http://wiki.manchester.ac.uk/blocks/index.php?action=history&feed=atom&title=M%2832%2C51%2C19%29
M(32,51,19) - Revision history
2026-05-06T16:15:05Z
Revision history for this page on the wiki
MediaWiki 1.30.1
http://wiki.manchester.ac.uk/blocks/index.php?title=M(32,51,19)&diff=1035&oldid=prev
CesareGArdito at 18:49, 8 December 2019
2019-12-08T18:49:44Z
<p></p>
<table class="diff diff-contentalign-left" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
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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 18:49, 8 December 2019</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l19" >Line 19:</td>
<td colspan="2" class="diff-lineno">Line 19:</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>|inertial-morita-inv? = Yes</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>|inertial-morita-inv? = Yes</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>|O-morita? = Yes</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>|O-morita? = Yes</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>|O-morita = <math>B_0(\<del class="diffchange diffchange-inline">cO</del>({\rm Aut}(SL_2(8)) \times (C_2)^2))</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>|O-morita = <math>B_0(\<ins class="diffchange diffchange-inline">mathcal{O}</ins>({\rm Aut}(SL_2(8)) \times (C_2)^2))</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>|decomp = See below.</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>|decomp = See below.</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>|O-morita-frob = 1</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>|O-morita-frob = 1</div></td></tr>
</table>
CesareGArdito
http://wiki.manchester.ac.uk/blocks/index.php?title=M(32,51,19)&diff=1033&oldid=prev
CesareGArdito: /* Other notatable representatives */
2019-12-08T18:49:05Z
<p><span dir="auto"><span class="autocomment">Other notatable representatives</span></span></p>
<table class="diff diff-contentalign-left" data-mw="interface">
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<col class="diff-content" />
<tr style="vertical-align: top;" lang="en-GB">
<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 18:49, 8 December 2019</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l38" >Line 38:</td>
<td colspan="2" class="diff-lineno">Line 38:</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;"></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;"></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>== Other notatable representatives ==</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>== Other notatable representatives ==</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><math>B_0(k((C_2)^2 \times {}^2G_2(q)))</math>, and the unique nonprincipal block with defect group <del class="diffchange diffchange-inline">$</del>(C_2)^5<del class="diffchange diffchange-inline">$ </del>of <del class="diffchange diffchange-inline">$</del>(C_2)^2 \times \operatorname{Co}_3<del class="diffchange diffchange-inline">$</del>.</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><math>B_0(k((C_2)^2 \times {}^2G_2(q)))</math>, and the unique nonprincipal block with defect group <ins class="diffchange diffchange-inline"><math></ins>(C_2)^5<ins class="diffchange diffchange-inline"></math> </ins>of <ins class="diffchange diffchange-inline"><math></ins>(C_2)^2 \times \operatorname{Co}_3<ins class="diffchange diffchange-inline"></math></ins>.</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> </div></td><td colspan="2"> </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;"></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;"></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>== Covering blocks and covered blocks ==</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>== Covering blocks and covered blocks ==</div></td></tr>
<!-- diff cache key wki_blocks:diff:version:1.11a:oldid:1032:newid:1033 -->
</table>
CesareGArdito
http://wiki.manchester.ac.uk/blocks/index.php?title=M(32,51,19)&diff=1032&oldid=prev
CesareGArdito: Created page with "{{blockbox |title = M(32,51,19) - <math>B_0(k({\rm Aut}(SL_2(8)) \times (C_2)^2))</math> |image = |representative = <math>B_0(k({\rm Aut}(SL_2(8)) \times (C_2)^2))</..."
