Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 1 / 19F / b

The translation subgroup $U$ is normal and $G/U\cong {\mathbb{F}}_p^{\times }$, via the upper-left entry $a$. Thus the coset representation is the regular representation of that cyclic quotient. If $\chi$ runs through its $p-1$ complex one-dimensional characters, inflated to $G$, thenEach occurs once: the regular character is $p-1$ at the quotient identity and zero elsewhere, which is also the sum of all quotient characters by finite character orthogonality.

Write $g=(a,b)$ for the affine map $z\mapsto az+b$. Conjugation by $(a,0)$ sends translation by $b$ to translation by $ab$, so it permutes transitively the nontrivial characters of $U$. Since $U$ is normal, the induced character is zero outside $U$. On a translation by $b$ it is the sum of all conjugates of $\psi$, henceFor $b\ne 0$ the sum of the nontrivial additive characters is minus one, since the sum including the trivial character is zero. Its character norm isBy the irreducibility criterion for characters, $\rho ={\mathrm{Ind}}_U^G\psi$ is irreducible of dimension $p-1$. All choices of nontrivial $\psi$ give the same representation.

There are the $p-1$ inflated quotient characters of dimension one and the single representation $\rho$ of dimension $p-1$ from part (ii). These are distinct: for $p>2$ their dimensions distinguish $\rho$, and for $p=2$ its nontrivial value on $U$ distinguishes it from the trivial quotient character. Their dimension squares sum toThe sum-of-squares theorem for irreducible representations leaves no room for any further irreducible. This includes $p=2$, when $G\cong C_2$ and both representations have dimension one.