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M(32,51,30) - Revision history
2024-03-29T10:54:15Z
Revision history for this page on the wiki
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CesareGArdito at 13:48, 9 December 2019
2019-12-09T13:48:48Z
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<td colspan="2" style="background-color: white; color:black; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: white; color:black; text-align: center;">Revision as of 13:48, 9 December 2019</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>{{blockbox</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>{{blockbox</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>|title = M(32,51,<del class="diffchange diffchange-inline">11</del>) - <math>k(((C_2)^5 : (C_{31}:C_5))</math>  </div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>|title = M(32,51,<ins class="diffchange diffchange-inline">30</ins>) - <math>k(((C_2)^5 : (C_{31}:C_5))</math>  </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>|image = &nbsp;  </div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>|image = &nbsp;  </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>|representative =  <math>k(((C_2)^5 : (C_{31}:C_5))</math></div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>|representative =  <math>k(((C_2)^5 : (C_{31}:C_5))</math></div></td></tr>
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CesareGArdito
http://wiki.manchester.ac.uk/blocks/index.php?title=M(32,51,30)&diff=1052&oldid=prev
CesareGArdito: Created page with "{{blockbox |title = M(32,51,11) - <math>k(((C_2)^5 : (C_{31}:C_5))</math> |image = |representative = <math>k(((C_2)^5 : (C_{31}:C_5))</math> |defect = (C2)%5E5|<ma..."
2019-12-09T13:48:00Z
<p>Created page with "{{blockbox |title = M(32,51,11) - <math>k(((C_2)^5 : (C_{31}:C_5))</math> |image = |representative = <math>k(((C_2)^5 : (C_{31}:C_5))</math> |defect = (C2)%5E5|<ma..."</p>
<p><b>New page</b></p><div>{{blockbox<br />
|title = M(32,51,11) - <math>k(((C_2)^5 : (C_{31}:C_5))</math> <br />
|image = &nbsp; <br />
|representative = <math>k(((C_2)^5 : (C_{31}:C_5))</math><br />
|defect = [[(C2)%5E5|<math>(C_2)^5</math>]]<br />
|inertialquotients = <math>C_{31}:C_5</math><br />
|k(B) = 16<br />
|l(B) = 11<br />
|k-morita-frob = 1 <br />
|Pic-k= &nbsp;<br />
|cartan = See below.<br />
|defect-morita-inv? = Yes<br />
|inertial-morita-inv? = Yes<br />
|O-morita? = Yes<br />
|O-morita = <math>\mathcal{O}(((C_2)^5 : (C_{31}:C_5))</math><br />
|decomp = See below.<br />
|O-morita-frob = 1<br />
|Pic-O = <br />
|PIgroup = <br />
|source? = No<br />
|sourcereps =<br />
|k-derived-known? = Yes<br />
|k-derived = [[M(32,51,31)]]<br />
|O-derived-known? = Yes<br />
|coveringblocks =<br />
|coveredblocks =<br />
|pcoveringblocks =<br />
}}<br />
<br />
<br />
== Basic algebra ==<br />
<br />
== Other notatable representatives ==<br />
<br />
== Covering blocks and covered blocks ==<br />
<br />
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>.<br />
<br />
If <math>b</math> is in M(32,51,30), then <math>B</math> is in [[M(32,51,22)]] or M(32,51,30).<br />
<br />
== Projective indecomposable modules ==<br />
<br />
Labelling the simple <math>B</math>-modules by <math>S_1, \dots, S_{11}</math>, the projective indecomposable modules have Loewy structure as follows:<br />
<br />
<math>\begin{array}{ccccc}<br />
\begin{array}{c}<br />
S_{1} \\<br />
S_{7} \\<br />
S_{9} S_{11} \\<br />
S_{8} S_{10} \\<br />
S_{6} \\<br />
S_{1} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{2} \\<br />
S_{7} \\<br />
S_{11} S_{9} \\<br />
S_{8} S_{10} \\<br />
S_{6} \\<br />
S_{2} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{3} \\<br />
S_{7} \\<br />
S_{9} S_{11} \\<br />
S_{10} S_{8} \\<br />
S_{6} \\<br />
S_{3} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{4} \\<br />
