Group Representation
Two representations are considered equivalent if they are similar . For example, performing similarity transformations of the above matrices by
This entry contributed by Todd Rowland
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Rowland, Todd . "Group Representation." From MathWorld --A Wolfram Web Resource, created by Eric W. Weisstein . https://mathworld.wolfram.com/GroupRepresentation.html
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In chemistry, a group representation can relate mathematical group elements to symmetric rotations and reflections of molecules. Representations of groups allow many group-theoretic problems to be reduced to problems in linear algebra.
Let $V$ be a nonzero (not necessarily finite dimensional) vector space over a field $k$ of characteristic $p$ and let $\rho:G\to \mathrm{GL}(V)$ be a representation of a finite $p$-group $G$ on $V$. Then, $V^G\neq 0$, i.e., there exists a nonzero vector $v\in V$ which is fixed by $\rho(g)$ for all $g\in G$. Proof.
We can now define a group representation. Definition 1.6.Let G be a group. A representation of G (also called a G-representation, or just a representation) is a pair (π,V) where V is a vector space and π: G →Autvect(V) is a group action. I.e., an action on the set V so that for each g ∈G, π(g) : V →V is a linear map. Remark 1.7.
Every linear representation of a compact group over a field of characteristic zero is a direct sum of irreducible representations. Or in the language of []-modules: If () =, the group algebra [] is semisimple, i.e. it is the direct sum of simple algebras.
Recall that a representation of a group G over a field k is a k-vector space V together with a group homomorphism δ : G ⊃ GL(V ). As we have explained above, a representation of a group G over k is the same thing as a representation of its group algebra k[G].
A representation of a group G is a group action of G on a vector space V by invertible linear maps. For example, the group of two elements Z_2= {0,1} has a representation phi by phi (0)v=v and phi (1)v=-v. A representation is a group homomorphism phi:G->GL (V).
Each representation of a group defines a representation of a subgroup, by restriction; that much is obvious. More subtly, each representation of the subgroup defines a representation of the full group, by a process called in- duction. This provides the most powerful tool we have for constructing group representations.
Representation theory reverses the question to “Given a group G, what objects X does it act on?” and attempts to answer this question by classifying such X up to isomorphism. Before restricting to the linear case, our main concern, let us remember another way to describe an action of G on X. Every g ∈ G defines a map a(g) : X → X by x 7→gx.
A representation of Q over a field k is an assignment of a k-vector space V i to every vertex i of Q, and of a linear operator A h : V i ⊃ V j to every directed edge h going from i to j (loops and multiple edges are allowed).
Representation theory is the study of the various ways a given group can be mapped into a general linear group. This information has proven to be e ective at providing insight into the structure of the given group as well as the objects on which the group acts.