Blocks with basic algebras of low dimension

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Blocks with basic algebras of dimension at most 12

In [Li18b] Markus Linckelmann calculated the [math]k[/math]-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 [math]\dim_k(Z(A))[/math] [math]l(A)[/math] Notes
1 M(1,1,1) [math]1[/math] [math]k1[/math] 1 1 Blocks of defect zero
2 M(2,1,1) [math]C_2[/math] [math]kC_2[/math] 2 1
3 M(3,1,1) [math]C_3[/math] [math]kC_3[/math] 3 1
4 M(4,1,1) [math]C_4[/math] [math]kC_4[/math] 4 1
4 M(4,2,1) [math]C_2 \times C_2[/math] [math]k(C_2 \times C_2)[/math] 4 1
5 M(5,1,1) [math]C_5[/math] [math]kC_5[/math] 5 1
6 M(3,1,2) [math]C_3[/math] [math]kS_3[/math] 3 2
7 M(5,1,3) [math]C_5[/math] [math]B_0(kA_5)[/math] 4 2
7 M(7,1,1) [math]C_7[/math] [math]kC_7[/math] 7 1
8 M(8,1,1) [math]C_8[/math] [math]kC_8[/math] 8 1
8 M(8,2,1) [math]C_4 \times C_2[/math] [math]k(C_4 \times C_2)[/math] 8 1
8 M(8,3,1) [math]D_8[/math] [math]kD_8[/math] 5 1
8 M(8,4,1) [math]Q_8[/math] [math]kQ_8[/math] 5 1
8 M(8,5,1) [math]C_2 \times C_2 \times C_2[/math] [math]k(C_2 \times C_2 \times C_2)[/math] 8 1
8 M(7,1,3) [math]C_7[/math] [math]B_0(kPSL_2(13))[/math] 5 2
9 M(9,1,1) [math]C_9[/math] [math]kC_9[/math] 9 1
9 M(9,1,3) [math]C_9[/math] [math]B_0(kSL_2(8))[/math] 6 2
9 M(9,2,1) [math]C_3 \times C_3[/math] [math]k(C_3 \times C_3)[/math] 9 1
9 M(9,2,23) [math]C_3 \times C_3[/math] Faithful block of [math]k((C_3 \times C_3):Q_8)[/math], in which [math]Z(Q_8)[/math] acts trivially 6 1 SmallGroup(72,24)
9 [math]C_3 \times C_3[/math]  ? 6 2 Unknown
10 M(5,1,2) [math]C_5[/math] [math]kD_{10}[/math] 4 2
10 M(11,1,3) [math]C_{11}[/math] [math]B_0(kSL_2(32))[/math] 7 2
11 M(8,3,3) [math]D_8[/math] [math]kS_4[/math] 5 2
11 M(7,1,6) [math]C_7[/math] [math]B_0(kA_7)[/math] 5 3
11 M(11,1,1) [math]C_{11}[/math] [math]kC_{11}[/math] 11 1
11 M(13,1,3) [math]C_{13}[/math] [math]B_0(kPSL_2(25))[/math] 8 2
12 M(4,2,3) [math]C_2 \times C_2[/math] [math]kA_4[/math] 4 3

The algebra [math]A[/math] of dimension 9 which may or may not be the basic algebra for a [math]k[/math]-block [math]B[/math] 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. [math]k[/math]: ab=c^3=d^2, cd=dc=0, ca=bc=da=bd=0

Cartan matrix: [math]\left( \begin{array}{cc} 5 & 1 \\ 1 & 2 \\ \end{array} \right)[/math]

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

Labelling the simple [math]B[/math]-modules by [math]S_1, S_2[/math], the projective indecomposable modules have Loewy structure as follows:

[math]\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} [/math]

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