Difference between revisions of "M(32,51,2)"

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(Covering blocks and covered blocks)
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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>.
 
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(32,51,2), then <math>B</math> is in [[M(32,51,1)]], M(32,51,2), [[M(32,51,8)]], [[M(31,51,13]] or [[M(32,51,32)]].
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If <math>b</math> is in M(32,51,2), then <math>B</math> is in [[M(32,51,1)]], M(32,51,2), [[M(32,51,8)]], [[M(31,51,13)]] or [[M(32,51,32)]].
  
 
== Projective indecomposable modules ==
 
== Projective indecomposable modules ==

Revision as of 14:25, 5 December 2019

M(32,51,1) - [math]k(A_4 \times (C_2)^3)[/math]
[[File: |250px]]
Representative: [math]k(A_4 \times (C_2)^3)[/math]
Defect groups: [math](C_2)^5[/math]
Inertial quotients: [math]C_3[/math]
[math]k(B)=[/math] 32
[math]l(B)=[/math] 3
[math]{\rm mf}_k(B)=[/math] 1
[math]{\rm Pic}_k(B)=[/math]  
Cartan matrix: [math]\left( \begin{array}{ccc} 16 & 8 & 8\\ 8 & 16 & 8\\ 8 & 8 & 16 \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} (A_4 \times (C_2)^3)[/math]
Decomposition matrices: See below.
[math]{\rm mf}_\mathcal{O}(B)=[/math] 1
[math]{\rm Pic}_{\mathcal{O}}(B)=[/math] [math]S_3 \times (C_2)^3 : GL_3(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: M(32,51,3)
[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(32,51,2), then [math]B[/math] is in M(32,51,1), M(32,51,2), M(32,51,8), M(31,51,13) or M(32,51,32).

Projective indecomposable modules

Labelling the simple [math]B[/math]-modules by [math]S_1, S_2, S_3[/math], the projective indecomposable modules have Loewy structure as follows:

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

Irreducible characters

All irreducible characters have height zero.

Decomposition matrix

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

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