Difference between revisions of "M(32,51,1)"

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(Created page with "{{blockbox |title = M(32,51,1) - <math>k((C_2)^5)</math> |image =   |representative = <math>k((C_2)^5)</math> |defect = <math>(C_2)^5</math> |inertialquot...")
 
 
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|image = &nbsp;  
 
|image = &nbsp;  
 
|representative =  <math>k((C_2)^5)</math>
 
|representative =  <math>k((C_2)^5)</math>
|defect = [[(C2)%5E4|<math>(C_2)^5</math>]]
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|defect = [[(C2)%5E5|<math>(C_2)^5</math>]]
 
|inertialquotients = <math>1</math>
 
|inertialquotients = <math>1</math>
 
|k(B) = 32
 
|k(B) = 32

Latest revision as of 14:17, 5 December 2019

M(32,51,1) - [math]k((C_2)^5)[/math]
[[File: |250px]]
Representative: [math]k((C_2)^5)[/math]
Defect groups: [math](C_2)^5[/math]
Inertial quotients: [math]1[/math]
[math]k(B)=[/math] 32
[math]l(B)=[/math] 1
[math]{\rm mf}_k(B)=[/math] 1
[math]{\rm Pic}_k(B)=[/math]  
Cartan matrix: [math]\left( \begin{array}{c} 32 \end{array} \right)[/math]
Defect group Morita invariant? Yes
Inertial quotient Morita invariant? Yes
[math]\mathcal{O}[/math]-Morita classes known? Yes
[math]\mathcal{O}[/math]-Morita classes: [math]\mathcal{O} ((C_2)^5)[/math]
Decomposition matrices: [math]\left( \begin{array}{c} 1 \\ 1 \\ \vdots \\ 1 \end{array}\right)[/math]
[math]{\rm mf}_\mathcal{O}(B)=[/math] 1
[math]{\rm Pic}_{\mathcal{O}}(B)=[/math] [math](C_2)^5:GL_5(2)[/math]
[math]PI(B)=[/math]
Source algebras known? No
Source algebra reps:
[math]k[/math]-derived equiv. classes known? Yes
[math]k[/math]-derived equivalent to: Forms a derived equivalence class
[math]\mathcal{O}[/math]-derived equiv. classes known? Yes
[math]p'[/math]-index covering blocks:
[math]p'[/math]-index covered blocks:
Index [math]p[/math] covering blocks:

These are nilpotent blocks.

Basic algebra

Other notatable representatives

Covering blocks and covered blocks

Let [math]N \triangleleft G[/math] with [math]p'[/math]-index and let [math]B[/math] be a block of [math]\mathcal{O} G[/math] covering a block [math]b[/math] of [math]\mathcal{O} N[/math].

If [math]b[/math] is in M(32,51,1), then [math]B[/math] is in M(32,51,1), M(32,51,2), M(32,51,4), M(32,51,6), M(32,51,8), M(16,14,11), M(16,14,13), M(16,14,17), M(16,14,20),M(16,14,22),M(16,14,24) or M(16,14,30).

Projective indecomposable modules

Labelling the unique simple [math]B[/math]-module by [math]S_1[/math], the unique projective indecomposable module has Loewy structure as follows:

[math]\begin{array}{c} S_1 \\ S_1 S_1 S_1 S_1 S_1 \\ S_1 S_1 S_1 S_1 S_1 S_1 S_1 S_1 S_1 S_1 \\ S_1 S_1 S_1 S_1 S_1 S_1 S_1 S_1 S_1 S_1 \\ S_1 S_1 S_1 S_1 S_1 \\ S_1 \\ \end{array} [/math]

Irreducible characters

All irreducible characters have height zero.

Back to [math](C_2)^5[/math]