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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 .
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.
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.
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 .
If is an irreducible polynomial of prime degree with rational coefficients and exactly two non-real roots, then the Galois group of is the full symmetric group. [2] For example, f ( x ) = x 5 − 4 x + 2 ∈ Q [ x ] {\displaystyle f(x)=x^{5}-4x+2\in \mathbb {Q} [x]} is irreducible from Eisenstein's criterion.
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).
Every polynomial equation in one variable has a Galois group, that is a certain permutation group on its roots. The axioms of a group formalize the essential aspects of symmetry. Symmetries form a group: they are closed because if you take a symmetry of an object, and then apply another symmetry, the result will still be a symmetry. The ...