189-570A: Higher Algebra I

Assignment 2

Due: Monday, October 7.

All groups $G$ are finite in this assignment.

1. Let $X$ be a (finite) permutation representation of a finite group $G$, and let $V= K[X]$ be the associated linear representation. Use character theory to show that the dimension of the space $V^G$ of $G$-invariant vectors is the number of orbits for $G$ acting on $X$. Use this to show that the average number of fixed points of $g$ acting on $X$ as $g$ ranges over $G$ is equal to the number of $G$-orbits on $X$.

You might want to compare (for yourself: no need to write anything up here) your approach to what you did in the first assignment, where you might have followed a more elementary path to this result that did not invoke character theory directly.

2. Let $G$ be a finite group, let $p$ be a prime dividing the cardinality of $G$, and let $K$ be a field of characteristic $p$. Show that the line spanned by $N = \sum_{g\in G} g$ is a $G$-stable line in $K[G]$ which does not admit a $G$-stable vector space complement in $K[G]$, and hence, that the regular representation of $G$ is never semi-simple under these conditions.

3. Show that any representation of $G$ whose character $\chi$ satisfies $\chi(g)= 0$ for all $g\ne 1$ is a direct sum of copies of the regular representation of $G$.

4. Let G be a finite group, and V a representation for G over a field of characteristic zero. Show that the set $g\in G$ such that $\chi_V(g) = \pm \chi_V(1)$ is a normal subgroup of G.

5. Let $D_8$ be the dihedral group of order $8$ and let $Q$ be the quaternion group of order 8 consisting of the elements $\{\pm 1, \pm i, \pm j,\pm k \}$, with multiplication as in the ring of Hamilton quaternions. Show that $D_8$ and $Q$ are not isomorphic to each other. Compute the character tables of $D_8$ and $Q$. What do you observe?

6. A linear representation is said to be faithful if the associated homomorphism from $G$ to ${\rm Aut}(V)$ is injective. Show that, if $G$ admits a faithful irreducible representation over an algebraically closed field of characteristic zero, then its center is cyclic.

7. If $H$ is a subgroup of $G$, show that every irreducible representation of $G$ occurs as a constituent of a representation induced from $H$.

8. If $V_1$ and $V_2$ are representations of two groups $G_1$ and $G_2$, then the tensor product $V_1\otimes V_2$ is a representation of $G_1\times G_2$ via the rule $$ (g_1,g_2) v_1\otimes v_2 = (g_1v_1)\otimes (g_2 v_2), \qquad (g_1,g_2)\in G_1\times G_2, \quad v_i\in V_i.$$ Show that, if $V_1$ and $V_2$ are irreducible, then so is $V_1\otimes V_2$, and that every irreducible representation of $G_1\times G_2$ arises in this way.

9. If $g$ is an element of a finite group $G$, show that the following properties are equivalent:
(a) $\chi(g)$ is rational, for all characters $\chi$ of $G$;
(b) $g$ is conjugate to $g^t$ for every $t$ which is relatively prime to the order of $G$.

10. Let $V$ be an irreducible representation of a finite group $G$ over an algebraically closed field of characteristic prime to the order of $G$, and let $\chi$ be the character of $V$. Show that the element $$ \pi_V:= \frac{\chi(1)}{\#G} \sum_{g\in G} \chi(g^{-1}) g$$ belongs to the center of the group ring $K[G]$ and that it is an idempotent: i.e., $\pi_V^2=\pi_V$.
If $W$ is any representation of $G$, show that it decomposes as a direct sum of representations $$ W = \oplus_{i=1}^h \pi_{V_i}(W), $$ where the $V_i$ range over the irreducible representations of $G$. Show that each $\pi_{V_i}(W)$ is isomorphic to a direct sum of copies of $V_i$ as a representation of $G$.