“representation”的概念、定义、习题解题方法及翻译-数学辞典

单词

representation

释义

representation

(representation theory) When a group G acts on a set S, there is an associated homomorphism ρ:G→Sym(S), where Sym(S) denotes the symmetry group of S defined by ρ(g)(s) = g·s. Here ρ is called a representation, and conversely each such representation gives an action of G on S. If the action is faithful, then the representation is one-to-one.

Commonly, a representation is defined as a homomorphism ρ:G→GL(V) or ρ:G→GL(n,F) associating each group element with a linear map or matrix. Such a representation can instead be treated as a module over the group algebra FG.

Representation theory, as applied to the study of groups and other algebraic structures, is a large part of abstract algebra. It has many applications, including in physics and chemistry, due to the rich theory of representations of Lie groups. See also character, character table, linear action.

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