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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 .
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.
Symmetric polynomials arise naturally in the study of the relation between the roots of a polynomial in one variable and its coefficients, since the coefficients can be given by polynomial expressions in the roots, and all roots play a similar role in this setting. From this point of view, the elementary symmetric polynomials are the most ...
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.
Any circulant is a matrix polynomial (namely, the associated polynomial) in the cyclic permutation matrix: = + + + + = (), where is given by the companion matrix = []. The set of n × n {\displaystyle n\times n} circulant matrices forms an n {\displaystyle n} - dimensional vector space with respect to addition and scalar multiplication.
S 2n, [2 +,2n +], (n×) of order 2n - gyro-n-gonal group (not to be confused with symmetric groups, for which the same notation is used; abstract group Z 2n); It has a 2n-fold rotoreflection axis, also called 2n-fold improper rotation axis, i.e., the symmetry group contains a combination of a reflection in the horizontal plane and a rotation by ...
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 quantum chemistry studies of atoms, molecules and solids.