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

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(Created page with "50px|left {{blockbox |title = M(16,2,2) - <math>k((C_4 \times C_4):C_3)</math> |image = M(4,2,3)quiver.png |representative = <math>k((C_4 \...")
 
 
(8 intermediate revisions by the same user not shown)
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[[Image:under-construction.png|50px|left]]
 
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{{blockbox
 
{{blockbox
 
|title = M(16,2,2) - <math>k((C_4 \times C_4):C_3)</math>  
 
|title = M(16,2,2) - <math>k((C_4 \times C_4):C_3)</math>  
 
|image = M(4,2,3)quiver.png
 
|image = M(4,2,3)quiver.png
 
|representative =  <math>k((C_4 \times C_4):C_3)</math>
 
|representative =  <math>k((C_4 \times C_4):C_3)</math>
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|defect = [[C2xC2|<math>C_4 \times C_4</math>]]
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|defect = [[C4xC4|<math>C_4 \times C_4</math>]]
 
|inertialquotients = <math>C_3</math>
 
|inertialquotients = <math>C_3</math>
 
|k(B) = 8
 
|k(B) = 8
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|Pic-k= <math></math>
 
|Pic-k= <math></math>
 
|cartan = <math>\left( \begin{array}{ccc}
 
|cartan = <math>\left( \begin{array}{ccc}
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2 & 1 & 1 \\
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6 & 5 & 5 \\
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1 & 2 & 1 \\
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5 & 6 & 5 \\
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1 & 1 & 2 \\
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5 & 5 & 6 \\
 
\end{array} \right)</math>
 
\end{array} \right)</math>
 
|defect-morita-inv? = Yes
 
|defect-morita-inv? = Yes
 
|inertial-morita-inv? = Yes
 
|inertial-morita-inv? = Yes
 
|O-morita? = Yes
 
|O-morita? = Yes
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|O-morita = <math>\mathcal{O}A_4</math>
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|O-morita = <math>\mathcal{O}((C_4 \times C_4):C_3)</math>
 
|decomp = <math>\left( \begin{array}{ccc}
 
|decomp = <math>\left( \begin{array}{ccc}
 
1 & 0 & 0 \\
 
1 & 0 & 0 \\
 
0 & 1 & 0 \\
 
0 & 1 & 0 \\
 
0 & 0 & 1 \\
 
0 & 0 & 1 \\
 +
1 & 1 & 1 \\
 +
1 & 1 & 1 \\
 +
1 & 1 & 1 \\
 +
1 & 1 & 1 \\
 
1 & 1 & 1 \\
 
1 & 1 & 1 \\
 
\end{array}\right)</math>
 
\end{array}\right)</math>
 
|O-morita-frob = 1
 
|O-morita-frob = 1
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|Pic-O = <math>\mathcal{T}(B)=S_3</math>
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|Pic-O = <math>C_2 \times S_3</math><ref>Proposition 4.3 of [[References|[BKL18]]]</ref>
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|source? = Yes
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|PIgroup = <math>S_3 \times D_8 \times C_2</math><ref>Using GAP, with code from [[References#R|[Ru11]]]</ref>
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|sourcereps = <math>\mathcal{O}A_4</math>
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|source? = No
 +
|sourcereps =
 
|k-derived-known? = Yes
 
|k-derived-known? = Yes
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|k-derived = [[M(4,2,2)]]
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|k-derived = Forms its own derived equivalence class
 
|O-derived-known? = Yes
 
|O-derived-known? = Yes
 +
|coveringblocks =
 +
|coveredblocks = [[M(16,2,1)]]
 +
|pcoveringblocks =
 
}}
 
}}
  
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'''Quiver:''' a:<1,2>, b:<2,3>, c:<3,1>, d:<2,1>, e:<3,2>, f:<1,3>
 
