189-457B: Algebra 4

Assignment 1

Due: Wednesday, January 24.

1. Let $$ G \ \ = \ \ D_{2n}\ \ = \ \ \{ 1, \ \sigma,\ \sigma^2, \ \ldots,\ \sigma^{n-1}, \ \ \tau, \ \tau\cdot \sigma,\ \ldots, \ \tau\cdot \sigma^{n-1} \}$$ be the dihedral group of order $2n$, where $n>1$ is an odd integer. Here $\sigma$ can be envisaged as a rotation of angle $2\pi/n$, preserving a regular polygon with $n$ sides centered at the origin, and $\tau$ as a reflection about one of its axes of symmetry, satisfying the basic commutation relation $\tau \sigma = \sigma^{-1} \tau$. (In particular, the elements $\tau \sigma^j$ are all of order two, as you should verify for yourselves, and correspond to the reflections in this dihedral group.)

Let $V$ be a complex representation of $G$, and $\rho: G \rightarrow {\rm Aut}_{\mathbb C}(V)$ be the associated homomorphism.

(a) Show that $V$ is a direct sum of eigenspaces for $\rho(\sigma)$, and describe what $\tau$ does to these eigenspaces.

(b) Use the understanding gained in (a) to show that, up to isomorphism, $G$ has exactly two distinct irreducible representations of dimension $1$, and $(n-1)/2$ representations of dimension $2$, and that these are all the irreducible representations of $G$.

2. By following a similar approach as was used in class to show that the quaternion group has a unique (up to isomorphism) faithful irreducible representation, show that the same assertion is true for the dihedral group of order $8$.

3. Recall the Frobenius group $F_{20}$ of order $20$ introduced last semester as the group of affine linear transformations on the field with $5$ elements. Show that $F_{20}$ has $4$ distinct one-dimensional representations and a unique faithful irreducible representation, which is of dimension $4$. (Hint: try to apply the methods used in questions 1 and 2.)

4. Let $p$ be a prime, let $G$ be a cyclic group of order $p$, and let $F$ be the finite field with $p$ elements. Construct a two-dimensional $F$-linear representation of $G$ which is not a direct sum of irreducible representations.

5. Let $F$ be a field. The Heisenberg group $H(F)$ attached to $F$ is defined to be $$ H(F) = \left\{ \left(\begin{array}{ccc} 1 & a & t \\ 0 & 1 & b \\ 0 & 0 & 1 \end{array} \right) \mbox{ with } a,b,t \in F \right\}.$$ Denoting by $[a,b;t]$ the Heisenberg matrix attached to the parameters $(a,b,t) \in F^3$, check that multiplication of $3\times 3$ matrices corresponds to the group law $$ [a_1, b_1; t_1] \cdot [a_2,b_2;t_2] = [a_1+a_2, b_1+b_2; t_1 + t_2 + a_1 b_2],$$ so that $H(F)$ is indeed a group.

(a) Show that the natural map $[a,b;t] \mapsto (a,b)$ is a group homomorphism to $F^2$ (endowed with the usual addition of vectors), and that the kernel of this homomorphism is the center of $H(F)$ and is isomorphic to the additive group of $F$.

(Cultural remark. The theory of groups is a standard instance where the principle of semisimplicity fails: $H(F)$ has a normal subbroup isomorphic to $F$ and the quotient $H(F)/F$ is isomorphic to $F^2$, but of course $H(F)$ is not isomorphic to the direct sum $(F^2) \oplus F = F^3$ as a group, indeed, it is non-abelian.)

Let $F_p = \mathbb Z/p\mathbb Z$ be the finite field with $p$ elements.

(b) Show that $H(F_p)$ is generated by the elements $$ A=[1,0;0], \quad B=[0,1;0], \qquad T = [0,0;1]$$ satisfying the relations $$ A^p=B^p=T^p=1, \qquad TA = AT, \quad TB = BT, \quad AB = TBA.$$ Deduce from this that every element of $H(F_p)$ is uniquely expressed as $$A^i B^j T^k, \qquad \mbox{ with } 0 \le i,j,k \le p-1.$$

6. Let $V$ be a complex irreducible representation of the Heisenberg group $H(F_p)$ introduced in the previous question. Show that there is a $p$-th root of unity in $\mathbb C$ for which $$ T v = \zeta v, \qquad \mbox{ for all } v\in V.$$ This $p$th root of unity is an important instrinsic invariant of the representation $V$.

