SD16 - Revision history

Charles Eaton: /* Blocks with defect group SD_{16} */ Added note about Monster group

2022-08-04T13:34:27Z

‎Blocks with defect group SD_{16}: Added note about Monster group

Charles Eaton: /* Blocks with defect group SD_{16} */ Removed M_{11}

2022-08-04T13:23:27Z

‎Blocks with defect group SD_{16}: Removed M_{11}

Charles Eaton: Updated following [Mac]

2022-08-04T13:20:25Z

Updated following [Mac]

Charles Eaton at 12:23, 16 November 2018

2018-11-16T12:23:03Z

Charles Eaton at 18:23, 14 November 2018

2018-11-14T18:23:50Z

Charles Eaton at 16:55, 14 November 2018

2018-11-14T16:55:33Z

Charles Eaton at 08:57, 6 November 2018

2018-11-06T08:57:02Z

Charles Eaton: Created page with "NOTITLE == Blocks with defect group $SD_{16}$ == left These are examples of tame blocks and were first..."

2018-11-05T22:26:27Z

Created page with "__NOTITLE__ == Blocks with defect group <math>SD_{16}</math> == 50px|left These are examples of tame blocks and were first..."

New page

__NOTITLE__

== Blocks with defect group <math>SD_{16}</math> ==

[[Image:under-construction.png|50px|left]]

These are examples of [[Tame blocks|tame blocks]] and were first classified over <math>k</math> by Erdmann (see [[References|[Er88c], [Er90b]]]). The classification with respect to <math>\mathcal{O}</math> is still unknown.

{| class="wikitable"
|-
! scope="col"| Class
! scope="col"| Representative
! scope="col"| # lifts / <math>\mathcal{O}</math>
! scope="col"| <math>k(B)</math>
! scope="col"| <math>l(B)</math>
! scope="col"| Inertial quotients
! scope="col"| <math>{\rm Pic}_\mathcal{O}(B)</math>
! scope="col"| <math>{\rm Pic}_k(B)</math>
! scope="col"| <math>{\rm mf_\mathcal{O}(B)}</math>
! scope="col"| <math>{\rm mf_k(B)}</math>
! scope="col"| 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> || ? ||7 ||2 ||<math>1</math> || || || ||1 || <math>SD(2 {\cal A})_1</math>
|-
|[[M(16,8,3)]] || || ? ||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(16,8,5)]] || || ? ||7 ||2 ||<math>1</math> || || || ||1 || <math>SD(2 {\cal B})_2</math>
|-
|[[M(16,8,6)]] || || ? ||7 ||3 ||<math>1</math> || || || ||1 || <math>SD(3 {\cal A})_1</math>
|-
|[[M(16,8,7)]] || || ? ||7 ||3 ||<math>1</math> || || || ||1 || <math>SD(3 {\cal B})_1</math>
|-
|[[M(16,8,8)]] || || ? ||7 ||3 ||<math>1</math> || || || ||1 || <math>SD(3 {\cal C})_2</math>
|-
|[[M(16,8,9)]] || || ? ||7 ||3 ||<math>1</math> || || || ||1 || <math>SD(3 {\cal D})</math>
|-
|[[M(16,8,10)]] || || ? ||7 ||3 ||<math>1</math> || || || ||1 || <math>SD(3 {\cal H})</math>
|}

@@ Line 42: / Line 42: @@
 | || || ? ||8 ||3 ||<math>1</math> || || || ||1 || <math>SD(3 {\cal B})_1</math>
 |-
-| || || ? ||8 ||3 ||<math>1</math> || || || ||1 || <math>SD(3 {\cal C})_2</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>
 |}
@@ Line 52: / Line 52: @@
 All Morita equivalence classes with three simple modules are derived equivalent over <math>k</math> by [[References|[Ho97] ]].
-<!--[[M(8,3,4)]], [[M(8,3,5)]] and [[M(8,3,6)]] are derived equivalent over <math>k</math> by [[References|[Li94b] ]].-->

@@ Line 38: / Line 38: @@
 |[[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>. Also <math>B_0(kM_{11})</math>. See note above.
+|[[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>

@@ Line 3: / Line 3: @@
 == 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]]]). It is not known (to the writer at least) whether there are blocks realising the classes <math>SD(3 {\cal C})_2</math> and <math>SD(3 {\cal H})</math>. Until this is resolved the labelling is provisional.
+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>.
 The classification with respect to <math>\mathcal{O}</math> is still unknown.
@@ Line 36: / Line 38: @@
 |[[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 B})_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>. Also <math>B_0(kM_{11})</math>. See note above.
 |-
-|[[M(16,8,8)]] || <math>B_0(kM_{11})</math> || ? ||8 ||3 ||<math>1</math> || || || ||1 || <math>SD(3 {\cal D})</math>
+| || || ? ||8 ||3 ||<math>1</math> || || || ||1 || <math>SD(3 {\cal B})_1</math>
 |-
 | || || ? ||8 ||3 ||<math>1</math> || || || ||1 || <math>SD(3 {\cal C})_2</math>
-|-
 |}

