# Exercises Weeks 3 & 4

From Mitchell’s talk on $\mathbb{C}[G]$-modules:

1. If $D$ is a division algebra, then check that every $D$-module is of the form $D^r$, and as a corollary, all $M_n(D)$-modules are of the form $\underbrace{D^r \oplus \dots \oplus D^r}_{n}$.
2. Prove the following lemma: If $F$ is a field, then $V$ is a (finitely generated) $F[G]$-module, then $V$ is a (finite dimensional) vector space over $F$.
3. Is the ring $\mathbb{Z}$ semisimple? (we did this one in class)
4. Prove the following theorem of Wedderburn: If $F$ is algebraically closed, then any semisimple algebra over $F$ deomposes as $\bigoplus_{i=1}^r M_{n_i}(F)$. As a corollary, this holds for $F[G]$. What does $r$ represent? What does this mean in the context of representations?

From my talk on representations of group products, and induced representations:

1. Let $\rho_1: G_1 \to GL(V_1), \rho_2: G_2 \to GL(V_2)$ be representations. Consider the representation $\rho_1 \oplus \rho_2 : G_1 \times G_2 \to GL(V_1 \oplus V_2)$ defined appropriately. Is it interesting?
2. Show that the character of $\rho_1 \otimes \rho_2$ is $\chi(g_1,g_2)=\chi_1(g_1) \chi_2(g_2)$.
3. We have shown that if $\rho_1, \rho_2$ are irreducible, $\rho_1 \otimes \rho_2$ is irreducible. Show then that in fact every irreducible representation of $G_1 \times G_2$ is of this form.
4. Verify and experiment with the examples in my notes (to be posted soon-ish).

From Reuben’s talk on Lie algebras:

1. Show there is a bijection between representations of $L$, and left $L$-modules.
2. Show that $\mathbb{R}^3$ under the cross product can be realised (by an isomorphism) as $\mathfrak{so}_2(\mathbb{R})$, the Lie subalgebra of $\mathfrak{gl}_2(\mathbb{R})$ containing matrices satisfying $X+X^T=0$.
3. Suppose there exist $G, \circ_1, \circ_2$ such that $(G, \circ_1)$ and $(G, \circ_2)$ are groups, and $(g \circ_1 h) \circ_2 k = g \circ_1 (h \circ_2 k)$. What can we say about $G$? Does it necessarily hold that $\circ_1 = \circ_2$?
4. Find an example of an indecomposable $L$-module which is not irreducible (i.e. contains nontrivial proper submodules).
5. Prove the following lemmas: For a Lie algebra $L$, and $m, n \geq 1$, $[L^m, L^n] \subseteq L^{m+n}$; and for $n \geq 0$, $L^{(n)} \subseteq L^{2^n}$. (It follows directly that nilpotent Lie algebras are solvable.)
6. Show that Lie algebras consisting of upper triangular matrices are solvable, and strictly upper triangular matrices are nilpotent.
7. Show that every irreducible representation of a solvable Lie algebra is one-dimensional.

And finally, from Chris’ talk on semisimple Lie algebras:

1. For $\mathfrak{g} = \mathfrak{gl}_n$, show that the radical is given by the centre, the subalgebra of nonzero scalar matrices, of the form $z I_n, z \in F^*$. Show also that $\mathfrak{g}/\textrm{Rad}(\mathfrak{g}) \simeq \mathfrak{sl}_n$.
2. Prove the following direction of the Cartan-Killing criterion: If $\mathfrak{g}$ is a semisimple Lie algebra, then its Killing form is nondegenerate.
3. Prove that $\mathfrak{sl}_n$, $\mathfrak{so}_n$$\mathfrak{sp}_{2n}$ are simple Lie algebras.

# Link to the complete* notes from talk 3 (characters, Schur’s Lemma, orthogonality)

* Not really “complete”.

Here you go.

The notes contain more detail on the standard representation of S3 and inter-relatedness (via characters) that arise between the different representations, their tensor products, direct sums and so on.

# Week 2 Exercises

Some questions we left open last time:

