\input amstex
\input amssym.def
\nopagenumbers
\magnification=\magstep1
\centerline{\bf HW \#4}
\bigskip
\centerline {\bf Math 506A}
\bigskip
\noindent 1. Prove or disprove:  if two representations $\pi,\pi'$ of a finite group $G$ are such that $\det\pi(g)=\det\pi'(g)$ for all $g\in G$, then $\pi$ and $\pi'$ are equivalent.
\vskip .3in
\noindent 2.  Let $\pi$ be an irreducible representation of a finite group $G$ with character $\chi$.  Show that $\chi(g)=\chi(1)$ if and only if $g$ lies in the kernel of the representation $\pi$, and that if $g\ne1$ then there is an irreducible character $\chi$ with $\chi(g)\ne\chi(1)$.  
\vskip .3in
\noindent 3.  Let $G$ be a finite group and $H$ a subgroup such that $g^{-1}Hg\cap H=1$ for $g\notin H$.  Show that the set $N$ of all elements in $G$ not conjugate to any element of $H$, together with 1, is a normal subgroup of $G$, using the following steps.  First show that if $c_H$ is a class function on $H$ with $c_H(1)=0$, then the induced function $c_G=\,$Ind$_H^G c_H$ has the same square length as $c_H$.  Next let $\chi_H$ be an irreducible character of $H$ of degree $d=\chi(1)$ and let $c_\chi$ be the function induced from  $\chi_H-d$ to $G$.  Use Frobenius reciprocity to show that $c_\chi$ takes the form $\chi_G-d=\chi_G-\chi_G(1)$ for some irreducible character $\chi_G$ of $G$.  Letting $\chi_H$ run through the irreducible characters of $H$, show that $N$ is the set of common zeros of the $c_\chi$ in $G$ and thus the intersection of the kernels of the representations $\pi_G$ with character $\chi_G$.
\vskip .3in
\noindent 4. (20 points)  For any partition $\lambda$ of $n$, the construction of the Specht module $S^\lambda$ makes sense over any basefield $k$.  Show that if $k$ has characteristic 3 and $\lambda=(2,1)$, then $V=S^\lambda$ is {\sl not} irreducible over $S_3$, by working out the actions of a 3-cycle and a transposition explicitly on the basis of $V$ parametrized by standard tableaux of shape $\lambda$.
\end