189-457B: Algebra 4

Assignment 2

Due: Wednesday, February 5.

1. Suppose that a finite group $G$ has an abelian subgroup $H$ of index $t$ in $G$, i.e., $\#(G/H) = t$. Show that any irreducible complex representation of $G$ has dimension $\le t$. (Hint: consider the orbit of a simultaneous eigenvector for $H$.)

2. Let $e_1,\ldots,e_t$ be a collection of unit vectors in a $d$-dimensional complex Hermitian space $V$ satisfying $$ \langle e_i,e_i\rangle = 1, \qquad | \langle e_i, e_j\rangle| = | \langle e_k, e_\ell\rangle|, \quad \mbox{ for all } i\ne j, \ \ k\ne \ell,$$ and let $\Pi_i$ be the orthogonal projection onto the complex line spanned by $e_i$, for $i=1,\ldots,t$.

(a) Show that the trace of $\Pi_j \Pi_i $ is equal to $1$ if $i=j$, and to $\alpha := |\langle e_i,e_j\rangle|^2$ if $i\ne j$.

(b) Use (a) to show that the $t$ projections $\Pi_1,\ldots, \Pi_t$ are linearly independent in ${\rm End_{\mathbb C}}(V)$, and conclude that $t\le d^2$.

(c) A SIC-POVM (``Symmetric Informationally Complete Positive Operator-Valued Measure") is a maximal collection of equiangular unit vectors, satisfying $t=d^2$. It is an important question in several branches of mathematics and physics, notably in quantum information theory, to exhibit such collections, which are not known to exist for general $d$. In virtually all examples that have been exhibited so far, the equiangular lines in a SIC-POVM form an orbit for the action of a finite group $G$ acting irreducibly on $V$. Assuming that this is the case, show that $\Pi_1+ \cdots + \Pi_t = dI$, where $I$ is the identity transformation on $V$. (Hint: show that $\Pi_1+\cdots+ \Pi_t$ commutes with the action of $G$, invoke Schur's lemma, and compute the trace of this endomorphism.)

(d) Using the result of (c), compute the real constant $\alpha = |\langle e_i,e_j\rangle|^2$ ($i\ne j$) of (a) in a SIC-POVM of dimension $d$ as a function of $d$, by calculating the trace of $(\Pi_1+\cdots+ \Pi_t) \Pi_j$ in two different ways.

3. Describe the character table of the quaternion group $Q$ of order $8$, and compare it with the character table of the dihedral group $D_8$ of order $8$ which we calculated in class. What do you observe?

4. Using the character table of $S_4$ constructed in class, describe which irreducible representations remain irreducible when restricted to the alternating group $A_4$ of order $12$, and which pairs of irreducible representations become isomorphic. Use this to write down the character table of $A_4$.

5. Use the work you did in Question 3 of Assignment 1 to write down the character table of the Frobenius group $F_{20}$ of order $20$, and check directly that all the orthonormality relations for characters proved in class are satisfied in this case.

6. Recall the Heisenberg group of order $p^3$ (for $p$ an odd prime) that was introduced in questions $5$-$7$ of Assignment 1. Use what was shown in that question to describe the character table of this Heisenberg group, and check directly that the sum of the squares of the dimensions of all its irreducible representations is equal to $p^3$, as predicted by the general theory.

7. Using the orthogonality relations for the rows of the character table shown in class, show that the columns satisfy similar orthogonality relations, namely, $$ \sum_{\chi} \chi(C_1) \bar\chi(C_2) = \left\{ \begin{array}{cl} 0 & \mbox{ if } C_1\ne C_2 \\ \#G/\# C & \mbox{ if } C_1 = C_2 = C, \end{array} \right. $$ where the sum ranges over the distinct characters of irreducible representations of $G$, and $C_1$ and $C_2$ are conjugacy classes in $G$ (and $\chi(C_1)$ is a shorthand for the value of $\chi$ on any element of $C_1$).

8. Show that $S_5$ has two distinct one-dimensional representations. Use the natural permutation action of $S_5$ on the set of $5$ elements to construct two further irreducible representations of $S_5$, of dimension $4$, and write down their characters as functions on the $7$ conjugacy classes of $S_5$.

9. Use the ``exotic" transitive action of $S_5$ on $6$ elements constructed last semester (from the action of $S_5$ on the coset space $S_5/F_{20}$) to to construct two further irreducible $5$-dimensional representation of $S_5$, and write down their characters. (Note: a lot of the work for this has already been carried out last semester; this is your chance to review it.)

10. Use the work you have done in Questions 8 and 9 to write down the first 6 rows of the character table of $S_5$. Conclude that $S_5$ has one further irreducible representation, and use the result in Question 7 to write down its character (and dimension).

Food for thought/ extra credit question: Can you give a simple explicit description of this seventh irreducible representation, whose existence emerges somewhat obliquely from the general theory we have developed?