SD16

Blocks with defect group [math]SD_{16}[/math]

These are examples of tame blocks and were first classified over [math]k[/math] by Erdmann (see [Er88c], [Er90b]). Further work was carried out in [Mac], where [math]SD(3 {\cal H})[/math] was eliminated, and the block [math]B_0(kPSL_3(3))[/math] reattributed to [math]SD(3 {\cal D})[/math], so that the class [math]SD(3 {\cal B})_1[/math] is now a class of algebras with no known block-realised representative. It is not known whether there are blocks realising the class [math]SD(3 {\cal C})_2[/math].

Until these remaining cases are resolved the labelling is provisional.

The classification with respect to [math]\mathcal{O}[/math] is still unknown.

CLASSIFICATION INCOMPLETE

Class	Representative	# lifts / [math]\mathcal{O}[/math]	[math]k(B)[/math]	[math]l(B)[/math]	Inertial quotients	[math]{\rm Pic}_\mathcal{O}(B)[/math]	[math]{\rm Pic}_k(B)[/math]	[math]{\rm mf_\mathcal{O}(B)}[/math]	[math]{\rm mf_k(B)}[/math]	Notes
M(16,8,1)	[math]kSD_{16}[/math]	1	7	1	[math]1[/math]				1
M(16,8,2)	[math]B_5(kPSU_3(5))[/math]	?	8	2	[math]1[/math]				1	[math]SD(2 {\cal A})_1[/math]
M(16,8,3)	[math]B_0(kM_{10})=B_0(k(A_6.2_3))[/math]	?	7	2	[math]1[/math]				1	[math]SD(2 {\cal A})_2[/math]
M(16,8,4)	[math]B_3(k(3.M_{10}))=B_3(k(3.A_6.2_3))[/math]	?	7	2	[math]1[/math]				1	[math]SD(2 {\cal B})_1[/math]
M(16,8,5)	[math]B_1(kPSL_3(11))[/math]	?	8	2	[math]1[/math]				1	[math]SD(2 {\cal B})_2[/math]
M(16,8,6)	[math]B_0(kPSU_3(5))[/math]	?	8	3	[math]1[/math]				1	[math]SD(3 {\cal A})_1[/math]
M(16,8,7)	[math]B_0(kPSL_3(3))[/math]	?	8	3	[math]1[/math]				1	[math]SD(3 {\cal D})[/math]. See note above.
		?	8	3	[math]1[/math]				1	[math]SD(3 {\cal B})_1[/math]
		?	8	3	[math]1[/math]				1	[math]SD(3 {\cal B})_2[/math] or [math]SD(3 {\cal C})_2[/math]

M(16,8,2) and M(16,8,5) are derived equivalent over [math]k[/math] by [Ho97] .

M(16,8,3) and M(16,8,4) are derived equivalent over [math]k[/math] by [Ho97] .

All Morita equivalence classes with three simple modules are derived equivalent over [math]k[/math] by [Ho97] .