7. Let $V$ be an irreducible representation of $H(F_p)$ on which $T$ acts by multiplication by $\zeta$, where $\zeta$ is a primitive $p$-th root of unity. Show that every $p$-th root of unity is an eigenvalue for $\rho(A)$. (Hint: if $v$ is an eigenvector for $A$, show that $Bv$ is also an eigenvector for $A$, and calculate its associated eigenvalue.)

Use this to conclude that $V$ is of dimension $p$, and that $V$ is in fact the unique irreducible representation of $H(F_p)$ on which $T$ acts as multiplication by $\zeta$.

( Cultural remark: The inherent rigidity in the representation theory of the Heisenberg group, that every irreducible representation is determined by how the center $F$ acts on $V$, remains true in a remarkable degree of generality. The Stone-Von Neumann Theorem asserts that given any additive unitary character $\psi: \mathbb R \rightarrow \mathbb C^\times$ (like $\psi(x) = e^{2\pi xy}$ for some $y\in \mathbb R$), there is a unique unitary representation of $H(\mathbb R)$ whose restriction to the center is given by $\psi$. This distinguished complex representation is infinite dimensional, and it construction involves non-trivial analytic difficulties which are not present when $\mathbb R$ is replaced by a finite field and the corresponding Heisenberg group is just a finite group.

8. Consider the element $$ \theta = \frac{1}{M} \sum_{1\le i \lt j \le n} (ij) \ \in \ \mathbb C[S_n], \qquad M:= \frac{n(n-1)}{2}$$ in the complex group ring of the symmetric group on $n$ elements. It models the probability distribution on $S_n$ in which each of the $M$ transpositions in $S_n$ is assigned an equal non-zero probability, and all other permutations are given the probability zero.

(a) Show that the coefficient $m_g(N)\in \mathbb R$ in $$\theta^N = \sum_{g\in S_n} m_g(N) \cdot g$$ can be interpreted as the probability of obtaining $g$ after choosing $N$ transpositions at random and composing them.

(b) The expectation that applying a random sequence of $N$ transpositions to (say) a deck of $n$ cards will eventually lead, once $N$ is large enough, to a deck that is thoroughly well suffled can be expressed in the statement that $m_g(N)$ tends to $1/n!$ when $N$ gets large. In other words, that $$ \lim_{N\rightarrow \infty} \theta^N \ \stackrel{?}{=} \ \frac{1}{n!} \sum_{g\in S_n} g \qquad \mbox{ in } \mathbb C[S_n].$$ Explain why this guess is wrong and formulate a better guess about the limit of $\theta^N$.

9. We keep the notations of question 8. Let $\varrho: S_n \rightarrow {\rm Aut}_{\mathbb C}(V)$ be a complex finite-dimensional representation of the symmetric group $S_n$, and consider $$ \varrho(\theta)\ = \ \frac{1}{M} \sum_{1\le i \lt j \le n} \varrho((ij)) \ \in \ {\rm End}_{\mathbb C}(V).$$ Show that the endomorphism $\varrho(\theta)$ is diagonalisable and has real eigenvalues. (Hint: evoke the structures we brought to light in our proof of Maschke's theorem in class, and the spectral theorem which you saw at the end of Algebra 2.)

10. With notations as in question $9$, show that the eigenvalues of $\varrho(\theta)$ lie in the closed interval $[-1,1]$. (Hint: use the Cauchy-Schwartz inequality, or the triangle inequality...) Conclude that $\varrho(\theta)^{2N}$ and $\varrho(\theta)^{2N+1}$ have limits $T_+, T_- \in {\rm End}_{\mathbb C}(V)$) as $N$ tends to $\infty$. Show that $T_+$ and $T_-$ are the sum and difference of two projectors, i.e., that $T_+ = p_+ + p_-$ and $T_- = p_+ - p_-$, where $p_+$ and $p_-$ are idempotents, namely, satisfy $T^2 = T$. Describe the images of $p_+$ and $p_-$, and try to identify the limit of $\theta^{2N}$ in the group ring of $S_n$, as $N\rightarrow \infty$.