1. Let $\rho: \mathbb{Z}/n \to \mathbb{C}^*$ be given by $1 \mapsto \zeta$ for some $n$-th root of unity $\zeta$. Show that $\rho^*$ is $1 \mapsto \bar{\zeta}$. Conclude that $\rho, \rho^*$ are not isomorphic.
2. Let $A$ be an $n \times n$ matrix of finite order ($A^m = I$ for some $m \in \mathbb{Z}$). Show that $tr(A^{-1}) = \overline{tr(A)}$, without using eigenvalues.
3. In the proof of Prop. 1 of Serre (see Section 2.1), he states that “… $\rho_s$ can be defined to be a unitary matrix”. Show that every representation of a finite group is unitary. (As part of this exercise, give an appropriate definition of a unitary representation.)
4. Prove that the character of $V_1 \otimes V_2$ is given by $\chi_1 \cdot \chi_2$. If $\rho$ is a representation, prove that the character of a dual representation is $\chi_{\rho^*} = \overline{\chi_\rho}$. Find the characters of $\textrm{Sym}^2(\rho)$ and $\rho \wedge \rho = \textrm{Alt}^2(\rho)$.
5. If $f: (\rho_1, V_1) \to (\rho_2, V_2)$ is a map of representations, show that the kernel and image of $f$ are subrepresentations of $(\rho_1, V_1), (\rho_2, V_2)$ respectively.
6. Midterm question: Give your own proof of Serre Thm. 3 (section 2.3), that characters of irreducible representations form an orthonormal system.
7. Prove Prop. 6 of Serre (section 2.5): If $(\rho, V)$ is an irreducible representation on $G$ of degree $n$ and character $\chi$, and $f$ is a class function on $G$, then $\rho_f := \sum_{t \in G} f(t)\rho(t)$ is a homothety of ratio $\frac{1}{n} \sum_{t \in G} f(t) \chi(t)$.
8. Prove that the number of degree 1 representations on a finite group $G$ is $|G^{ab}|$.

# Future talk Schedule

Hey guys,

Sorry this is coming through a bit late, been a crazy week!

The talk schedule which was decided on Wednesday afternoon is as follows:

1. Reuben, Intro to Lie algebra, nilpotent and solvable (Serre 1, FH 9.1,9.2)
2. Chris, Semi-simple Lie algebras (Serre 2, FH 9.3,9.4)
3. Pat, Lie algebras of dim 1,2,3 (FH 10)
4. Stephen, Rep theory of sl(2,C) (Serre 4, FH 11)
5. Reuben, Cartan Subalgebras (Serre 3, FH 14 & App. D)
6. Mitchell, Root Systems I (Serre 5, FH 21)
7. Jackson, Structure of semi-simple Lie algebras
8. Alex, Reps of semi-simple Lie algebras

That covers the talks that should take us to the end of the course. Other things that were mentioned on Wednesday were that the midterm would assess questions on reps of finite groups, and then the final project would be about Lie algebras.

(edit 26/3: Swapped Reuben and Alex)

# Progress on irreps of abelian groups

I can’t sleep and I’ve had an idea, so here’s another post. We’ve discussed a few times the possibility of proving that every irreducible representation of an abelian group is 1-dimensional, by elementary means.

Alex proved this statement in his talk today (as a corollary of some wider considerations), using the character machinery that he and Pat outlined, however I think I’ve got a proof that circumvents the need for this. I claim very little originality – the key idea is stolen right from the proof of ‘Schur’s lemma’ in the form that it is presented in MATH3103. Here goes:

Suppose $G$ is an abelian group, $V$ is a finite dimensional vector space over $\mathbb{C}$ and $\rho: G \to GL(V)$ is an irreducible representation. Fix $g$ in $G$ and let $\lambda$ be an eigenvalue of $\rho(g)$ whose corresponding eigenspace is $E_\lambda \subset V$. Then if $v$ is an element of $E_\lambda$, for any $h$ in $G$ we have: $\rho(g)\rho(h)v = \rho(gh)v = \rho(hg)v = \rho(h)\rho(g)v = \lambda \rho(h)v$, so $\rho(h)v$ is also an element of $E_\lambda$. Therefore $E_\lambda$ is closed under the action of $G$ and forms a sub-representation of $\rho$. By irreducibility it must hold that $E_\lambda = V$, so $\rho(g) = \lambda I$, where $I$ is the identity on $V$. Since $g$ was arbitrary we may conclude that $\rho$ sends every group element to some scalar multiple of the identity and it’s clear then that if dim$V \geq 2$ any 1-dimensional subspace of $V$ constitutes a strict sub-representation of $\rho$. This violates irreducibility, so dim$V =1$.

The other nice thing about this argument is that it seems to work for infinite $G$. Thoughts?

# Norms come cheap

We had a brief discussion today about when exactly the norm on a vector space is induced by an inner product, that was left unresolved. The following Wikipedia article treats the question well enough, for those who care:

http://en.wikipedia.org/wiki/Parallelogram_law

Tl;dr the answer is almost never – most norms you can think of are not born of an inner product.

This is quite an important fact I think, particularly in infinite dimensions where Hilbert spaces have a richer algebraic structure than their lesser counterparts, Banach spaces. For example this natural isomorphism we discussed, between a vector space and it’s double dual still ‘works’ for a Hilbert space (sort of – we distinguish between the algebraic dual and the continuous or topological dual of an infinite dimensional topological vector space, and I’m talking here about the latter), but one can only say in general that a Banach space is contained in it’s double dual.