D32

Blocks with defect group [math]D_{32}[/math]

These are examples of tame blocks and were first classified over [math]k[/math] by Erdmann (see [Er87] ). The classification with respect to [math]\mathcal{O}[/math] is still unknown. Note that the class [math]D(3 {\cal B})_1[/math] is only realised for defect groups [math]D_8[/math] (see [Er87, Proposition 7.5.1]).

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(32,18,1)	[math]k(D_{32})[/math]	1	11	1	[math]1[/math]				1
M(32,18,2)	[math]B_0(kPGL_2(17))[/math]	?	11	2	[math]1[/math]				1	[math]D(2 {\cal A})[/math]
M(32,18,3)	[math]B_0(kPGL_2(47))[/math]	?	11	2	[math]1[/math]				1	[math]D(2 {\cal B})[/math]
M(32,18,4)	[math]B_0(kPSL_2(97))[/math]	?	11	3	[math]1[/math]				1	[math]D(3 {\cal A})_1[/math]
M(32,18,5)	[math]B_0(kPSL_2(31))[/math]	?	11	3	[math]1[/math]				1	[math]D(3 {\cal K})[/math]

M(32,18,2) and M(32,18,3) are derived equivalent over [math]k[/math] by [Ho97] .

M(32,18,4) and M(32,18,5) are derived equivalent over [math]k[/math] by [Li94b] .