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The following polynomials in two variables X 1 and X 2 are symmetric: + + + + (+) as is the following polynomial in three variables X 1, X 2, X 3: . There are many ways to make specific symmetric polynomials in any number of variables (see the various types below).
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 ...
(For example, if the variables x, y, and z are permuted in all 6 possible ways in the polynomial x + y − z then we get a total of 3 different polynomials: x + y − z, x + z − y, and y + z − x. Note that 3 is a factor of 6.) The number of such polynomials is the index in the symmetric group S n of the subgroup H of permutations that ...
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.
Cyclic group, a group generated by a single element; Cyclic homology, an approximation of K-theory used in non-commutative differential geometry; Cyclic module, a module generated by a single element; Cyclic notation, a way of writing permutations; Cyclic number, a number such that cyclic permutations of the digits are successive multiples of ...
That is, any symmetric polynomial P is given by an expression involving only additions and multiplication of constants and elementary symmetric polynomials. There is one elementary symmetric polynomial of degree d in n variables for each positive integer d ≤ n, and it is formed by adding together all distinct products of d distinct variables.