Difference between revisions of "MNA(2,1)"

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([Sa16] reference)
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== Blocks with defect group <math>MNA(2,1)=\langle x,y|x^4=y^2=[x,y]^2=[x,[x,y]]=[y,[x,y]]=1 \rangle</math> ==
 
== Blocks with defect group <math>MNA(2,1)=\langle x,y|x^4=y^2=[x,y]^2=[x,[x,y]]=[y,[x,y]]=1 \rangle</math> ==
  
The defect groups are minimal nonabelian <math>2</math>-groups. The invariants <math>k(B)</math>, <math>l(B)</math> and <math>k_i(B)</math> for all <math>i</math> are determined in [[References#S|[Sa11]]]. The Cartan matrices are also determined up to equivalence of quadratic forms. These results do not rely on the [[Glossary#CFSG|CFSG]]. The automorphism group of <math>MNA(2,1)</math> is a <math>2</math>-group, but by [[References#S|[Sa14,12.7]]] there exists precisely one non-nilpotent fusion system for blocks with this defect group, realised in SmallGroup(48,30) <math>\cong A_4:C_4</math>.
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The defect groups are minimal nonabelian <math>2</math>-groups. The invariants <math>k(B)</math>, <math>l(B)</math> and <math>k_i(B)</math> for all <math>i</math> are determined in [[References#S|[Sa11]]]. The Cartan matrices are also determined up to equivalence of quadratic forms. These results do not rely on the [[Glossary#CFSG|CFSG]]. The automorphism group of <math>MNA(2,1)</math> is a <math>2</math>-group, but by [[References#S|[Sa14,12.7]]] there exists precisely one non-nilpotent fusion system for blocks with this defect group, realised in SmallGroup(48,30) <math>\cong A_4:C_4</math>. By [[References#S|[Sa16]]] all non-nilpotent blocks with this defect group are [[Glossary#Isotypy|isotypic]].  
  
 
'''<pre style="color: red">CLASSIFICATION INCOMPLETE</pre>'''
 
'''<pre style="color: red">CLASSIFICATION INCOMPLETE</pre>'''

Revision as of 22:17, 2 January 2019

Blocks with defect group [math]MNA(2,1)=\langle x,y|x^4=y^2=[x,y]^2=[x,[x,y]]=[y,[x,y]]=1 \rangle[/math]

The defect groups are minimal nonabelian [math]2[/math]-groups. The invariants [math]k(B)[/math], [math]l(B)[/math] and [math]k_i(B)[/math] for all [math]i[/math] are determined in [Sa11]. The Cartan matrices are also determined up to equivalence of quadratic forms. These results do not rely on the CFSG. The automorphism group of [math]MNA(2,1)[/math] is a [math]2[/math]-group, but by [Sa14,12.7] there exists precisely one non-nilpotent fusion system for blocks with this defect group, realised in SmallGroup(48,30) [math]\cong A_4:C_4[/math]. By [Sa16] all non-nilpotent blocks with this defect group are isotypic.

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(16,3,1) [math]k(MNA(2,1))[/math] 1 10 1 [math]1[/math] 1 1
M(16,3,2) [math]k(SmallGroup(48,30))[/math]  ? 10 2 [math]1[/math] 1 1