Difference between revisions of "SD16"
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== Blocks with defect group <math>SD_{16}</math> == | == Blocks with defect group <math>SD_{16}</math> == | ||
| + | These are examples of [[Tame blocks|tame blocks]] and were first classified over <math>k</math> by Erdmann (see [[References|[Er88c], [Er90b]]]). Further work was carried out in [[References#M|[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. | |
| + | '''<pre style="color: red">CLASSIFICATION INCOMPLETE</pre>''' | ||
{| class="wikitable" | {| class="wikitable" | ||
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|[[M(16,8,1)]] || <math>kSD_{16}</math> || 1 ||7 ||1 ||<math>1</math> || || || ||1 || | |[[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> || ? || | + | |[[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)]] || | + | |[[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)]] || || ? ||7 ||2 ||<math>1</math> || || || ||1 || <math>SD(2 {\cal B})_1</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)]] || || ? || | + | |[[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)]] || || ? || | + | |[[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)]] || || ? || | + | |[[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><ref>See discussion in [[References#M|[Mac]]]: A block of the Monster group could be in one or other of these classes.</ref> |
| − | + | ||
| − | |||
|} | |} | ||
| − | + | [[M(16,8,2)]] and [[M(16,8,5)]] are derived equivalent over <math>k</math> by [[References|[Ho97] ]]. | |
| + | |||
| + | [[M(16,8,3)]] and [[M(16,8,4)]] are derived equivalent over <math>k</math> by [[References|[Ho97] ]]. | ||
| + | |||
| + | All Morita equivalence classes with three simple modules are derived equivalent over <math>k</math> by [[References|[Ho97] ]]. | ||
| + | |||
| + | |||
| + | == Notes == | ||
| − | + | <references /> | |
Latest revision as of 13:34, 4 August 2022
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][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] .