S_{7} \\<br />
S_{11} S_{9} \\<br />
S_{10} S_{8} \\<br />
S_{6} \\<br />
S_{4} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{5} \\<br />
S_{7} \\<br />
S_{9} S_{11} \\<br />
S_{8} S_{10} \\<br />
S_{6} \\<br />
S_{5} \\<br />
\end{array}<br />
\end{array}<br />
</math><br />
<br />
<br>&nbsp; <br><br />
<br />
<math>\begin{array}{ccccc}<br />
\begin{array}{c}<br />
S_{6} \\<br />
S_{5} S_{4} S_{1} S_{2} S_{3} S_{9} S_{6} S_{8} S_{7} \\<br />
S_{11} S_{9} S_{7} S_{7} S_{8} S_{10} S_{11} S_{6} S_{7} S_{7} \\<br />
S_{6} S_{8} S_{11} S_{10} S_{10} S_{9} S_{11} S_{11} S_{9} S_{9} \\<br />
S_{8} S_{8} S_{6} S_{10} S_{10} \\<br />
S_{6} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{7} \\<br />
S_{9} S_{11} S_{9} S_{11} S_{7} \\<br />
S_{7} S_{11} S_{11} S_{8} S_{10} S_{9} S_{10} S_{10} S_{8} S_{8} \\<br />
S_{6} S_{7} S_{6} S_{9} S_{6} S_{8} S_{10} S_{10} S_{11} S_{6} \\<br />
S_{5} S_{1} S_{2} S_{4} S_{3} S_{6} S_{9} S_{7} S_{8} \\<br />
S_{7} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{8} \\<br />
S_{10} S_{6} S_{7} S_{6} S_{11} \\<br />
S_{3} S_{5} S_{4} S_{2} S_{1} S_{11} S_{6} S_{10} S_{9} S_{7} S_{8} S_{8} S_{9} S_{9} \\<br />
S_{8} S_{6} S_{11} S_{11} S_{7} S_{7} S_{8} S_{7} S_{10} S_{10} \\<br />
S_{6} S_{9} S_{11} S_{9} S_{10} \\<br />
S_{8} \\<br />
\end{array}<br />
\end{array}<br />
</math><br />
<br />
<br>&nbsp; <br><br />
<br />
<math>\begin{array}{ccccc}<br />
\begin{array}{c}<br />
S_{9} \\<br />
S_{7} S_{8} S_{10} S_{11} S_{8} \\<br />
S_{9} S_{11} S_{7} S_{6} S_{11} S_{9} S_{6} S_{6} S_{10} S_{10} \\<br />
S_{3} S_{4} S_{2} S_{5} S_{1} S_{8} S_{6} S_{8} S_{8} S_{10} S_{7} S_{11} S_{9} S_{9} \\<br />
S_{11} S_{7} S_{7} S_{10} S_{6} \\<br />
S_{9} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{10} \\<br />
S_{9} S_{10} S_{6} S_{6} S_{8} \\<br />
S_{4} S_{2} S_{3} S_{5} S_{1} S_{9} S_{11} S_{6} S_{6} S_{10} S_{7} S_{7} S_{8} S_{8} \\<br />
S_{7} S_{6} S_{8} S_{9} S_{11} S_{10} S_{9} S_{11} S_{7} S_{7} \\<br />
S_{10} S_{11} S_{11} S_{9} S_{8} \\<br />
S_{10} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{11} \\<br />
S_{10} S_{8} S_{9} S_{10} S_{11} \\<br />
S_{6} S_{10} S_{10} S_{11} S_{6} S_{8} S_{7} S_{9} S_{8} S_{6} \\<br />
S_{5} S_{3} S_{1} S_{4} S_{2} S_{9} S_{6} S_{6} S_{10} S_{8} S_{11} S_{9} S_{7} S_{7} \\<br />
S_{8} S_{11} S_{7} S_{7} S_{9} \\<br />
S_{11} \\<br />
\end{array}<br />
\end{array}<br />
</math><br />
<br />
== Irreducible characters ==<br />
<br />
All irreducible characters have height zero.<br />
<br />
== Cartan matrix ==<br />
<math>\left( \begin{array}{ccc}<br />
2 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 \\<br />
0 & 2 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 \\<br />
0 & 0 & 2 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 \\<br />
0 & 0 & 0 & 2 & 0 & 1 & 1 & 1 & 1 & 1 & 1 \\<br />
0 & 0 & 0 & 0 & 2 & 1 & 1 & 1 & 1 & 1 & 1 \\<br />
1 & 1 & 1 & 1 & 1 & 6 & 5 & 5 & 5 & 5 & 5 \\<br />
1 & 1 & 1 & 1 & 1 & 5 & 6 & 5 & 5 & 5 & 5 \\<br />
1 & 1 & 1 & 1 & 1 & 5 & 5 & 6 & 5 & 5 & 5 \\<br />
1 & 1 & 1 & 1 & 1 & 5 & 5 & 5 & 6 & 5 & 5 \\<br />
1 & 1 & 1 & 1 & 1 & 5 & 5 & 5 & 5 & 6 & 5 \\<br />
1 & 1 & 1 & 1 & 1 & 5 & 5 & 5 & 5 & 5 & 6 <br />
\end{array} \right)</math><br />
<br />
== Decomposition matrix ==<br />
<br />
<math>\left( \begin{array}{ccc}<br />
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\<br />
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\<br />
1 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 \\<br />
0 & 1 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 \\<br />
0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\<br />
0 & 0 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 \\<br />
0 & 0 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 <br />
\end{array}\right)</math><br />
<br />
[[(C2)%5E5|Back to <math>(C_2)^5</math>]]</div>
CesareGArdito