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Permutation Representation (symmetric Group)
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. The symmetric group S''n'' has order ''n''!. Its conjugacy classes are labeled by partitions of ''n''. Therefore according to the representation theory of a finite group, the number of inequivalent irreducible representations, over the complex numbers, is equal to the number of partitions of ''n''. Unlike the general situation for finite groups, there is in fact a natural way to parametrize irreducible representations by the same set that parametrizes conjugacy classes, namely by partitions of ''n'' or equivalently Young diagrams of size ''n''. Each such irreducible representation can in fact be realized over the integers (every permutation acting by a m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Permutations RGB
In mathematics, a permutation of a Set (mathematics), set can mean one of two different things: * an arrangement of its members in a sequence or linear order, or * the act or process of changing the linear order of an ordered set. An example of the first meaning is the six permutations (orderings) of the set : written as tuples, they are (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1). Anagrams of a word whose letters are all different are also permutations: the letters are already ordered in the original word, and the anagram reorders them. The study of permutations of finite sets is an important topic in combinatorics and group theory. Permutations are used in almost every branch of mathematics and in many other fields of science. In computer science, they are used for analyzing sorting algorithms; in quantum physics, for describing states of particles; and in biology, for describing RNA sequences. The number of permutations of distinct objects is &nbs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |