Difference between revisions of "M(32,51,7)"

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(Created page with "{{blockbox |title = M(32,51,7) - <math>B_0(k(\SL_2(8) \times (C_2)^2))</math> |image =   |representative = <math>B_0(k(\SL_2(8) \times (C_2)^2))</math> |defect = (C2...")
 
 
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{{blockbox
 
{{blockbox
|title = M(32,51,7) - <math>B_0(k(\SL_2(8) \times (C_2)^2))</math>  
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|title = M(32,51,7) - <math>B_0(k(SL_2(8) \times (C_2)^2))</math>  
 
|image = &nbsp;  
 
|image = &nbsp;  
|representative =  <math>B_0(k(\SL_2(8) \times (C_2)^2))</math>
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|representative =  <math>B_0(k(SL_2(8) \times (C_2)^2))</math>
 
|defect = [[(C2)%5E5|<math>(C_2)^5</math>]]
 
|defect = [[(C2)%5E5|<math>(C_2)^5</math>]]
 
|inertialquotients = <math>C_7</math>
 
|inertialquotients = <math>C_7</math>
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|inertial-morita-inv? = Yes
 
|inertial-morita-inv? = Yes
 
|O-morita? = Yes
 
|O-morita? = Yes
|O-morita = <math>\mathcal{O} B_0(\mathcal{O}(\SL_2(8) \times (C_2)^2))</math>
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|O-morita = <math>\mathcal{O} B_0(\mathcal{O}(SL_2(8) \times (C_2)^2))</math>
 
|decomp = See below.
 
|decomp = See below.
 
|O-morita-frob = 1
 
|O-morita-frob = 1

Latest revision as of 12:14, 6 December 2019

M(32,51,7) - [math]B_0(k(SL_2(8) \times (C_2)^2))[/math]
[[File: |250px]]
Representative: [math]B_0(k(SL_2(8) \times (C_2)^2))[/math]
Defect groups: [math](C_2)^5[/math]
Inertial quotients: [math]C_7[/math]
[math]k(B)=[/math] 32
[math]l(B)=[/math] 7
[math]{\rm mf}_k(B)=[/math] 1
[math]{\rm Pic}_k(B)=[/math]  
Cartan matrix: [math]\left( \begin{array}{ccccccc} 32 & 16 & 16 & 16 & 8 & 8 & 8 \\ 16 & 16 & 8 & 8 & 0 & 4 & 8 \\ 16 & 8 & 16 & 8 & 8 & 0 & 4 \\ 16 & 8 & 8 & 16 & 4 & 8 & 0 \\ 8 & 0 & 8 & 4 & 8 & 0 & 0 \\ 8 & 4 & 0 & 8 & 0 & 8 & 0 \\ 8 & 8 & 4 & 0 & 0 & 0 & 8 \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} B_0(\mathcal{O}(SL_2(8) \times (C_2)^2))[/math]
Decomposition matrices: See below.
[math]{\rm mf}_\mathcal{O}(B)=[/math] 1
[math]{\rm Pic}_{\mathcal{O}}(B)=[/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: M(32,51,6)
[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:


Basic algebra

Other notatable representatives

Covering blocks and covered blocks

Let [math]N \triangleleft G[/math] with prime [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(32,51,6), then [math]B[/math] is in M(32,51,7), M(32,51,15), M(31,51,19) or M(32,51,21).

Projective indecomposable modules

Labelling the simple [math]B[/math]-modules by [math]S_1, \dots, S_7[/math], the projective indecomposable modules have Loewy structure as follows:

[math]\begin{array}{cccc} \begin{array}{c} S_1 \\ S_1 S_1 S_3 S_2 S_4 \\ S_1 S_1 S_1 S_1 S_4 S_3 S_2 S_3 S_4 S_2 S_5 S_6 S_7 \\ S_1 S_1 S_1 S_1 S_1 S_1 S_2 S_2 S_4 S_3 S_2 S_4 S_3 S_4 S_3 S_7 S_5 S_6 S_5 S_6 S_7 \\ S_1 S_1 S_1 S_1 S_1 S_1 S_2 S_3 S_4 S_3 S_2 S_4 S_3 S_4 S_2 S_2 S_4 S_3 S_5 S_5 S_7 S_7 S_6 S_6 \\ S_1 S_1 S_1 S_1 S_1 S_1 S_3 S_4 S_4 S_2 S_4 S_2 S_3 S_3 S_2 S_7 S_6 S_5 S_7 S_6 S_5 \\ S_1 S_1 S_1 S_1 S_3 S_2 S_3 S_2 S_4 S_4 S_6 S_5 S_7 \\ S_1 S_1 S_2 S_3 S_4 \\ S_1 \\ \end{array} & \begin{array}{c} S_2 \\ S_1 S_2 S_2 S_5 \\ S_1 S_1 S_3 S_2 S_2 S_4 S_5 S_5 \\ S_1 S_1 S_1 S_4 S_3 S_3 S_4 S_2 S_2 S_5 S_7 \\ S_1 S_1 S_1 S_1 S_2 S_4 S_2 S_4 S_3 S_3 S_7 S_7 \\ S_1 S_1 S_1 S_2 S_3 S_4 S_3 S_2 S_4 S_5 S_7 \\ S_1 S_1 S_4 S_2 S_2 S_3 S_5 S_5 \\ S_1 S_2 S_2 S_5 \\ S_2 \\ \end{array} & \begin{array}{c} S_3 \\ S_1 S_3 S_3 S_6 \\ S_1 S_1 S_2 S_4 S_3 S_3 S_6 S_6 \\ S_1 S_1 S_1 S_2 S_4 S_3 S_2 S_4 S_3 S_6 S_5 \\ S_1 S_1 S_1 S_1 S_4 S_2 S_3 S_3 S_2 S_4 S_5 S_5 \\ S_1 S_1 S_1 S_2 S_4 S_2 S_4 S_3 S_3 S_5 S_6 \\ S_1 S_1 S_4 S_2 S_3 S_3 S_6 S_6 \\ S_1 S_3 S_3 S_6 \\ S_3 \\ \end{array} \end{array}[/math]


 

[math] \begin{array}{ccc} \begin{array}{c} S_4 \\ S_1 S_4 S_4 S_7 \\ S_1 S_1 S_2 S_4 S_4 S_3 S_7 S_7 \\ S_1 S_1 S_1 S_3 S_3 S_2 S_2 S_4 S_4 S_7 S_6 \\ S_1 S_1 S_1 S_1 S_3 S_2 S_2 S_4 S_3 S_4 S_6 S_6 \\ S_1 S_1 S_1 S_4 S_2 S_3 S_3 S_2 S_4 S_7 S_6 \\ S_1 S_1 S_3 S_4 S_2 S_4 S_7 S_7 \\ S_1 S_4 S_4 S_7 \\ S_4 \\ \end{array} & \begin{array}{c} S_5 \\ S_2 S_5 S_5 \\ S_1 S_2 S_2 S_5 \\ S_1 S_1 S_3 S_2 \\ S_1 S_1 S_3 S_3 \\ S_1 S_1 S_2 S_3 \\ S_1 S_2 S_2 S_5 \\ S_2 S_5 S_5 \\ S_5 \\ \end{array} & \begin{array}{c} S_6 \\ S_3 S_6 S_6 \\ S_1 S_3 S_3 S_6 \\ S_1 S_1 S_4 S_3 \\ S_1 S_1 S_4 S_4 \\ S_1 S_1 S_3 S_4 \\ S_1 S_3 S_3 S_6 \\ S_3 S_6 S_6 \\ S_6 \\ \end{array} & \begin{array}{c} S_7 \\ S_4 S_7 S_7 \\ S_1 S_4 S_4 S_7 \\ S_1 S_1 S_2 S_4 \\ S_1 S_1 S_2 S_2 \\ S_1 S_1 S_4 S_2 \\ S_1 S_4 S_4 S_7 \\ S_4 S_7 S_7 \\ S_7 \\ \end{array} \end{array} [/math]

Irreducible characters

All irreducible characters have height zero.

Decomposition matrix

[math]\left( \begin{array}{ccccccc} 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 & 0 & 1 & 0 \\ 1 & 0 & 1 & 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 1 \\ 1 & 1 & 0 & 0 & 0 & 0 & 1 \\ 1 & 0 & 1 & 0 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 1 \\ 1 & 0 & 1 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 1 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 1 & 0 & 1 & 0 \\ 1 & 0 & 1 & 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 & 1 & 0 \\ 1 & 1 & 0 & 1 & 0 & 1 & 0 \\ 1 & 0 & 1 & 1 & 1 & 0 & 0 \\ 1 & 0 & 1 & 1 & 1 & 0 & 0 \\ 1 & 0 & 1 & 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 & 1 & 0 \\ 1 & 1 & 0 & 1 & 0 & 1 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 1 \\ 1 & 1 & 1 & 0 & 0 & 0 & 1 \\ 1 & 1 & 1 & 0 & 0 & 0 & 1 \\ 1 & 1 & 1 & 0 & 0 & 0 & 1 \end{array}\right)[/math]

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