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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.

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][1]

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] .


  1. See discussion in [Mac]: A block of the Monster group could be in one or other of these classes.