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One context in which symmetric polynomial functions occur is in the study of monic univariate 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.
The cycle index polynomial of a permutation group is the average of the cycle index monomials of its elements. The phrase cycle indicator is also sometimes used in place of cycle index . Knowing the cycle index polynomial of a permutation group, one can enumerate equivalence classes due to the group 's action .
For every symmetric group other than S 6, there is no other conjugacy class consisting of elements of order 2 that has the same number of elements as the class of transpositions. Or as follows: Each permutation of order two (called an involution ) is a product of k > 0 disjoint transpositions, so that it has cyclic structure 2 k 1 n −2 k .
Aside from polynomial functions, tensors that act as functions of several vectors can be symmetric, and in fact the space of symmetric -tensors on a vector space is isomorphic to the space of homogeneous polynomials of degree on . Symmetric functions should not be confused with even and odd functions, which have a different sort of symmetry.
Galois' theory originated in the study of symmetric functions – the coefficients of a monic polynomial are (up to sign) the elementary symmetric polynomials in the roots. For instance, ( x – a )( x – b ) = x 2 – ( a + b ) x + ab , where 1, a + b and ab are the elementary polynomials of degree 0, 1 and 2 in two variables.
Cyclic order, a ternary relation defining a way to arrange a set of objects in a circle; Cyclic permutation, a permutation with one nontrivial orbit; Cyclic polygon, a polygon which can be given a circumscribed circle; Cyclic shift, also known as circular shift; Cyclic symmetry, n-fold rotational symmetry of 3-dimensional space
A polynomial code of length is cyclic if and only if its generator polynomial divides Since g ( x ) {\displaystyle g(x)} is the minimal polynomial with roots α c , … , α c + d − 2 , {\displaystyle \alpha ^{c},\ldots ,\alpha ^{c+d-2},} it suffices to check that each of α c , … , α c + d − 2 {\displaystyle \alpha ^{c},\ldots ,\alpha ...
The complete homogeneous symmetric polynomials are characterized by the following identity of formal power series in t: = (, …,) = = = = = (this is called the generating function, or generating series, for the complete homogeneous symmetric polynomials).