Difference between revisions of "D8*C4"

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== Blocks with defect group <math>D_8 * C_4</math> ==
 
== Blocks with defect group <math>D_8 * C_4</math> ==
  
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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.
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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.
  
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'''<pre style="color: red">CLASSIFICATION INCOMPLETE</pre>'''
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'''<pre style="color: red">CLASSES NOT CLASSIFIED</pre>'''
  
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<!--{| class="wikitable"
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{| class="wikitable"
 
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! scope="col"| Class
 
! scope="col"| Class
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|[[M(16,9,1)]] || <math>kSD_{16}</math> || 1 ||7 ||1 ||<math>1</math> || || || ||1 ||
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|[[M(16,13,1)]] || <math>k(Q_8*C_4)</math> || 1 ||10 ||1 ||<math>1</math> || || ||1 ||1 ||
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|[[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>
 
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|[[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>
 
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|[[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>
 
 
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|[[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>
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|[[M(16,13,2)]] || <math>B_0(k(SL_2(5)*C_4))</math> || ? ||14 ||3 ||<math>1</math> || || ||1 ||1 ||  
 
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|[[M(16,9,6)]] || <math>B_0(kSL_2(7))</math> || 1 ||9 ||3 ||<math>1</math> || || || ||1 || <math>Q(3 {\cal K})</math>
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|[[M(16,13,3)]] || <math>k(SL_2(3)*C_4)</math> || ? ||14 ||3 ||<math>1</math> || || ||1 ||1 ||  
 
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[[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).
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If <math>B</math> is not nilpotent, then <math>k_0(B)=8, k_1(B)=6, l(B)=3</math>.
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[[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>.-->
 
  
 
== Notes ==
 
== Notes ==
  
 
<references />
 
<references />

Latest revision as of 11:47, 15 August 2019

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 [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. 

CLASSES NOT CLASSIFIED


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(16,13,1) [math]k(Q_8*C_4)[/math] 1 10 1 [math]1[/math] 1 1
M(16,13,2) [math]B_0(k(SL_2(5)*C_4))[/math]  ? 14 3 [math]1[/math] 1 1
M(16,13,3) [math]k(SL_2(3)*C_4)[/math]  ? 14 3 [math]1[/math] 1 1

If [math]B[/math] is not nilpotent, then [math]k_0(B)=8, k_1(B)=6, l(B)=3[/math].

Notes

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