Q16 - Revision history

Charles Eaton: Removed incorrect statement of Holm result.

2025-04-02T15:05:50Z

Removed incorrect statement of Holm result.

Charles Eaton: Q(2B)_1

2025-04-02T15:02:39Z

Q(2B)_1

Charles Eaton: /* Blocks with defect group Q_{16} */

2018-11-26T11:43:10Z

‎Blocks with defect group Q_{16}

Charles Eaton: Created page with "NOTITLE == Blocks with defect group $Q_{16}$ == These are examples of tame blocks and were first classified over $k$ by Erdmann (se..."

2018-11-26T08:56:27Z

Created page with "__NOTITLE__ == Blocks with defect group <math>Q_{16}</math> == These are examples of tame blocks and were first classified over <math>k</math> by Erdmann (se..."

New page

__NOTITLE__

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

These are examples of [[Tame blocks|tame blocks]] and were first classified over <math>k</math> by Erdmann (see [[References|[Er88a], [Er88b]]]) with some exceptions. It is not known which algebras in the infinite families <math>Q(2 {\cal A})</math> and <math>Q(2 {\cal B})_1</math> are realised by blocks, and as such Donovan's conjecture is still open for <math>Q_{16}</math> for blocks with two simple modules. Until this is resolved the labelling is provisional.

For blocks with three simple modules the <math>k</math>-Morita equivalence classes lift to unique <math>\mathcal{O}</math>-classes by [[References|[Ei16]]], but otherwise the classification with respect to <math>\mathcal{O}</math> is still unknown.

'''<pre style="color: red">CLASSIFICATION INCOMPLETE</pre>'''

{| 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(16,9,1)]] || <math>kSD_{16}</math> || 1 ||7 ||1 ||<math>1</math> || || || ||1 ||
|-
|[[M(16,9,2)]] || <math>B_0(k \tilde{S}_5)</math> || ? ||8 ||2 ||<math>1</math> || || || ||1 || <math>Q(2 {\cal A})</math>
|-
|[[M(16,9,3)]] || <math>B_0(k \tilde{S}_4)</math> || ? ||8 ||2 ||<math>1</math> || || || ||1 || <math>Q(2 {\cal B})_1</math>
|-
|[[M(16,9,4)]] || <math>B_0(kSL_2(9))</math> || 1 ||9 ||3 ||<math>1</math> || || || ||1 || <math>Q(3 {\cal A})_2</math>
|-
|[[M(16,9,5)]] || <math>B_0(k(2.A_7))</math> || 1 ||10 ||3 ||<math>1</math> || || || ||1 || <math>Q(3 {\cal B})</math>
|-
|[[M(16,9,6)]] || <math>B_0(kSL_2(7))</math> || 1 ||9 ||3 ||<math>1</math> || || || ||1 || <math>Q(3 {\cal K})</math>
|}

@@ Line 37: / Line 37: @@
 |}
-[[M(16,9,2)]] and [[M(16,9,3)]] are derived equivalent over <math>k</math> by [[References|[Ho97]]], in which it is further proved that ''all'' blocks with defect group <math>Q_{16}</math> and two simple modules are derived equivalent (irrespective of the unknown cases in the classification).
+[[M(16,9,2)]] and [[M(16,9,3)]] are derived equivalent over <math>k</math> by [[References|[Ho97]]].
 [[M(16,9,4)]], [[M(16,9,5)]] and [[M(16,9,6)]] are derived equivalent over <math>\mathcal{O}</math> by [[References|[Ei16]]]<ref>This result was obtained over <math>k</math> in [[References|[Ho97]]]</ref>.

@@ Line 3: / Line 3: @@
 == Blocks with defect group <math>Q_{16}</math> ==
-These are examples of [[Tame blocks|tame blocks]] and were first classified over <math>k</math> by Erdmann (see [[References|[Er88a], [Er88b]]]) with some exceptions. It is not known which algebras in the infinite families  <math>Q(2 {\cal A})</math> and <math>Q(2 {\cal B})_1</math> are realised by blocks, and as such Donovan's conjecture is still open for <math>Q_{16}</math> for blocks with two simple modules. 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|[Er88a], [Er88b]]]) with some exceptions.<ref>Note that as in [[References|[Ho97]]] the class <math>Q(2 {\cal B})_2</math> cannot be realised by a block.</ref> It is not known which algebras in the infinite families  <math>Q(2 {\cal A})</math> and <math>Q(2 {\cal B})_1</math> are realised by blocks, and as such Donovan's conjecture is still open for <math>Q_{16}</math> for blocks with two simple modules. Until this is resolved the labelling is provisional.
 For blocks with three simple modules the <math>k</math>-Morita equivalence classes lift to unique <math>\mathcal{O}</math>-classes by [[References|[Ei16]]], but otherwise the classification with respect to <math>\mathcal{O}</math> is still unknown.

@@ Line 26: / Line 26: @@
 |[[M(16,9,1)]] || <math>kSD_{16}</math> || 1 ||7 ||1 ||<math>1</math> || || || ||1 ||
 |-
-|[[M(16,9,2)]] || <math>B_0(k \tilde{S}_5)</math> || ? ||8 ||2 ||<math>1</math> || || || ||1 || <math>Q(2 {\cal A})</math>
+|[[M(16,9,2)]] || <math>B_0(k \tilde{S}_5)</math><ref>This is the double cover SmallGroup(240,89)</ref> || ? ||8 ||2 ||<math>1</math> || || || ||1 || <math>Q(2 {\cal A})</math>
 |-
-|[[M(16,9,3)]] || <math>B_0(k \tilde{S}_4)</math> || ? ||8 ||2 ||<math>1</math> || || || ||1 || <math>Q(2 {\cal B})_1</math>
+|[[M(16,9,3)]] || <math>B_0(k \tilde{S}_4)</math><ref>This is the double cover SmallGroup(48,28)</ref> || ? ||8 ||2 ||<math>1</math> || || || ||1 || <math>Q(2 {\cal B})_1</math>
 |-
 |[[M(16,9,4)]] || <math>B_0(kSL_2(9))</math> || 1 ||9 ||3 ||<math>1</math> || || || ||1 || <math>Q(3 {\cal A})_2</math>
@@ Line 37: / Line 37: @@
 |}
-<!--[[M(16,8,2)]] and [[M(16,8,5)]] are derived equivalent over <math>k</math> by [[References|[Ho97] ]].
+[[M(16,9,2)]] and [[M(16,9,3)]] are derived equivalent over <math>k</math> by [[References|[Ho97]]], in which it is further proved that ''all'' blocks with defect group <math>Q_{16}</math> and two simple modules are derived equivalent (irrespective of the unknown cases in the classification).
-[[M(16,8,3)]] and [[M(16,8,4)]] are derived equivalent over <math>k</math> by [[References|[Ho97] ]].
+== Notes ==
-All Morita equivalence classes with three simple modules are derived equivalent over <math>k</math> by [[References|[Ho97] ]].
+<references />

Q16 - Revision history

Charles Eaton: Removed incorrect statement of Holm result.

Charles Eaton: Q(2B)_1

Charles Eaton: /* Blocks with defect group Q_{16} */

Charles Eaton: Created page with "__NOTITLE__ == Blocks with defect group Q_{16} == These are examples of tame blocks and were first classified over k by Erdmann (se..."

Charles Eaton: Created page with "NOTITLE == Blocks with defect group $Q_{16}$ == These are examples of tame blocks and were first classified over $k$ by Erdmann (se..."