7: Representations of Finite Groups
- Page ID
- 180020
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\dsum}{\displaystyle\sum\limits} \)
\( \newcommand{\dint}{\displaystyle\int\limits} \)
\( \newcommand{\dlim}{\displaystyle\lim\limits} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\(\newcommand{\longvect}{\overrightarrow}\)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\ket}[1]{\left| #1 \right>}\)
\(\newcommand{\bra}[1]{\left< #1 \right|}\)
\(\newcommand{\braket}[2]{\left< #1 \vphantom{#2} \right| \left. #2 \vphantom{#1} \right>}\)
\(\newcommand{\braopket}[3]{\left< #1 \vphantom{#2}\vphantom{#3} \right| #2 \vphantom{#1}\vphantom{#3} \left| #3 \vphantom{#1}\vphantom{#2} \right>}\)
\(\newcommand{\qmvec}[1]{\mathbf{\vec{#1}}}\)
\(\newcommand{\op}[1]{\hat{\mathbf{#1}}}\)
\(\newcommand{\expect}[1]{\langle #1 \rangle}\)
\(\newcommand{\dfn}[1]{\emph{\textbf{#1}}}\)
Throughout this chapter, \(G\) is a finite group and \(F\) is a field. All vector spaces are finite-dimensional.
An \(F\)-algebra is a ring \(A\) containing \(F\) in its centre and finite dimensional as an \(F\)-vector space. We do not assume \(A\) to be commutative; for example, \(A\) could be the matrix algebra \(M_{n}(F)\). Let \(\{e_{1},\ldots,e_{n}\}\) be a basis for \(A\) as an \(F\)-vector space; then \(e_{i}e_{j}=\sum\nolimits_{k}a_{ij}^{k}e_{k}\)for some \(a_{ij}^{k}\in F\), called the structure constants of \(A\) relative to the basis; once a basis has been chosen, the algebra \(A\) is uniquely determined by its structure constants.
All \(A\)-modules are finite-dimensional when regarded as \(F\)-vector spaces. For an \(A\)-module \(V\), \(mV\) denotes the direct sum of \(m\) copies of \(V\).
The opposite \(A^{\mathrm{opp}}\) of an \(F\)-algebra \(A\) is the same \(F\)-algebra as \(A\) but with the multiplication reversed, i.e., \(A^{\mathrm{opp}}=(A,+,\cdot^{\prime })\) with \(a\cdot^{\prime}b=ba\). In other words, there is a one-to-one correspondence \(a\leftrightarrow a^{\prime}\colon A\leftrightarrow A^{\mathrm{opp}}\) which is an isomorphism of \(F\)-vector spaces and has the property that \(a^{\prime}b^{\prime}=(ba)^{\prime}\).
An \(A\)-module \(M\) is simple if it is nonzero and contains no submodules except \(0\) and \(M\), and it is semisimple if it is isomorphic to a direct sum of simple modules.