http://wiki.manchester.ac.uk/blocks/index.php?title=M(32,51,23)&feed=atom&action=historyM(32,51,23) - Revision history2024-03-28T16:24:11ZRevision history for this page on the wikiMediaWiki 1.30.1http://wiki.manchester.ac.uk/blocks/index.php?title=M(32,51,23)&diff=1043&oldid=prevCesareGArdito: 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>..."2019-12-09T12:47:48Z<p>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 = <a href="/blocks/index.php/(C2)%5E5" title="(C2)^5"><math>(C_2)^5</math></a>..."</p>
<p><b>New page</b></p><div>{{blockbox<br />
|title = M(32,51,23) - <math>B_0(k(SL_2(32)))</math> <br />
|image = &nbsp; <br />
|representative = <math>B_0(k(SL_2(32)))</math><br />
|defect = [[(C2)%5E5|<math>(C_2)^5</math>]]<br />
|inertialquotients = <math>C_{31}</math><br />
|k(B) = 32<br />
|l(B) = 31<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>B_0(\mathcal{O}(SL_2(32)))</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(32,51,22)]]<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(32,51,23), then <math>B</math> is in M(32,51,23) or [[M(32,51,31)]].<br />
<br />
== Projective indecomposable modules ==<br />
<br />
== Irreducible characters ==<br />
<br />
All irreducible characters have height zero.<br />
<br />
== Cartan matrix ==<br />
<math>\left( \begin{array}{ccccccccccccccccccccccccccccccc}<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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<br />
\end{array} \right)</math><br />
<br />
== Decomposition matrix ==<br />
<br />
<math>\left( \begin{array}{ccccccccccccccccccccccccccccccc}<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 <br />
\end{array}\right)</math><br />
<br />
[[(C2)%5E5|Back to <math>(C_2)^5</math>]]</div>CesareGArdito