Blocks with basic algebras of low dimension

Blocks with basic algebras of dimension at most 16

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 was identified. This final case was ruled out by Linckelmann and Murphy in [LM20]. Using the classification of finite simple groups, the basic algebras of dimension 13 or 14 for blocks of finite groups were calculated by Sambale in [Sa20]. Later Benson and Sambale in [BS23] gave a classification for dimensions 15 and 16, except for one unsettled case of a block with defect group [math]C_{13}[/math] in dimension 15.

The results are incorporated into the table below.

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)
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
13	M(16,7,3)	[math]D_{16}[/math]	[math]B_0(kPGL_2(7))[/math]	7	2
13	M(16,8,4)	[math]SD_{16}[/math]	[math]B_3(k(3.M_{10}))[/math]	7	2
13	M(7,1,7)	[math]C_7[/math]	[math]B_{15}(k6.A_7)[/math]	5	3
13	M(13,1,1)	[math]C_{13}[/math]	[math]kC_{13}[/math]	13	1
13	M(13,1,?)	[math]C_{13}[/math]	[math]B_0(kPSL_3(3))[/math]	7	3
13	M(17,1,?)	[math]C_{17}[/math]	[math]B_0(kPSL_2(16))[/math]	10	2
14	M(5,1,5)	[math]C_5[/math]	[math]B_0(kS_5)[/math]	5	4
14	M(7,1,2)	[math]C_7[/math]	[math]kD_{14}[/math]	5	2
14	M(7,1,5)	[math]C_7[/math]	[math]B_0(kPSL_3(3))[/math]	5	3
14	M(19,1,?)	[math]C_{19}[/math]	[math]B_0(kPSL_2(37))[/math]	11	2
15	M(19,1,?)	[math]C_{19}[/math]	[math]B_0(kGL_3(7))[/math]
15		[math]C_{13}[/math]	??