Difference between revisions of "M(32,51,5)"

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A block with defect group [[(C2)%5E5|<math>(C_2)^5</math>]] and inertial quotient <math>C_5</math> is Morita equivalent to M(32,51,5).  
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A block with defect group [[(C2)%5E5|<math>(C_2)^5</math>]] and inertial quotient <math>C_5</math> is in this Morita equivalence class.
  
 
It is unknown whether this Morita equivalence class contains blocks with inertial quotient <math>C_7:C_3</math> (with action as in [[M(32,51,20)]]).
 
It is unknown whether this Morita equivalence class contains blocks with inertial quotient <math>C_7:C_3</math> (with action as in [[M(32,51,20)]]).

Latest revision as of 15:54, 9 December 2019

M(32,51,5) - [math]k((C_2)^4 : C_5) \times C_2)[/math]
[[File: |250px]]
Representative: [math]k((C_2)^4 : C_5) \times C_2)[/math]
Defect groups: [math](C_2)^5[/math]
Inertial quotients: [math]C_5[/math], ?
[math]k(B)=[/math] 16
[math]l(B)=[/math] 5
[math]{\rm mf}_k(B)=[/math] 1
[math]{\rm Pic}_k(B)=[/math]  
Cartan matrix: [math]\left( \begin{array}{ccc} 8 & 6 & 6 & 6 & 6 \\ 6 & 8 & 6 & 6 & 6 \\ 6 & 6 & 8 & 6 & 6 \\ 6 & 6 & 6 & 8 & 6 \\ 6 & 6 & 6 & 6 & 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} ((C_2)^4 : C_5) \times C_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: 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:

A block with defect group [math](C_2)^5[/math] and inertial quotient [math]C_5[/math] is in this Morita equivalence class.

It is unknown whether this Morita equivalence class contains blocks with inertial quotient [math]C_7:C_3[/math] (with action as in M(32,51,20)).

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(32,51,5), then [math]B[/math] is in M(32,51,1), M(32,51,5), or M(32,51,11).

Projective indecomposable modules

Labelling the simple [math]B[/math]-modules by [math]S_1, S_2, S_3, S_4, S_5[/math], the projective indecomposable modules have Loewy structure as follows:

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

Irreducible characters

All irreducible characters have height zero.

Decomposition matrix

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

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