7.13: Exercises

[x40]Let \(C\) be an \(n\times r\) matrix with coefficients in a field \(F\). Show that

\[\{M\in M_{n}(F)\mid MC=0\} \nonumber \]

is a left ideal in \(M_{n}(F)\), and that every left ideal is of this form for some \(C\).

[x41]This exercise shows how to recover a finite group \(G\) from its category of representations over a field \(k\). Let \(S\) be a finite set, and let \(A\) be the set of maps \(S\rightarrow k\).

1. Show that \(A\) becomes a commutative ring with the product

   \[(f_{1}f_{2})(g)=f_{1}(g)f_{2}(g),\quad f_{1}\text{, }f_{2}\in A,\quad g\in S. \nonumber \]

   Moreover, when we identify \(c\in k\) with the constant function, \(A\) becomes a \(k\)-algebra.

2. Show that

   \[A\simeq\prod\nolimits_{s\in S}k_{s}\quad\quad\text{(product of copies of }k\text{ indexed by the elements of }S\text{)}, \nonumber \]

   and that the \(k_{s}\) are exactly the minimal \(k\)-subalgebras of \(A\). Deduce that \(\End_{k\text{-alg}}(A)\simeq\Sym(S)\).

3. Let \((f_{1},f_{2})\in A\times A\) act on \(S\times S\) by \((f_{1}% ,f_{2})(s_{1},s_{2})=f_{1}(s_{1})f_{2}(s_{2})\); show that this defines a bijection \(A\otimes A\simeq\Map(S\times S,k)\). Now take \(S=G\).

4. Show that the map \(r_{A}\colon G\rightarrow\End_{k\text{-linear}}(A)\),

   \[(r_{A}(g)f)(g^{\prime})=f(gg^{\prime}),\quad f\in A,\quad g,g^{\prime}\in G \nonumber \]

   is a representation of \(G\) (this is the regular representation).

5. Define \(\Delta\colon A\rightarrow A\otimes A\) by \(\Delta(f)(g_{1}% ,g_{2})=f(g_{1}g_{2})\). Show that, for any homomorphism \(\alpha\colon A\rightarrow A\) of \(k\)-algebras such \((1\otimes\alpha)\circ\Delta=\Delta \circ\alpha\), there exists a unique element \(g\in G\) such that \(\alpha(f)=gf\) for all \(f\in A\). [Hint: Deduce from (b) that there exists a bijection \(\phi\colon G\rightarrow G\) such that \(\left( \alpha f\right) (g)=f(\phi g)\) for all \(g\in G\). From the hypothesis on \(\alpha\), deduce that \(\phi (g_{1}g_{2})=g_{1}\cdot\phi(g_{2})\) for all \(g_{1},g_{2}\in G(R)\). Hence \(\phi(g)=g\cdot\phi(e)\) for all \(g\in G\). Deduce that \(\alpha(f)=\phi(e)f\) for all \(f\in A\).]

6. Show that the following maps are \(G\)-equivariant

   \[\begin{aligned} e\colon & k\rightarrow A\quad\quad\text{(trivial representation on }k\text{; }r_{A}\text{ on }A)\\ m\colon & A\otimes A\rightarrow A\quad\quad\text{(}r_{A}\otimes r_{A}\text{ on }A\otimes A\text{; }r_{A}\text{ on }A)\\ \Delta\colon & A\rightarrow A\otimes A\quad\quad\text{(}r_{A}\text{ on }A\text{; }1\otimes r_{A}\text{ on }A\otimes A).\end{aligned} \nonumber \]

7. Suppose that we are given, for each finite-dimensional representation \((V,r_{V})\), a \(k\)-linear map \(\lambda_{V}\). If the family \((\lambda_{V})\) satisfies the conditions

   1. for all representations \(V\), \(W\), \(\lambda_{V\otimes W}=\lambda _{V}\otimes\lambda_{W};\)

   2. for \(k\) with its trivial representation, \(\lambda_{k}=\id_{k}\);

   3. for all \(G\)-equivariant maps \(\alpha\colon V\rightarrow W\), \(\lambda _{W}\circ\alpha=\alpha\circ\lambda_{V};\)

   then there exists a unique \(g\in G(R)\) such that \(\lambda_{V}% =r_{V}(g)\) for all \(V\). [Hint: show that \(\lambda_{A}\) satisfies the conditions of (d).]

For a historical account of the representation theory of finite groups, emphasizing the work of “the four principal contributors to the theory in its formative stages: Ferdinand Georg Frobenius, William Burnside, Issai Schur, and Richard Brauer”, see .