Blocks with basic algebras of low dimension

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.