'''Quiver:''' a:<1,2>, b:<2,3>, c:<3,1>, d:<2,1>, e:<3,2>, f:<1,3>
  
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'''Relations w.r.t. <math>k</math>:''' ab=bc=ca=0, df=fe=ed=0, ad=fc, be=da, cf=eb
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'''Relations w.r.t. <math>k</math>:''' abca=bcab=cabc=0, dfed=fedf=edfe=0, ad=fc, be=da, cf=eb
  
 
== Other notatable representatives ==
 
== Other notatable representatives ==
−
 
−
Block number 2 of <math>k PSL_3(7)</math> in the labelling used in [http://www.math.rwth-aachen.de/~MOC/decomposition/tex/L3(7)/]
 
−
 
−
 
−
== Covering blocks and covered blocks ==
 
−
 
−
Let <math>N \triangleleft G</math> with <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>.
 
−
 
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If <math>b</math> lies in M(4,2,3), then <math>B</math> must lie in [[M(4,2,1)]] or M(4,2,3). For example consider blocks of <math>PSL_3(7) \triangleleft PGL_3(7)</math>.
 
−
 
−
If <math>B</math> lies in M(4,2,3), then <math>b</math> must lie in [[M(4,2,1)]] or M(4,2,3). For example consider the principal blocks of <math>O_2(A_4) \triangleleft A_4</math>.
 
  
 
== Projective indecomposable modules ==
 
== Projective indecomposable modules ==
  
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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}
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   \begin{array}{c}
−
     & S_1 & \\
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     1 \\
−
      S_2 & & S_3 \\
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    2 \ 3  \\
−
     & S_1 & \\
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    3 \ 1 \ 2 \\
 +
    1 \ 2 \ 3 \ 1 \\
 +
    3 \ 1 \ 2 \\
 +
    2 \ 3 \\
 +
     1 \\
 
   \end{array},
 
   \end{array},
 
&
 
&
−
   \begin{array}{ccc}
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   \begin{array}{c}
−
     & S_2 & \\
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     2 \\
−
      S_1 & & S_3 \\
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    1 \ 3  \\
−
     & S_2 & \\
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    3 \ 2 \ 1 \\
 +
    2 \ 1 \ 3 \ 2 \\
 +
    3 \ 2 \ 1 \\
 +
    1 \ 3 \\
 +
     2 \\
 
   \end{array},   
 
   \end{array},   
 
&  
 
&  
−
   \begin{array}{ccc}
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   \begin{array}{c}
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     & S_3 & \\
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     3 \\
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      S_1 & & S_2 \\
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    1 \ 2  \\
−
     & S_3 & \\
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    2 \ 3 \ 1 \\
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   \end{array}  
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    3 \ 1 \ 2 \ 3 \\
 +
    2 \ 3 \ 1 \\
 +
    1 \ 2 \\
 +
     2 \\
 +
   \end{array}
 
\end{array}
 
\end{array}
 
</math>
 
</math>
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All irreducible characters have height zero.
 
All irreducible characters have height zero.
  
 +
[[C4xC4|Back to <math>C_4 \times C_4</math>]]
 +
 +
== Notes ==
 +
<references />
  
−
[[C2xC2|Back to <math>C_2 \times C_2</math>]]
 
  
−
[[Category: Morita equivalence classes|4,2,3]]
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[[Category: Morita equivalence classes|16,2,2]]
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[[Category: Blocks with defect group C2xC2]]
+
[[Category: Blocks with defect group C4xC4]]
 
[[Category: Tame blocks|4,2,3]]
 
[[Category: Tame blocks|4,2,3]]

Latest revision as of 09:57, 28 July 2019

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]C_2 \times S_3[/math][1]
[math]PI(B)=[/math] [math]S_3 \times D_8 \times C_2[/math][2]
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:

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_4 \times C_4[/math]

Notes

  1. ↑ Proposition 4.3 of [BKL18]
  2. ↑ Using GAP, with code from [Ru11]
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