http://wiki.manchester.ac.uk/blocks/index.php?title=M(16,14,3)&feed=atom&action=history
M(16,14,3) - Revision history
2024-03-28T22:11:37Z
Revision history for this page on the wiki
MediaWiki 1.30.1
http://wiki.manchester.ac.uk/blocks/index.php?title=M(16,14,3)&diff=945&oldid=prev
CesareGArdito at 14:02, 27 November 2019
2019-11-27T14:02:42Z
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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 14:02, 27 November 2019</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l17" >Line 17:</td>
<td colspan="2" class="diff-lineno">Line 17:</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><del class="diffchange diffchange-inline">B_0(</del>\mathcal{O} (C_2 \times C_2 \times A_4)</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>\mathcal{O} (C_2 \times C_2 \times A_4)</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>
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CesareGArdito
http://wiki.manchester.ac.uk/blocks/index.php?title=M(16,14,3)&diff=943&oldid=prev
CesareGArdito at 13:56, 27 November 2019
2019-11-27T13:56:16Z
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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 13:56, 27 November 2019</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1" >Line 1:</td>
<td colspan="2" class="diff-lineno">Line 1:</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>{{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(16,14,<del class="diffchange diffchange-inline">2</del>) - <math>k(C_2 \times C_2 \times A_4)</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(16,14,<ins class="diffchange diffchange-inline">3</ins>) - <math>k(C_2 \times C_2 \times A_4)</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 \times C_2 \times A_4)</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 \times C_2 \times A_4)</math></div></td></tr>
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CesareGArdito
http://wiki.manchester.ac.uk/blocks/index.php?title=M(16,14,3)&diff=942&oldid=prev
CesareGArdito at 13:48, 27 November 2019
2019-11-27T13:48:31Z
<p></p>
<table class="diff diff-contentalign-left" data-mw="interface">
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<col class="diff-marker" />
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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, 27 November 2019</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l25" >Line 25:</td>
<td colspan="2" class="diff-lineno">Line 25:</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>|sourcereps =</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>|sourcereps =</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>|k-derived-known? = 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>|k-derived-known? = 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>|k-derived = <del class="diffchange diffchange-inline">Forms a derived equivalence class</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>|k-derived = <ins class="diffchange diffchange-inline">[[M(16,14,2)]]</ins></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-derived-known? = 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-derived-known? = 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>|coveringblocks =</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>|coveringblocks =</div></td></tr>
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CesareGArdito
http://wiki.manchester.ac.uk/blocks/index.php?title=M(16,14,3)&diff=939&oldid=prev
CesareGArdito: Created page with "{{blockbox |title = M(16,14,2) - <math>k(C_2 \times C_2 \times A_4)</math> |image = |representative = <math>k(C_2 \times C_2 \times A_4)</math> |defect = (C2)%5E4|..."
2019-11-27T13:46:52Z
<p>Created page with "{{blockbox |title = M(16,14,2) - <math>k(C_2 \times C_2 \times A_4)</math> |image = |representative = <math>k(C_2 \times C_2 \times A_4)</math> |defect = (C2)%5E4|..."</p>
<p><b>New page</b></p><div>{{blockbox<br />
|title = M(16,14,2) - <math>k(C_2 \times C_2 \times A_4)</math> <br />
|image = &nbsp; <br />
|representative = <math>k(C_2 \times C_2 \times A_4)</math><br />
|defect = [[(C2)%5E4|<math>(C_2)^4</math>]]<br />
|inertialquotients = <math>C_3</math><br />
|k(B) = 16<br />
|l(B) = 3<br />
|k-morita-frob = 1 <br />
|Pic-k= &nbsp;<br />
|cartan = <math>\left( \begin{array}{ccc}<br />
8 & 4 & 4\\<br />
4 & 8 & 4 \\<br />
4 & 4 & 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(\mathcal{O} (C_2 \times C_2 \times A_4)</math><br />
|decomp = See below<br />
|O-morita-frob = 1<br />
|Pic-O = <math>(C_2 \times C_2):S_3 \times S_3</math><br />
|PIgroup = <br />
|source? = No<br />
|sourcereps =<br />
|k-derived-known? = Yes<br />
|k-derived = Forms a derived equivalence class<br />
|O-derived-known? = Yes<br />
|coveringblocks =<br />
|coveredblocks =<br />
|pcoveringblocks =<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 <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(16,14,3), then <math>B</math> is in [[M(16,14,1)]], M(16,14,3), [[M(16,14,8)]], [[M(16,14,16)]].<br />
<br />
== Projective indecomposable modules ==<br />
<br />
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:<br />
<br />
<math>\begin{array}{ccc}<br />
\begin{array}{c}<br />
S_1 \\<br />
S_1 S_1 S_2 S_3 \\<br />
S_1 S_1 S_2 S_2 S_3 S_3 \\ <br />
S_1 S_1 S_2 S_3 \\<br />
S_1 \\<br />
<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_2 \\<br />
S_1 S_2 S_2 S_3 \\<br />
S_1 S_1 S_2 S_2 S_3 S_3 \\ <br />
S_1 S_2 S_2 S_3 \\<br />
S_2 \\<br />
<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_3 \\<br />
S_1 S_2 S_3 S_3 \\<br />
S_1 S_1 S_2 S_2 S_3 S_3 \\ <br />
S_1 S_2 S_3 S_3 \\<br />
S_3 \\<br />
<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 />
<math>\left( \begin{array}{ccc}<br />
1 &0 &0 \\<br />
1 &0 &0 \\<br />
1 &0 &0 \\<br />
1 &0 &0 \\<br />
0 &0 &1 \\<br />
0 &1 &0 \\<br />
0 &0 &1 \\<br />
0 &1 &0 \\<br />
0 &0 &1 \\<br />
0 &1 &0 \\<br />
0 &0 &1 \\<br />
0 &1 &0 \\<br />
1 &1 &1 \\<br />
1 &1 &1 \\<br />
1 &1 &1 \\<br />
1 &1 &1<br />
\end{array}\right)</math><br />
<br />
<br />
[[(C2)%5E4|Back to <math>(C_2)^4</math>]]</div>
CesareGArdito