matrix representationMatrixRepresentation2004-12-14 20:34:262005-02-18 12:00:20Definitiongroup representation\usepackage{graphicx}
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\newtheorem{dfn}{Definition}A matrix representation of a group $G$ is a group homomorphism between $G$ and $GL_n(\C)$, that is, a function
\[ X:G\to GL_n(\C)\]
such that
\begin{itemize}
\item $X(gh)=X(g)X(h)$,
\item $X(e)=I$
\end{itemize}
Notice that this definition is equivalent to the group representation definition when the vector space $V$ is finite dimensional over $\C$. The parameter $n$ (or in the case of a group representation, the dimension of $V$) is called the \emph{degree} of the representation.
\begin{thebibliography}{9}
\bibitem{sagan} Bruce E. Sagan. \emph{The Symmetric Group: Representations, Combinatorial Algorithms and Symmetric Functions.} 2a Ed. 2000. Graduate Texts in Mathematics. Springer.
\end{thebibliography}