Difference between revisions of "M(16,2,2)"

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(Projective indecomposable modules)
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== Projective indecomposable modules ==
 
== 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:
+
Labelling the simple <math>B</math>-modules by <math>1,2,3</math>, the projective indecomposable modules have Loewy structure as follows:
  
 
<math>\begin{array}{ccc}
 
<math>\begin{array}{ccc}
   \begin{array}{ccc}
+
   \begin{array}{c}
     & S_1 & \\
+
     1 \\
      S_2 & & S_3 \\
+
    2 \ 3  \\
     & S_1 & \\
+
    3 \ 1 \ 2 \\
 +
    1 \ 2 \ 3 \ 1 \\
 +
    3 \ 1 \ 2 \\
 +
    2 \ 3 \\
 +
     1 \\
 
   \end{array},
 
   \end{array},
 
&
 
&
   \begin{array}{ccc}
+
   \begin{array}{c}
     & S_2 & \\
+
     2 \\
      S_1 & & S_3 \\
+
    1 \ 3  \\
     & S_2 & \\
+
    3 \ 2 \ 1 \\
 +
    2 \ 1 \ 3 \ 2 \\
 +
    3 \ 2 \ 1 \\
 +
    1 \ 3 \\
 +
     2 \\
 
   \end{array},   
 
   \end{array},   
 
&  
 
&  
   \begin{array}{ccc}
+
   \begin{array}{c}
     & S_3 & \\
+
     3 \\
      S_1 & & S_2 \\
+
    1 \ 2  \\
     & S_3 & \\
+
    2 \ 3 \ 1 \\
   \end{array}  
+
    3 \ 1 \ 2 \ 3 \\
 +
    2 \ 3 \ 1 \\
 +
    1 \ 2 \\
 +
     2 \\
 +
   \end{array}
 
\end{array}
 
\end{array}
</math> -->
+
</math>
  
 
== Irreducible characters ==
 
== Irreducible characters ==

Revision as of 08:42, 4 October 2018

Under-construction.png
M(16,2,2) - [math]k((C_4 \times C_4):C_3)[/math]
M(4,2,3)quiver.png
Representative: [math]k((C_4 \times C_4):C_3)[/math]
Defect groups: [math]C_4 \times C_4[/math]
Inertial quotients: [math]C_3[/math]
[math]k(B)=[/math] 8
[math]l(B)=[/math] 3
[math]{\rm mf}_k(B)=[/math] 1
[math]{\rm Pic}_k(B)=[/math] [math][/math]
Cartan matrix: [math]\left( \begin{array}{ccc} 6 & 5 & 5 \\ 5 & 6 & 5 \\ 5 & 5 & 6 \\ \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_4 \times C_4):C_3)[/math]
Decomposition matrices: [math]\left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ \end{array}\right)[/math]
[math]{\rm mf}_\mathcal{O}(B)=[/math] 1
[math]{\rm Pic}_{\mathcal{O}}(B)=[/math] [math]S_3[/math]
[math]PI(B)=[/math] {{{PIgroup}}}
Source algebras known? No
Source algebra reps:
[math]k[/math]-derived equiv. classes known? Yes
[math]k[/math]-derived equivalent to: Forms its own derived equivalence class
[math]\mathcal{O}[/math]-derived equiv. classes known? Yes
[math]p'[/math]-index covering blocks:
[math]p'[/math]-index covered blocks: M(16,2,1)
Index [math]p[/math] covering blocks: {{{pcoveringblocks}}}

Basic algebra

Quiver: a:<1,2>, b:<2,3>, c:<3,1>, d:<2,1>, e:<3,2>, f:<1,3>

Relations w.r.t. [math]k[/math]: abca=bcab=cabc=0, dfed=fedf=edfe=0, ad=fc, be=da, cf=eb

Other notatable representatives

Projective indecomposable modules

Labelling the simple [math]B[/math]-modules by [math]1,2,3[/math], the projective indecomposable modules have Loewy structure as follows:

[math]\begin{array}{ccc} \begin{array}{c} 1 \\ 2 \ 3 \\ 3 \ 1 \ 2 \\ 1 \ 2 \ 3 \ 1 \\ 3 \ 1 \ 2 \\ 2 \ 3 \\ 1 \\ \end{array}, & \begin{array}{c} 2 \\ 1 \ 3 \\ 3 \ 2 \ 1 \\ 2 \ 1 \ 3 \ 2 \\ 3 \ 2 \ 1 \\ 1 \ 3 \\ 2 \\ \end{array}, & \begin{array}{c} 3 \\ 1 \ 2 \\ 2 \ 3 \ 1 \\ 3 \ 1 \ 2 \ 3 \\ 2 \ 3 \ 1 \\ 1 \ 2 \\ 2 \\ \end{array} \end{array} [/math]

Irreducible characters

All irreducible characters have height zero.


Back to [math]C_2 \times C_2[/math]