189-457B: Algebra 4
Assignment 2
Due: Wednesday, February 7.
1. Let $G$ be a finite group and let $X$ be a finite $G$-set.
Show that the dimension of the $G$-invariant subspace of the
linear representation $\mathbb C^X$ is equal to the number of $G$-orbits
of $G$ acting on $X$.
2. By specialising
the formula for the
dimension of the subspace $V^G$ of $G$-invariant vectors
shown in class (in terms of the character of $G$ attached to
a representation $V$)
to the case $V= \mathbb C^X$,
give a new proof of the Burnside
lemma proved last semester.
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 question $7$ (b) 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?