2019-12-08T18:48:41Z
<p>Created page with "{{blockbox |title = M(32,51,19) - <math>B_0(k({\rm Aut}(SL_2(8)) \times (C_2)^2))</math> |image = |representative = <math>B_0(k({\rm Aut}(SL_2(8)) \times (C_2)^2))</..."</p>
<p><b>New page</b></p><div>{{blockbox<br />
|title = M(32,51,19) - <math>B_0(k({\rm Aut}(SL_2(8)) \times (C_2)^2))</math> <br />
|image = &nbsp; <br />
|representative = <math>B_0(k({\rm Aut}(SL_2(8)) \times (C_2)^2))</math><br />
|defect = [[(C2)%5E5|<math>(C_2)^5</math>]]<br />
|inertialquotients = <math>C_7:C_3</math><br />
|k(B) = 32<br />
|l(B) = 5<br />
|k-morita-frob = 1 <br />
|Pic-k= &nbsp;<br />
|cartan = <math>\left( \begin{array}{ccccc}<br />
16 & 8 & 8 & 16 & 8 \\<br />
8 & 16 & 8 & 16 & 8 \\<br />
8 & 8 & 16 & 16 & 8 \\<br />
16 & 16 & 16 & 32 & 12 \\<br />
8 & 8 & 8 & 12 & 8<br />
\end{array} \right)</math><br />
|defect-morita-inv? = Yes<br />
|inertial-morita-inv? = Yes<br />
|O-morita? = Yes<br />
|O-morita = <math>B_0(\cO({\rm Aut}(SL_2(8)) \times (C_2)^2))</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(32,51,17)]], [[M(32,51,18)]]<br />
|O-derived-known? = Yes<br />
|coveringblocks =<br />
|coveredblocks =<br />
|pcoveringblocks =<br />
}}<br />
<br />
<br />
== Basic algebra ==<br />
<br />
== Other notatable representatives ==<br />
<math>B_0(k((C_2)^2 \times {}^2G_2(q)))</math>, and the unique nonprincipal block with defect group $(C_2)^5$ of $(C_2)^2 \times \operatorname{Co}_3$.<br />
<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(32,51,19), then <math>B</math> is in [[M(32,51,7)]], M(32,51,19), [[M(32,51,28)]] or [[M(32,51,34)]].<br />
<br />
== Projective indecomposable modules ==<br />
<br />
Labelling the simple <math>B</math>-modules by <math>S_1, \dots, S_5</math>, the projective indecomposable modules have Loewy structure as follows:<br />
<br />
<math>\begin{array}{ccc}<br />
\begin{array}{c}<br />
S_{1} \\<br />
S_{1} S_{1} S_{4} \\<br />
S_{3} S_{1} S_{1} S_{2} S_{4} S_{4} S_{5} \\<br />
S_{3} S_{2} S_{1} S_{2} S_{1} S_{3} S_{4} S_{4} S_{4} S_{5} S_{5} \\<br />
S_{3} S_{1} S_{2} S_{1} S_{2} S_{3} S_{4} S_{4} S_{4} S_{4} S_{5} S_{5} \\<br />
S_{2} S_{3} S_{1} S_{1} S_{2} S_{3} S_{4} S_{4} S_{4} S_{5} S_{5} \\<br />
S_{1} S_{3} S_{2} S_{1} S_{4} S_{4} S_{5} \\<br />
S_{1} S_{1} S_{4} \\<br />
S_{1} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{2} \\<br />
S_{2} S_{2} S_{4} \\<br />
S_{2} S_{3} S_{2} S_{1} S_{4} S_{4} S_{5} \\<br />
S_{1} S_{3} S_{3} S_{2} S_{2} S_{1} S_{4} S_{4} S_{4} S_{5} S_{5} \\<br />
S_{2} S_{1} S_{3} S_{2} S_{3} S_{1} S_{4} S_{4} S_{4} S_{4} S_{5} S_{5} \\<br />
S_{2} S_{1} S_{1} S_{3} S_{3} S_{2} S_{4} S_{4} S_{4} S_{5} S_{5} \\<br />
S_{2} S_{2} S_{3} S_{1} S_{4} S_{4} S_{5} \\<br />
S_{2} S_{2} S_{4} \\<br />
S_{2} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{3} \\<br />
