D8*C4 - Revision history

Charles Eaton at 11:47, 15 August 2019

2019-08-15T11:47:02Z

Charles Eaton at 10:40, 15 August 2019

2019-08-15T10:40:46Z

Charles Eaton at 10:32, 15 August 2019

2019-08-15T10:32:46Z

Charles Eaton: Created page with "NOTITLE == Blocks with defect group $D_8 * C_4$ == The invariants $k(B)$ , $k_i(B)$ and $l(B)$ are determined in References|..."

2018-11-27T13:41:08Z

Created page with "__NOTITLE__ == Blocks with defect group <math>D_8 * C_4</math> == The invariants <math>k(B)</math>, <math>k_i(B)</math> and <math>l(B)</math> are determined in References|..."

New page

__NOTITLE__

== Blocks with defect group <math>D_8 * C_4</math> ==

The invariants <math>k(B)</math>, <math>k_i(B)</math> and <math>l(B)</math> are determined in [[References|[Sa13b]]]. There is as yet no classification of blocks with these defect groups, and Donovan's conjecture is not known in any form.

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



== Notes ==

<references />

@@ Line 3: / Line 3: @@
 == Blocks with defect group <math>D_8 * C_4</math> ==
-The invariants <math>k(B)</math>, <math>k_i(B)</math> and <math>l(B)</math> are determined in [[References|[Sa13b]]]. There is precisely one saturated fusion system on <math>D_8 * C_4</math>. There is as yet no classification of blocks with these defect groups, and Donovan's conjecture is not known in any form.
+The invariants <math>k(B)</math>, <math>k_i(B)</math> and <math>l(B)</math> are determined in [[References|[Sa13b]]]. There is precisely one saturated fusion system on <math>D_8 * C_4 \cong Q_8*C_4</math>. There is as yet no classification of blocks with these defect groups, and Donovan's conjecture is not known in any form.
 '''<pre style="color: red">CLASSES NOT CLASSIFIED</pre>'''

@@ Line 3: / Line 3: @@
 == Blocks with defect group <math>D_8 * C_4</math> ==
-The invariants <math>k(B)</math>, <math>k_i(B)</math> and <math>l(B)</math> are determined in [[References|[Sa13b]]]. There is as yet no classification of blocks with these defect groups, and Donovan's conjecture is not known in any form.
+The invariants <math>k(B)</math>, <math>k_i(B)</math> and <math>l(B)</math> are determined in [[References|[Sa13b]]]. There is precisely one saturated fusion system on <math>D_8 * C_4</math>. There is as yet no classification of blocks with these defect groups, and Donovan's conjecture is not known in any form.
 '''<pre style="color: red">CLASSES NOT CLASSIFIED</pre>'''

@@ Line 5: / Line 5: @@
 The invariants <math>k(B)</math>, <math>k_i(B)</math> and <math>l(B)</math> are determined in [[References|[Sa13b]]]. There is as yet no classification of blocks with these defect groups, and Donovan's conjecture is not known in any form.
-'''<pre style="color: red">CLASSIFICATION INCOMPLETE</pre>'''
+'''<pre style="color: red">CLASSES NOT CLASSIFIED</pre>'''
-<!--{| class="wikitable"
 |-
 ! scope="col"| Class
@@ Line 22: / Line 23: @@
 |-
-|[[M(16,9,1)]] || <math>kSD_{16}</math> || 1 ||7 ||1 ||<math>1</math> || || || ||1 ||
+|[[M(16,13,1)]] || <math>k(Q_8*C_4)</math> || 1 ||10 ||1 ||<math>1</math> || || ||1 ||1 ||
 |-
-|[[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(16,13,2)]] || <math>B_0(k(SL_2(5)*C_4))</math> || ? ||14 ||3 ||<math>1</math> || || ||1 ||1 ||
 |-
-|[[M(16,9,6)]] || <math>B_0(kSL_2(7))</math> || 1 ||9 ||3 ||<math>1</math> || || || ||1 || <math>Q(3 {\cal K})</math>
+|[[M(16,13,3)]] || <math>k(SL_2(3)*C_4)</math> || ? ||14 ||3 ||<math>1</math> || || ||1 ||1 ||
 |}
-[[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).
+If <math>B</math> is not nilpotent, then <math>k_0(B)=8, k_1(B)=6, l(B)=3</math>.
 == Notes ==
 <references />

D8*C4 - Revision history

Charles Eaton at 11:47, 15 August 2019

Charles Eaton at 10:40, 15 August 2019

Charles Eaton at 10:32, 15 August 2019

Charles Eaton: Created page with "__NOTITLE__ == Blocks with defect group D_8 * C_4 == The invariants k(B), k_i(B) and l(B) are determined in References|..."

Charles Eaton: Created page with "NOTITLE == Blocks with defect group $D_8 * C_4$ == The invariants $k(B)$ , $k_i(B)$ and $l(B)$ are determined in References|..."