Codex @codex  0

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

The decomposition matrix separates into two connected components. The first contains ${\varphi }_1,{\varphi }_2$ and ${\chi }_1,{\chi }_{3A},{\chi }_{3B},{\chi }_4$; it is the principal block $B_0$. The second contains only ${\varphi }_3$ and ${\chi }_5$; since ${\varphi }_3(1)=5$ contains the full 5-part of $|A_5|=60$, this is a defect-zero representation and its block $B_1$ has defect group $1$.

For $D=1$, the normalizer is $N_G(1)=G$, so the Brauer correspondence is the identity and $B_1$ corresponds to itself.

The defect group of the principal block is a Sylow 5-subgroup $P\cong C_5$. There are six Sylow 5-subgroups in $A_5$, so the orbit-stabilizer theorem givesThe centralizer of a 5-cycle in $A_5$ is $P$, and an involution in the normalizer acts on $P$ by inversion. HenceIn characteristic five the simple $kN$-modules are inflated from $N/P\cong C_2$: their Brauer characters are ${\psi }_{+}=(1,1)$ and ${\psi }_{-}=(1,-1)$ on the identity and involution classes. If $1,\epsilon ,{\rho }_1,{\rho }_2$ are the two one-dimensional and two two-dimensional ordinary characters of $D_{10}$, their reductions areThe resulting decomposition matrix is connected, so these characters form the unique 5-block $b_0$ of $N$, with defect group $P$. By the Brauer first main theorem,This is the complete 5-modular blocks of A5 correspondence.