S_{3} S_{3} S_{4} \\<br />
S_{1} S_{3} S_{3} S_{2} S_{4} S_{4} S_{5} \\<br />
S_{1} S_{3} S_{2} S_{2} S_{1} S_{3} S_{4} S_{4} S_{4} S_{5} S_{5} \\<br />
S_{3} S_{2} S_{1} S_{2} S_{1} S_{3} S_{4} S_{4} S_{4} S_{4} S_{5} S_{5} \\<br />
S_{1} S_{2} S_{2} S_{3} S_{3} S_{1} S_{4} S_{4} S_{4} S_{5} S_{5} \\<br />
S_{2} S_{1} S_{3} S_{3} S_{4} S_{4} S_{5} \\<br />
S_{3} S_{3} S_{4} \\<br />
S_{3} \\<br />
\end{array}<br />
\end{array}</math><br />
<br />
<br>&nbsp;<br><br />
<br />
<math><br />
\begin{array}{cc}<br />
\begin{array}{c}<br />
S_{4} \\<br />
S_{2} S_{1} S_{3} S_{4} S_{4} S_{5} \\<br />
S_{1} S_{2} S_{1} S_{3} S_{2} S_{3} S_{4} S_{4} S_{4} S_{4} S_{5} S_{5} \\<br />
S_{2} S_{1} S_{3} S_{1} S_{1} S_{2} S_{2} S_{3} S_{3} S_{4} S_{4} S_{4} S_{4} S_{4} S_{4} S_{5} S_{5} \\<br />
S_{1} S_{2} S_{2} S_{3} S_{1} S_{1} S_{3} S_{1} S_{3} S_{2} S_{3} S_{2} S_{4} S_{4} S_{4} S_{4} S_{4} S_{4} S_{5} S_{5} \\<br />
S_{3} S_{2} S_{1} S_{3} S_{2} S_{1} S_{1} S_{3} S_{2} S_{4} S_{4} S_{4} S_{4} S_{4} S_{4} S_{5} S_{5} \\<br />
S_{3} S_{2} S_{1} S_{1} S_{3} S_{2} S_{4} S_{4} S_{4} S_{4} S_{5} S_{5} \\<br />
S_{2} S_{1} S_{3} S_{4} S_{4} S_{5} \\<br />
S_{4} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{5} \\<br />
S_{4} S_{5} S_{5} \\<br />
S_{1} S_{2} S_{3} S_{4} S_{4} S_{5} \\<br />
S_{1} S_{3} S_{3} S_{2} S_{2} S_{1} S_{4} S_{4} \\<br />
S_{1} S_{2} S_{2} S_{3} S_{1} S_{3} S_{4} S_{4} \\<br />
S_{3} S_{1} S_{2} S_{1} S_{3} S_{2} S_{4} S_{4} \\<br />
S_{3} S_{2} S_{1} S_{4} S_{4} S_{5} \\<br />
S_{4} S_{5} S_{5} \\<br />
S_{5} \\<br />
\end{array}<br />
\end{array}<br />
</math><br />
<br />
== Irreducible characters ==<br />
<br />
All irreducible characters have height zero.<br />
<br />
== Decomposition matrix ==<br />
<br />
<math>\left( \begin{array}{ccccc}<br />
1 & 0 & 0 & 0 & 0 \\<br />
1 & 0 & 0 & 0 & 0 \\<br />
1 & 0 & 0 & 0 & 0 \\<br />
1 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 1 & 0 & 0 \\<br />
0 & 0 & 1 & 0 & 0 \\<br />
0 & 1 & 0 & 0 & 0 \\<br />
0 & 1 & 0 & 0 & 0 \\<br />
0 & 0 & 1 & 0 & 0 \\<br />
0 & 0 & 1 & 0 & 0 \\<br />
0 & 1 & 0 & 0 & 0 \\<br />
0 & 1 & 0 & 0 & 0 \\<br />
1 & 0 & 0 & 1 & 0 \\<br />
1 & 0 & 0 & 1 & 0 \\<br />
1 & 0 & 0 & 1 & 0 \\<br />
1 & 0 & 0 & 1 & 0 \\<br />
0 & 0 & 1 & 1 & 0 \\<br />
0 & 0 & 1 & 1 & 0 \\<br />
0 & 0 & 1 & 1 & 0 \\<br />
0 & 0 & 1 & 1 & 0 \\<br />
0 & 1 & 0 & 1 & 0 \\<br />
0 & 1 & 0 & 1 & 0 \\<br />
0 & 1 & 0 & 1 & 0 \\<br />
0 & 1 & 0 & 1 & 0 \\<br />
1 & 1 & 1 & 1 & 1 \\<br />
1 & 1 & 1 & 1 & 1 \\<br />
1 & 1 & 1 & 1 & 1 \\<br />
1 & 1 & 1 & 1 & 1 \\<br />
1 & 1 & 1 & 2 & 1 \\<br />
1 & 1 & 1 & 2 & 1 \\<br />
1 & 1 & 1 & 2 & 1 \\<br />
1 & 1 & 1 & 2 & 1 <br />
\end{array}\right)</math><br />
<br />
[[(C2)%5E5|Back to <math>(C_2)^5</math>]]</div>
CesareGArdito