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{\bf{\centerline{Homework \#3 Mathematics 8669 Due Monday April 4, 2016}}}
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1. Find the character table of the symmetric group $S_4.$
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2. What is the orthogonality relation for the characters of 
the cyclic group of order $n$?
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3. Show that for any character $\chi$ and $g\in G$, $|\chi(g)|\le \chi(e)={\text{ dimension}}(\chi).$
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4. Let $G$ be a finite group of order $p^2$, $p$ a prime. 
By considering the possible dimensions of the irreducible representations 
of $G$, prove that $G$ is abelian. 
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5. Prove that the sum of any row of the character table of $G$ 
is a non-negative integer (see problem 9 for notation),
$$
\sum_{i=1}^s \chi^L(K_i) {\text{ is a non-negative integer.}}
$$
(Hint: Consider the character $\chi^C$ obtained by letting $G$ 
at on itself by conjugation. What is $<\chi^L,\chi^C>$?) 
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6. The dihedral group $D_n$ is the group of order $2n$ which is 
the symmetry group of the regular $n$-gon. It may be given in 
terms of reflections and rotations as the set 
$$
D_n=\{ r^k: 0\le k\le n-1\}\cup  \{ sr^k: 0\le k\le n-1\}
$$
where $r^n=e$, $s^2=e$, $srs=r^{-1}$. Show that the set of 
irreducible representations of $D_n$ consists of 
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(a) four 1-dimensional representations, and $(n/2-1)$ 2-dimensional representations 
for $n$ even 
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(b)  two 1-dimensional representations, and $(n-1)/2$ 2-dimensional representations 
for $n$ odd.
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You should be able to construct these characters, using induced characters from the cyclic subgroup of order $n$.
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7. Let $S_n$ act on the set $\{1,\cdots, n\}$ by the natural action of permutations. 
Let $V$ be the $n$-dimensional vector space over $\Bbb{C}$ whose basis is 
$\{1,\cdots, n\}.$  
In this problem you will prove that $V$ decomposes into two irreducibles: the identity of dimension 1, and another irreducible of dimension $n-1.$ Let $\chi^V$ denote 
the permutation character on $V$.
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(a) Recall that the exponential formula says that
$$
\sum_{n=0}^\infty \frac{t^n}{n!} \sum_{g\in S_n} x_1^{\# 1{\text{-cycles of }}g}=
exp\left(tx_1+\sum_{k=2}^\infty t^k/k\right).
$$
Show that this formula implies that the average number of fixed points of 
$g\in S_n$ is 1 for $n\ge 1$, and that the average of 
$({\text{number of fixed points}})^2$ is 
2 for $n\ge 2.$ 
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(b) Use part (a) to show that $\theta=\chi^V-\chi^{id}$ satisfies 
$<\theta,\theta>=1$, and conclude that $\theta$ is irreducible.
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8. Suppose that $V$ and $W$ are finite dimensional 
representations of $G$ over $\Bbb{C}$, with bases $\{v_1,\cdots, v_n\}$ and 
$\{w_1,\cdots, w_n\}.$ Let $V\otimes W$ be the $mn$ dimensional 
$\Bbb{C}$-vector space whose 
basis is $v_i\otimes w_j$, $1\le i\le n$, $1\le j\le m.$ Let $G$ act on 
$V\otimes W$ by 
$$
\sum_{i,j} c_{ij}v_i\otimes w_j\xrightarrow{\rho(g)}  
\sum_{i,j} c_{ij}\rho_1(g)v_i\otimes \rho_2(g)w_j.
$$ 
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(a) Check that $\rho(g)$ is a non-singular linear transfromation on $V\otimes W$, and
$\rho(g_1g_2)=\rho(g_1)\rho(g_2).$
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(b) By explicitly computing traces, show that 
$$
\chi^{V\otimes W}(g)=\chi^V(g)\chi^W(g).
$$
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9. We proved in class that the center of the group algebra has dimension equal 
to the number of conjugacy classes of $G$. In this problem you will give 
two different bases for this center. Let $K_1,\cdots , K_s$ be 
the conjugacy classes of the finite group $G$. Let 
$\{\chi^L: L {\text{ is irreducible}}\}$ be the irreducible characters for $G$.
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(a) Put 
$$
\hat K_i=\sum_{g\in K_i} g \in C[G], \quad 1\le i\le s.
$$
Show that $g\hat K_i= \hat K_i g$ for all $g\in G$, and conclude that 
$\{\hat K_1, \cdots,\hat K_s\}$ is a basis for the center of $C[G].$
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(b)  For an irreducible $L$ of $G$ put
$$
e_L= \frac{dim(L)}{|G|} \sum_{g\in G} \chi^L(g^{-1}) g \in C[G].
$$
Show that $g e_L= e_L g$ for all $g\in G$, and conclude that 
$\{e_L: L {\text{ is irreducible}}\}$ is a basis for the center of $C[G].$
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(c) Using the orthogonality relation for matrix elements, show that
$e_Ke_L=e_K\delta_{KL}$ and find $\sum_{K}e_K.$ 
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(d) For each irreducible $L$, define a map 
$\phi_L:Center(\Bbb{C}[G])\to \Bbb{C}$ by 
$$
\phi_L(z)=\frac{1}{dim(L)}\sum_{g\in G} z(g)\chi^L(g). 
$$
Show that $\phi_L$ is an algebra homomorphism by checking that
$\phi_L(e_K)=\delta_{KL}$ and using part (c).
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10. In the notation of Problem 9, 
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(a) prove that there exists non-negative integers $\alpha_{ij}^k$ such that
$$
\hat K_i \hat K_j =\sum_{k=1}^s \alpha_{ij}^k\hat K_k.
$$  
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(b) prove that there exists non-negative integers $\beta_{LJ}^K$ such that
$$
\chi^L(g) \chi^J(g)=\sum_{K} \beta_{LJ}^K\chi^K(g).
$$  
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11.  You may use the following fact from algebra: If $R$ is a commutative ring 
and $x\in R$, then 
$x$ is integral over $\Bbb{Z}$ if, and only if, the subring $\Bbb{Z}[x]$ of $R$ 
generated by $x$ is finitely generated, if, and only if, $R$ contains a 
finitely generated $Z$-submodule which contains $\Bbb{Z}[x].$ Prove that $\hat K_i$ is 
integral over $\Bbb{Z}$, that is $\hat K_i$ satisfies a polynomial equation with 
integral coefficients whose leading term has coefficient 1.
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12. This problem uses problems 9 and 11 to show $dim(L)$ divides $|G|$ 
for an irreducible $L$.
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(a) Let $z\in Center(\Bbb{C}[G]),$ so that $z$ may be considered as a 
function on the conjugacy classes $K_i.$ If $z(K_i)$ is integral 
over $\Bbb{Z}$ for $1\le i\le s$, verify that
$$
X=\sum_{g\in G} z(g) g=\sum_{i=1}^s z(K_i) K_i
$$ 
is also integral over $\Bbb{Z}$. 
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(b) Since $\phi_L$ is an algebra homomorphism, conclude that 
$\phi_L(X)$ is also integral over $\Bbb{Z}.$
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(c) Finally prove that $dim(L)$ divides $|G|$ by choosing 
$z(K_i)=\chi^L(g^{-1}),$ and noting that 
$$
\phi_L(X)=\frac{|G|}{dim(L)}<X_L,X_L>=\frac{|G|}{dim(L)}
$$
is integral over $\Bbb{Z}.$
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