meaning, for all there exists a unique such that . The set of all permutations of the elements of is denoted by . Example For there are different permutations: In general, if then . Definition 2 Let be a polynomial. Let us define: Properties Let be polynomials. Then we have: such that . Proof By definition, the permutation is applied on the variable indexes only. First, let be monomials of the form We can generalize by induction for , such that are monomials. Same as before, let be monomials of the form Again, We can generalize by induction for , such that are monomials: By definition we get: Definition 3 Let be a polynomial. Then it is called symmetric if for all permutations . Examples A symmetric polynomial: A non-symmetric polynomial: Properties The sum and product of symmetric polynomials is a symmetric polynomial. Let be a polynomial in variables , and let be symmetric polynomials in variables . Then is also symmetric in variables . Proof This follows from the properties in definition 2 and the symmetric polynomial definition above. By definition we get: Permutations and Symmetry Monomial Ordering Elementary Symmetric Polynomials