Difference between revisions of "M(8,5,7)"

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(Created page with "{{blockbox |title = M(8,5,7) - <math>B_0(kJ_1)</math> |image = M(8,5,7)quiver.png |representative = <math>B_0(kJ_1)</math> |defect = C2xC2xC2|<math>C_2 \times C_2 \times C...")
 
(Added PIMs)
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<math>a_5b_6c_1=b_5c_7a_4=c_5a_2b_3</math>
 
<math>a_5b_6c_1=b_5c_7a_4=c_5a_2b_3</math>
 
<math>a_6b_7c_2=b_6c_1a_5=c_6a_3b_4</math>
 
<math>a_6b_7c_2=b_6c_1a_5=c_6a_3b_4</math>
<math>a_7b_1c_3=b_7c_2a_6=c_7a_4b_5</math>
+
<math>a_7b_1c_3=b_7c_2a_6=c_7a_4b_5</math> -->
  
 
== Other notatable representatives ==
 
== Other notatable representatives ==
Line 86: Line 86:
 
== 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>1,2,3,4,5</math>, the projective indecomposable modules have Loewy structure as follows:
  
<math>\begin{array}{ccccccc}
+
<math>\begin{array}{ccccc}
 
   \begin{array}{c}
 
   \begin{array}{c}
     S_1 \\
+
     1 \\
     S_2 S_3 S_5 \\
+
     3 \ 4 \ 5 \\
     S_4 S_6 S_7 \\   
+
     1 \ 2 \ 1 \ 2 \ 1 \\   
     S_1 \\
+
    3 \ 3 \ 4 \ 4 \ 5 \ 5 \\
 +
    1 \ 2 \ 1 \ 2 \ 1 \\
 +
    3 \ 4 \ 5 \\
 +
     1 \\
 
   \end{array},
 
   \end{array},
 
&
 
&
   \begin{array}{ccc}
+
   \begin{array}{c}
     S_2 \\
+
     2 \\
     S_3 S_4 S_6 \\
+
     3 \ 4 \\
     S_1 S_5 S_7 \\   
+
     1 \ 2 \ 1 \\   
     S_2 \\  
+
    3 \ 4 \ 5 \\
 +
    1 \ 2 \ 1 \\
 +
    3 \ 4 \\
 +
     2 \\
 
   \end{array},   
 
   \end{array},   
 
&  
 
&  
   \begin{array}{ccc}
+
   \begin{array}{c}
     S_3 \\
+
     3 \\
     S_4 S_5 S_7 \\
+
     1 \ 2 \\
     S_1 S_2 S_6 \\   
+
     3 \ 4 \ 5 \\   
     S_3 \\  
+
    1 \ 2 \ 1 \\
 +
    3 \ 4 \ 5 \\
 +
    1 \ 2 \\
 +
     3 \\
 
   \end{array}
 
   \end{array}
 
,   
 
,   
 
&  
 
&  
   \begin{array}{ccc}
+
   \begin{array}{c}
     S_4 \\
+
     4 \\
     S_5 S_6 S_1 \\
+
     1 \ 2 \\
     S_2 S_3 S_7 \\
+
     3 \ 4 \ 5 \\   
    S_4 \\  
+
    1 \ 2 \ 1 \\  
  \end{array},  
+
     3 \ 4 \ 5 \\
&
+
     1 \ 2 \\
  \begin{array}{ccc}
+
     4 \\
    S_5 \\
 
    S_6 S_7 S_2 \\
 
     S_1 S_3 S_4 \\ 
 
    S_3 \\  
 
  \end{array}, 
 
&
 
  \begin{array}{ccc}
 
    S_6 \\
 
     S_7 S_1 S_3 \\
 
    S_2 S_4 S_5 \\
 
     S_6 \\  
 
 
   \end{array},   
 
   \end{array},   
 
&  
 
&  
 
   \begin{array}{ccc}
 
   \begin{array}{ccc}
     S_7 \\
+
     5 \\
     S_1 S_2 S_4 \\
+
     1 \ 5 \\
     S_3 S_5 S_6 \\   
+
     3 \ 4 \\   
     S_7 \\  
+
    1 \ 2 \ 1 \\
   \end{array}  
+
    3 \ 4 \ 5 \\
 +
    1 \\
 +
     5 \\
 +
   \end{array}  
 
\end{array}
 
\end{array}
</math> -->
+
</math>
  
 
== Irreducible characters ==
 
== Irreducible characters ==

Revision as of 17:38, 24 September 2018

M(8,5,7) - [math]B_0(kJ_1)[/math]
M(8,5,7)quiver.png
Representative: [math]B_0(kJ_1)[/math]
Defect groups: [math]C_2 \times C_2 \times C_2[/math]
Inertial quotients: [math]C_7:C_3[/math]
[math]k(B)=[/math] 8
[math]l(B)=[/math] 5
[math]{\rm mf}_k(B)=[/math] 1
[math]{\rm Pic}_k(B)=[/math]  
Cartan matrix: [math]\left( \begin{array}{ccccc} 8 & 4 & 4 & 4 & 4 \\ 4 & 4 & 3 & 3 & 1 \\ 4 & 3 & 4 & 2 & 2 \\ 4 & 3 & 2 & 4 & 2 \\ 4 & 1 & 2 & 2 & 4 \\ \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]B_0(\mathcal{O}J_1)[/math]
Decomposition matrices: [math]\left( \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 1 \\ 1 & 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 \\ 1 & 1 & 1 & 1 & 0 \\ 1 & 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 & 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]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: M(8,5,6), M(8,5,8)
[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: {{{pcoveringblocks}}}


Basic algebra

Other notatable representatives

Projective indecomposable modules

Labelling the simple [math]B[/math]-modules by [math]1,2,3,4,5[/math], the projective indecomposable modules have Loewy structure as follows:

[math]\begin{array}{ccccc} \begin{array}{c} 1 \\ 3 \ 4 \ 5 \\ 1 \ 2 \ 1 \ 2 \ 1 \\ 3 \ 3 \ 4 \ 4 \ 5 \ 5 \\ 1 \ 2 \ 1 \ 2 \ 1 \\ 3 \ 4 \ 5 \\ 1 \\ \end{array}, & \begin{array}{c} 2 \\ 3 \ 4 \\ 1 \ 2 \ 1 \\ 3 \ 4 \ 5 \\ 1 \ 2 \ 1 \\ 3 \ 4 \\ 2 \\ \end{array}, & \begin{array}{c} 3 \\ 1 \ 2 \\ 3 \ 4 \ 5 \\ 1 \ 2 \ 1 \\ 3 \ 4 \ 5 \\ 1 \ 2 \\ 3 \\ \end{array} , & \begin{array}{c} 4 \\ 1 \ 2 \\ 3 \ 4 \ 5 \\ 1 \ 2 \ 1 \\ 3 \ 4 \ 5 \\ 1 \ 2 \\ 4 \\ \end{array}, & \begin{array}{ccc} 5 \\ 1 \ 5 \\ 3 \ 4 \\ 1 \ 2 \ 1 \\ 3 \ 4 \ 5 \\ 1 \\ 5 \\ \end{array} \end{array} [/math]

Irreducible characters

All irreducible characters have height zero.

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