Blocks with basic algebras of low dimension

Blocks with basic algebras of dimension at most 12

In [Li18b] Markus Linckelmann calculated the $k$-algebras of dimension at most twelve which occur as basic algebras of blocks of finite groups, with the exception of one case of dimension 9 where no block with that basic algebra is identified.

Dimension Class Defect group Representative $\dim_k(Z(A))$ $l(A)$ Notes
1 M(1,1,1) $1$ $k1$ 1 1 Blocks of defect zero
2 M(2,1,1) $C_2$ $kC_2$ 2 1
3 M(3,1,1) $C_3$ $kC_3$ 3 1
4 M(4,1,1) $C_4$ $kC_4$ 4 1
4 M(4,2,1) $C_2 \times C_2$ $k(C_2 \times C_2)$ 4 1
5 M(5,1,1) $C_5$ $kC_5$ 5 1
6 M(3,1,2) $C_3$ $kS_3$ 3 2
7 M(5,1,3) $C_5$ $B_0(kA_5)$ 4 2
7 M(7,1,1) $C_7$ $kC_7$ 7 1
8 M(8,1,1) $C_8$ $kC_8$ 8 1
8 M(8,2,1) $C_4 \times C_2$ $k(C_4 \times C_2)$ 8 1
8 M(8,3,1) $D_8$ $kD_8$ 5 1
8 M(8,4,1) $Q_8$ $kQ_8$ 5 1
8 M(8,5,1) $C_2 \times C_2 \times C_2$ $k(C_2 \times C_2 \times C_2)$ 8 1
8 M(7,1,3) $C_7$ $B_0(kPSL_2(13))$ 5 2
9 M(9,1,1) $C_9$ $kC_9$ 9 1
9 M(9,1,3) $C_9$ $B_0(kSL_2(8))$ 6 2
9 M(9,2,1) $C_3 \times C_3$ $k(C_3 \times C_3)$ 9 1
9 M(9,2,23) $C_3 \times C_3$ Faithful block of $k((C_3 \times C_3):Q_8)$, in which $Z(Q_8)$ acts trivially 6 1 SmallGroup(72,24)
9 $C_3 \times C_3$  ? 6 2 Unknown
10 M(5,1,2) $C_5$ $kD_{10}$ 4 2
10 M(11,1,3) $C_{11}$ $B_0(kSL_2(32))$ 7 2
11 M(8,3,3) $D_8$ $kS_4$ 5 2
11 M(7,1,6) $C_7$ $B_0(kA_7)$ 5 3
11 M(11,1,1) $C_{11}$ $kC_{11}$ 11 1
11 M(13,1,3) $C_{13}$ $B_0(kPSL_2(25))$ 8 2
12 M(4,2,3) $C_2 \times C_2$ $kA_4$ 4 3

The algebra $A$ of dimension 9 which may or may not be the basic algebra for a $k$-block $B$ of a finite group has the following structure.

Quiver: a:<1,2>, b:<2,1>, c:<1,1>, d:<1,1>

Relations w.r.t. $k$: ab=c^3=d^2, cd=dc=0, ca=bc=da=bd=0

Cartan matrix: $\left( \begin{array}{cc} 5 & 1 \\ 1 & 2 \\ \end{array} \right)$

An $\mathcal{O}$-block corresponding to $B$ must have decomposition matrix $\left( \begin{array}{cc} 1 & 0 \\ 1 & 0 \\ 1 & 0 \\ 1 & 0 \\ 0 & 1 \\ 1 & 1 \\ \end{array}\right)$

Labelling the simple $B$-modules by $S_1, S_2$, the projective indecomposable modules have Loewy structure as follows:

$\begin{array}{cc} \begin{array}{ccc} & S_1 & \\ S_2 & \begin{array}{c} S_1 \\ S_1 \\ \end{array} & S_1 \\ & S_1 & \\ \end{array} , & \begin{array}{c} S_2 \\ S_1 \\ S_2 \\ \end{array} \end{array}$

By [Ki84] the inertial quotient of $B$ must either be $C_2$ (acting with no non-trivial fixed points) or $D_8$ (with a non-trivial 2-cocyle). By [Ko03] $B$ cannot be Morita equivalent to a principal block.