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The symmetric group on a set of size n is the Galois group of the general polynomial of degree n and plays an important role in Galois theory. In invariant theory, the symmetric group acts on the variables of a multi-variate function, and the functions left invariant are the so-called symmetric functions.
The alternating group, symmetric group, and their double covers are related in this way, and have orthogonal representations and covering spin/pin representations in the corresponding diagram of orthogonal and spin/pin groups. Explicitly, S n acts on the n-dimensional space R n by permuting coordinates (in matrices, as permutation matrices).
The group of all permutations of a set M is the symmetric group of M, often written as Sym(M). [1] The term permutation group thus means a subgroup of the symmetric group. If M = {1, 2, ..., n} then Sym(M) is usually denoted by S n, and may be called the symmetric group on n letters. By Cayley's theorem, every group is isomorphic to some ...
In mathematics, specifically in representation theory, the Frobenius formula, introduced by G. Frobenius, computes the characters of irreducible representations of the symmetric group S n. Among the other applications, the formula can be used to derive the hook length formula.
In mathematics, a Young tableau (/ t æ ˈ b l oʊ, ˈ t æ b l oʊ /; plural: tableaux) is a combinatorial object useful in representation theory and Schubert calculus.It provides a convenient way to describe the group representations of the symmetric and general linear groups and to study their properties.
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 .
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. [1] [2]
In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient space which takes the object to itself, and which preserves all the relevant structure of the object.