189-457B: Algebra 4

Assignment 3

Due: Wednesday, February 19.

1. Show that if $g\ne 1$ is an element of a finite group $G$, there is an irreducible representation $\varrho$ of $G$ for which $\varrho(g) \ne I$.

2. Let $g_1$ and $g_2$ be elements of a finite group $G$. Show that $g_1$ is conjugate to $g_2$ if and only if $\chi(g_1)=\chi(g_2)$ for all characters $\chi$ of irreducible representations of $G$. (This implies in particular that $g$ is the identity if $\chi(g)=\chi(1)$ for all irreducible characters of $G$.)

3. Show that the following three conditions on a finite group $G$ are equivalent:

(a) all elements of $G$ are conjugate to their inverses.
(b) all the characters of representations of $G$ take real values;
(c) all the characters of irreducible representations of $G$ take real values.

(Hint: (a) $\Rightarrow$ (b) $\Rightarrow$ (c) is the most straightforward; Q2 is germane for the implication (c) $\Rightarrow$ (a).)

4. Show that condition $(a)$ of Q3 is:

(a) only satisfied for an abelian group $G$ if every element of $G$ is of order $1$ or $2$;
(b) always satisfied if $G=S_n$ is the symmetric group on $n$ elements;
(c) true for the simple group $G=A_5$ of order $60$, but not for the simple group $G=GL_3(2)$ of order $168$. (The latter point will be developped further in the next few questions.)

5. Recall the following partial character table which we constructed in class for the simple group $G={\rm GL}_3(\mathbb Z/2\mathbb Z)$ of order $168$:

$$ \begin{array}{c|rrrrrr} & 1 & 21 & 56 & 42 & 24 & 24 \\ & 1A & 2A & 3A & 4A & 7A & 7B \\ \hline \chi_1 & 1 & 1 & 1& 1 & 1 & 1 \\ \chi_2 & 6 & 2 & 0& 0 & -1 & -1 \\ \chi_3 & 7 & -1 & 1 & -1 & 0 & 0 \\ \chi_4 & 8 & 0 & -1 & 0 & 1 & 1 \\ \chi_5 & 3 & & & & & \\ \chi_6 & 3 & & & & & \end{array} $$

(a) Show that an element of order $7$ in $G$ is never conjugate to its inverse, and conclude that $\chi_6(g) = \overline{\chi_5(g)}$, for all $g\in G$.
(b) Show that $\chi_5(g) = \chi_6(g)$ whenever $g$ is not of order $7$, and that $$ \chi_5(7A) = \overline{\chi_5(7B)} = \overline{\chi_6(7A)} = \chi_6(7B).$$
(c) Use the results you have obtained in (a) and (b) to complete the above character table for $G$.

6. Let $G$ be a finite group, let $\varrho:G \rightarrow {\rm GL}_n(\mathbb C)$ be an irreducible representation of $G$, let $\chi$ be its associated character, and let $f:G\rightarrow {\mathbb C}$ be a class function on $G$. Show that $$ \frac{1}{\#G} \sum_{g\in G} \overline{f(g)} \varrho(g) = \frac{\langle f,\chi\rangle}{n} I_n,$$ where $I_n$ is the $n\times n$ identity matrix.
(Hint: Show that the left-hand matrix is $G$-equivariant, invoke Schur's Lemma, and compare the traces on both sides.)

7. With the notations as in Q6, let $\chi$ be the character of the irreducible representation $\rho$, and let $\alpha: G \rightarrow {\rm GL}_N(\mathbb C)$ be any (not necessarily irreducible) representation of $G$. Using the results of Q6, show that the matrix $$ M_\chi:= \frac{\chi(1)}{\# G} \sum_{g\in G} \chi(g) \cdot \alpha(g)$$ is an idempotent matrix (i.e., satisfies $M^2 = M$) and describe its image.

8. Two subgroups $H_1$ and $H_2$ of a finite group $G$ are said to be almost conjugate if $$ \# (H_1 \cap C) = \#(H_2 \cap C),$$ for all conjugacy classes $C$ of $G$.

(a) Show that if $H_1$ and $H_2$ are conjugate to each other, then they are almost conjugate (explaining the terminology);
(b) Show that if $H_1$ and $H_2$ are almost conjugate, then the linear representations $\mathbb C[G/H_1]$ and $\mathbb C[G/H_2]$ are isomorphic as representations of $G$. (Hint: compare the characters of these two representaions.)

9. Let $G = {\rm GL}_3(\mathbb Z/2\mathbb Z)$ be our favorite finite simple group of order $168$, let $H_1$ be the stabiliser in $G$ of a non-zero vector in $V:=(\mathbb Z/2\mathbb Z)^3$, , and let $H_2$ the stabiliser in $G$ of a two-dimensional subspace of $V$. Recall that last semester you showed that $H_1$ and $H_2$ are not conjugate to each other (cf.~Q9 of Asst 2 of Math 456A). Show that nevertheless, $H_1$ and $H_2$ are almost conjugate to each other in the sense of Q8.

10. Retaining the notations of Q9, let $X_1 = G/H_1 = V-\{0\}$ be the set of one-dimensional subspaces of $V$ and let $X_2=G/H_2$ be the set of two-dimensional subspaces of $V$. Show that the maps $$ A: \mathbb C[X_1] \rightarrow \mathbb C[X_2], \qquad B: \mathbb C[X_2] \rightarrow \mathbb C[X_1]$$ given by $$ A(v) = \sum_{W \ni v} [W], \qquad B(W) = \sum_{v\in W} [v],$$ are $G$-equivariant isomorphisms. Note here that the sums in the above definitions of $A$ and $B$ are to be understood as formal sums of elements in the sets $X_2$ and $X_1$ respectively, and not as sums in $V$ (for the definition of $B$ in particular). (Hint: one can show that $A$ and $B$ are invertible, by showing that zero is not an eigenvalue of $AB$ or $BA$; more precisely these transformations each have exactly two eigenvalues which are both non-zero. This can perhaps best be shown by expressing $AB$ and $BA$ as suitable linear combination of the identity and a linear transformation of rank one.)