Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 138 / 4 / c / Solution

Let $U\le {\mathrm{SL}}_2(p)$ be the Sylow p-subgroup of upper unitriangular matrices. It is cyclic of order $p$, generated byRealize $S^r(V_2)$ as the homogeneous polynomials of degree $r$ in $X,Y$, with $u$ acting by $X\mapsto X$ and $Y\mapsto X+Y$. For $0\le r<p$, the only vectors fixed by $u$ are the multiples of $X^r$: successive comparison of the coefficients of $Y^r,Y^{r-1},\dots$ proves this. Thus the nilpotent operator $u-1$ has one-dimensional kernel. Its Jordan normal form therefore has a single block, soThis also follows from the indecomposable modules of a cyclic p-group in characteristic p.

For $r=p-1$, the restriction has dimension $p$ and is the regular $kU$-module, hence is projective. Since $[G:U]$ is prime to $p$, part (b)(ii) makes $S^{p-1}(V_2)$ a simple projective $kG$-module. It is therefore a defect-zero representation and lifts to an ordinary irreducible representation of the same dimension. Consequently