CesareGArdito: Created page with "{{blockbox |title = M(32,51,23) - $B_0(k(SL_2(32)))$ |image = |representative = $B_0(k(SL_2(32)))$ |defect = $(C_2)^5$ ..."

2019-12-09T12:47:48Z

Created page with "{{blockbox |title = M(32,51,23) - <math>B_0(k(SL_2(32)))</math> |image = |representative = <math>B_0(k(SL_2(32)))</math> |defect = <math>(C_2)^5</math>..."

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{{blockbox
|title = M(32,51,23) - <math>B_0(k(SL_2(32)))</math>
|image =  
|representative = <math>B_0(k(SL_2(32)))</math>
|defect = [[(C2)%5E5|<math>(C_2)^5</math>]]
|inertialquotients = <math>C_{31}</math>
|k(B) = 32
|l(B) = 31
|k-morita-frob = 1
|Pic-k=  
|cartan = See below.
|defect-morita-inv? = Yes
|inertial-morita-inv? = Yes
|O-morita? = Yes
|O-morita = <math>B_0(\mathcal{O}(SL_2(32)))</math>
|decomp = See below.
|O-morita-frob = 1
|Pic-O =
|PIgroup =
|source? = No
|sourcereps =
|k-derived-known? = Yes
|k-derived = [[M(32,51,22)]]
|O-derived-known? = Yes
|coveringblocks =
|coveredblocks =
|pcoveringblocks =
}}

== 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(32,51,23), then <math>B</math> is in M(32,51,23) or [[M(32,51,31)]].

== Projective indecomposable modules ==

== Irreducible characters ==

All irreducible characters have height zero.

