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M(9,1,3) - Revision history
2024-03-29T02:21:28Z
Revision history for this page on the wiki
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http://wiki.manchester.ac.uk/blocks/index.php?title=M(9,1,3)&diff=275&oldid=prev
Charles Eaton: Created page with "{{blockbox |title = M(9,1,3) - <math>B_0(kPSL_2(8))</math> |image = M(5,1,3)quiver.png |representative = <math>B_0(kPSL_2(8))</math> |defect = <math>C_9</math> |inert..."
2018-09-05T20:56:38Z
<p>Created page with "{{blockbox |title = M(9,1,3) - <math>B_0(kPSL_2(8))</math> |image = M(5,1,3)quiver.png |representative = <math>B_0(kPSL_2(8))</math> |defect = <a href="/blocks/index.php/C9" title="C9"><math>C_9</math></a> |inert..."</p>
<p><b>New page</b></p><div>{{blockbox<br />
|title = M(9,1,3) - <math>B_0(kPSL_2(8))</math> <br />
|image = M(5,1,3)quiver.png<br />
|representative = <math>B_0(kPSL_2(8))</math><br />
|defect = [[C9|<math>C_9</math>]]<br />
|inertialquotients = <math>C_2</math><br />
|k(B) = 6<br />
|l(B) = 2<br />
|k-morita-frob = 1 <br />
|Pic-k=<br />
|cartan = <math>\left( \begin{array}{cc}<br />
2 & 1 \\<br />
1 & 5 \\<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} PSL_2(8))</math><br />
|decomp = <math>\left( \begin{array}{cc}<br />
1 & 0 \\<br />
0 & 1 \\<br />
0 & 1 \\<br />
0 & 1 \\<br />
0 & 1 \\<br />
1 & 1 \\<br />
\end{array}\right)</math><br />
|O-morita-frob = 1<br />
|Pic-O = <!-- <math>\mathcal{T}(B)=C_4</math> -->|source? = Yes<br />
|sourcereps = <br />
|k-derived-known? = Yes<br />
|k-derived = [[M(9,1,2)]]<br />
|O-derived-known? = Yes<br />
}}<br />
<br />
== Basic algebra ==<br />
<br />
'''Quiver:''' a:<1,2>, b:<2,1>, c:<2,2><br />
<br />
'''Relations w.r.t. <math>k</math>:''' ac=cb=ba-c^4=0<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> lies in M(5,1,3), then <math>B</math> must lie in M(5,1,3) or [[M(5,1,5)]]. <span style="color: red">Examples needed.</span><br />
<br />
If <math>B</math> lies in M(5,1,3), then <math>b</math> must lie in [[M(5,1,1)]], M(5,1,2) or [[M(5,1,4)]]. <span style="color: red">Examples needed.</span><br />
--><br />
<br />
<br />
== Projective indecomposable modules ==<br />
<br />
Labelling the simple <math>B</math>-modules by <math>S_1, S_2</math>, the projective indecomposable modules have Loewy structure as follows:<br />
<br />
<math>\begin{array}{cc}<br />
\begin{array}{c}<br />
S_1 \\<br />
S_2 \\<br />
S_1 \\<br />
\end{array}, & <br />
\begin{array}{ccc}<br />
& S_2 & \\<br />
S_1 & & \begin{array}{c} S_2 \\ S_2 \\ S_2 \\ \end{array} \\<br />
& S_2 & \\ <br />
\end{array}<br />
\end{array}<br />
</math><br />
<br />
== Irreducible characters ==<br />
<br />
All irreducible characters have height zero.<br />
<br />
[[C9|Back to <math>C_9</math>]]</div>
Charles Eaton