@@ Line 3: / Line 3: @@
 == 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]]]). It is not known (to me at least) whether there are blocks realising the classes M(16,8,8) and M(16,8,10).
+These are examples of [[Tame blocks|tame blocks]] and were first classified over <math>k</math> by Erdmann (see [[References|[Er88c], [Er90b]]]). It is not known (to the writer at least) whether there are blocks realising the classes <math>SD(3 {\cal C})_2</math> and <math>SD(3 {\cal H})</math>. Until this is resolved the labelling is provisional.
 The classification with respect to <math>\mathcal{O}</math> is still unknown.
@@ Line 38: / Line 38: @@
 |[[M(16,8,7)]] || <math>B_0(kPSL_3(3))</math> || ? ||8 ||3 ||<math>1</math> || || || ||1 || <math>SD(3 {\cal B})_1</math>
 |-
-|[[M(16,8,8)]] || || ? ||8 ||3 ||<math>1</math> || || || ||1 || <math>SD(3 {\cal C})_2</math>
+|[[M(16,8,8)]] || <math>B_0(kM_{11})</math> || ? ||8 ||3 ||<math>1</math> || || || ||1 || <math>SD(3 {\cal D})</math>
 |-
-|[[M(16,8,9)]] || <math>B_0(kM_{11})</math> || ? ||8 ||3 ||<math>1</math> || || || ||1 || <math>SD(3 {\cal D})</math>
+| || || ? ||8 ||3 ||<math>1</math> || || || ||1 || <math>SD(3 {\cal C})_2</math>
 |-
-|[[M(16,8,10)]] || || ? ||8 ||3 ||<math>1</math> || || || ||1 || <math>SD(3 {\cal H})</math>
+| || || ? ||8 ||3 ||<math>1</math> || || || ||1 || <math>SD(3 {\cal H})</math>
 |}

@@ Line 3: / Line 3: @@
 == Blocks with defect group <math>SD_{16}</math> ==
-[[Image:under-construction.png|50px|left]]
+The classification with respect to <math>\mathcal{O}</math> is still unknown.
 {| class="wikitable"
@@ Line 34: / Line 34: @@
 |[[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)]] || || ? ||8 ||3 ||<math>1</math> || || || ||1 || <math>SD(3 {\cal A})_1</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)]] || || ? ||8 ||3 ||<math>1</math> || || || ||1 || <math>SD(3 {\cal B})_1</math>
+|[[M(16,8,7)]] || <math>B_0(kPSL_3(3))</math> || ? ||8 ||3 ||<math>1</math> || || || ||1 || <math>SD(3 {\cal B})_1</math>
 |-
 |[[M(16,8,8)]] || || ? ||8 ||3 ||<math>1</math> || || || ||1 || <math>SD(3 {\cal C})_2</math>
 |-
-|[[M(16,8,9)]] || || ? ||8 ||3 ||<math>1</math> || || || ||1 || <math>SD(3 {\cal D})</math>
+|[[M(16,8,9)]] || <math>B_0(kM_{11})</math> || ? ||8 ||3 ||<math>1</math> || || || ||1 || <math>SD(3 {\cal D})</math>
 |-
 |[[M(16,8,10)]] || || ? ||8 ||3 ||<math>1</math> || || || ||1 || <math>SD(3 {\cal H})</math>
@@ Line 48: / Line 48: @@
 [[M(16,8,3)]] and [[M(16,8,4)]] are derived equivalent over <math>k</math> by [[References|[Ho97] ]].
 <!--[[M(8,3,4)]], [[M(8,3,5)]] and [[M(8,3,6)]] are derived equivalent over <math>k</math> by [[References|[Li94b] ]].-->

@@ Line 26: / Line 26: @@
 |[[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> || ? ||7 ||2 ||<math>1</math> || || || ||1 || <math>SD(2 {\cal A})_1</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)]] ||  || ? ||7 ||2 ||<math>1</math> || || || ||1 || <math>SD(2 {\cal A})_2</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)]] || || ? ||7 ||2 ||<math>1</math> || || || ||1 || <math>SD(2 {\cal B})_2</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)]] || || ? ||7 ||3 ||<math>1</math> || || || ||1 || <math>SD(3 {\cal A})_1</math>
+|[[M(16,8,6)]] || || ? ||8 ||3 ||<math>1</math> || || || ||1 || <math>SD(3 {\cal A})_1</math>
 |-
-|[[M(16,8,7)]] || || ? ||7 ||3 ||<math>1</math> || || || ||1 || <math>SD(3 {\cal B})_1</math>
+|[[M(16,8,7)]] || || ? ||8 ||3 ||<math>1</math> || || || ||1 || <math>SD(3 {\cal B})_1</math>
 |-
-|[[M(16,8,8)]] || || ? ||7 ||3 ||<math>1</math> || || || ||1 || <math>SD(3 {\cal C})_2</math>
+|[[M(16,8,8)]] || || ? ||8 ||3 ||<math>1</math> || || || ||1 || <math>SD(3 {\cal C})_2</math>
 |-
-|[[M(16,8,9)]] || || ? ||7 ||3 ||<math>1</math> || || || ||1 || <math>SD(3 {\cal D})</math>
+|[[M(16,8,9)]] || || ? ||8 ||3 ||<math>1</math> || || || ||1 || <math>SD(3 {\cal D})</math>
 |-
-|[[M(16,8,10)]] || || ? ||7 ||3 ||<math>1</math> || || || ||1 || <math>SD(3 {\cal H})</math>
+|[[M(16,8,10)]] || || ? ||8 ||3 ||<math>1</math> || || || ||1 || <math>SD(3 {\cal H})</math>
 |}
-<!-- [[M(8,3,2)]] and [[M(8,3,3)]] are derived equivalent over <math>k</math> by [[References|[Ho97] ]].
+[[M(16,8,2)]] and [[M(16,8,5)]] are derived equivalent over <math>k</math> by [[References|[Ho97] ]].
-[[M(8,3,4)]], [[M(8,3,5)]] and [[M(8,3,6)]] are derived equivalent over <math>k</math> by [[References|[Li94b] ]].-->
+[[M(16,8,3)]] and [[M(16,8,4)]] are derived equivalent over <math>k</math> by [[References|[Ho97] ]].