== Cartan matrix ==
<math>\left( \begin{array}{ccccccccccccccccccccccccccccccc}
32 & 16 & 16 & 16 & 16 & 16 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 4 & 4 & 4 & 4 & 4 & 4 & 4 & 4 & 4 & 4 & 2 & 2 & 2 & 2 & 2 \\
16 & 16 & 8 & 8 & 8 & 8 & 4 & 4 & 8 & 4 & 4 & 8 & 4 & 4 & 8 & 0 & 0 & 4 & 2 & 4 & 2 & 2 & 4 & 0 & 0 & 2 & 0 & 1 & 2 & 0 & 0 \\
16 & 8 & 16 & 8 & 8 & 8 & 4 & 4 & 8 & 0 & 8 & 4 & 8 & 4 & 4 & 4 & 2 & 4 & 4 & 0 & 2 & 0 & 2 & 2 & 4 & 0 & 1 & 0 & 0 & 2 & 0 \\
16 & 8 & 8 & 16 & 8 & 8 & 8 & 4 & 4 & 8 & 4 & 8 & 4 & 0 & 4 & 4 & 4 & 2 & 2 & 4 & 0 & 4 & 0 & 2 & 2 & 0 & 0 & 0 & 0 & 1 & 2 \\
16 & 8 & 8 & 8 & 16 & 8 & 4 & 8 & 4 & 4 & 8 & 4 & 4 & 8 & 0 & 4 & 2 & 0 & 4 & 2 & 4 & 2 & 0 & 0 & 2 & 4 & 0 & 2 & 0 & 0 & 1 \\
16 & 8 & 8 & 8 & 8 & 16 & 8 & 8 & 4 & 4 & 4 & 4 & 0 & 4 & 4 & 8 & 4 & 2 & 0 & 2 & 4 & 0 & 2 & 4 & 0 & 2 & 2 & 0 & 1 & 0 & 0 \\
8 & 4 & 4 & 8 & 4 & 8 & 8 & 4 & 2 & 4 & 2 & 4 & 0 & 0 & 2 & 4 & 4 & 1 & 0 & 2 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
8 & 4 & 4 & 4 & 8 & 8 & 4 & 8 & 2 & 2 & 4 & 2 & 0 & 4 & 0 & 4 & 2 & 0 & 0 & 1 & 4 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 \\
8 & 8 & 8 & 4 & 4 & 4 & 2 & 2 & 8 & 0 & 4 & 4 & 4 & 2 & 4 & 0 & 0 & 4 & 2 & 0 & 1 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
8 & 4 & 0 & 8 & 4 & 4 & 4 & 2 & 0 & 8 & 0 & 4 & 0 & 0 & 2 & 2 & 2 & 0 & 0 & 4 & 0 & 4 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 2 \\
8 & 4 & 8 & 4 & 8 & 4 & 2 & 4 & 4 & 0 & 8 & 2 & 4 & 4 & 0 & 2 & 1 & 0 & 4 & 0 & 2 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 \\
8 & 8 & 4 & 8 & 4 & 4 & 4 & 2 & 4 & 4 & 2 & 8 & 2 & 0 & 4 & 0 & 0 & 2 & 1 & 4 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
8 & 4 & 8 & 4 & 4 & 0 & 0 & 0 & 4 & 0 & 4 & 2 & 8 & 2 & 2 & 0 & 0 & 2 & 4 & 0 & 0 & 0 & 1 & 0 & 4 & 0 & 0 & 0 & 0 & 2 & 0 \\
8 & 4 & 4 & 0 & 8 & 4 & 0 & 4 & 2 & 0 & 4 & 0 & 2 & 8 & 0 & 2 & 0 & 0 & 2 & 0 & 4 & 0 & 0 & 0 & 1 & 4 & 0 & 2 & 0 & 0 & 0 \\
8 & 8 & 4 & 4 & 0 & 4 & 2 & 0 & 4 & 2 & 0 & 4 & 2 & 0 & 8 & 0 & 0 & 4 & 0 & 2 & 0 & 1 & 4 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 \\
8 & 0 & 4 & 4 & 4 & 8 & 4 & 4 & 0 & 2 & 2 & 0 & 0 & 2 & 0 & 8 & 4 & 0 & 0 & 0 & 2 & 0 & 0 & 4 & 0 & 1 & 2 & 0 & 0 & 0 & 0 \\
4 & 0 & 2 & 4 & 2 & 4 & 4 & 2 & 0 & 2 & 1 & 0 & 0 & 0 & 0 & 4 & 4 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
4 & 4 & 4 & 2 & 0 & 2 & 1 & 0 & 4 & 0 & 0 & 2 & 2 & 0 & 4 & 0 & 0 & 4 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
4 & 2 & 4 & 2 & 4 & 0 & 0 & 0 & 2 & 0 & 4 & 1 & 4 & 2 & 0 & 0 & 0 & 0 & 4 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 \\
4 & 4 & 0 & 4 & 2 & 2 & 2 & 1 & 0 & 4 & 0 & 4 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 4 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
4 & 2 & 2 & 0 & 4 & 4 & 0 & 4 & 1 & 0 & 2 & 0 & 0 & 4 & 0 & 2 & 0 & 0 & 0 & 0 & 4 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 \\
4 & 2 & 0 & 4 & 2 & 0 & 0 & 0 & 0 & 4 & 0 & 2 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 2 & 0 & 4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 \\
4 & 4 & 2 & 0 & 0 & 2 & 0 & 0 & 2 & 0 & 0 & 0 & 1 & 0 & 4 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 4 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 \\
4 & 0 & 2 & 2 & 0 & 4 & 2 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 4 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 0 & 0 & 2 & 0 & 0 & 0 & 0 \\
4 & 0 & 4 & 2 & 2 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 4 & 1 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 4 & 0 & 0 & 0 & 0 & 2 & 0 \\
4 & 2 & 0 & 0 & 4 & 2 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 4 & 0 & 1 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 4 & 0 & 2 & 0 & 0 & 0 \\
2 & 0 & 1 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 2 & 0 & 0 & 0 & 0 \\
2 & 1 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 2 & 0 & 0 & 0 \\
2 & 2 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 \\
2 & 0 & 2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 2 & 0 \\
2 & 0 & 0 & 2 & 1 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2
\end{array} \right)</math>

== Decomposition matrix ==

<math>\left( \begin{array}{ccccccccccccccccccccccccccccccc}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 1 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
1 & 1 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 1 & 1 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
1 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
1 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 0 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 1 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
1 & 0 & 1 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
1 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0
\end{array}\right)</math>

[[(C2)%5E5|Back to <math>(C_2)^5</math>]]

M(32,51,23) - Revision history

CesareGArdito: Created page with "{{blockbox |title = M(32,51,23) - B_0(k(SL_2(32))) |image = |representative = B_0(k(SL_2(32))) |defect = (C_2)^5..."

CesareGArdito: Created page with "{{blockbox |title = M(32,51,23) - $B_0(k(SL_2(32)))$ |image = |representative = $B_0(k(SL_2(32)))$ |defect = $(C_2)^5$ ..."