Charles Eaton: Created page with "{{blockbox |title = M(9,1,3) - $B_0(kPSL_2(8))$ |image = M(5,1,3)quiver.png |representative = $B_0(kPSL_2(8))$ |defect = $C_9$ |inert..."

2018-09-05T20:56:38Z

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..."

New page

{{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 = [[C9|<math>C_9</math>]]
|inertialquotients = <math>C_2</math>
|k(B) = 6
|l(B) = 2
|k-morita-frob = 1
|Pic-k=
|cartan = <math>\left( \begin{array}{cc}
2 & 1 \\
1 & 5 \\
\end{array} \right)</math>
|defect-morita-inv? = Yes
|inertial-morita-inv? = Yes
|O-morita? = Yes
|O-morita = <math>B_0(\mathcal{O} PSL_2(8))</math>
|decomp = <math>\left( \begin{array}{cc}
1 & 0 \\
0 & 1 \\
0 & 1 \\
0 & 1 \\
0 & 1 \\
1 & 1 \\
\end{array}\right)</math>
|O-morita-frob = 1
|Pic-O = |source? = Yes
|sourcereps =
|k-derived-known? = Yes
|k-derived = [[M(9,1,2)]]
|O-derived-known? = Yes
}}

== Basic algebra ==

'''Quiver:''' a:<1,2>, b:<2,1>, c:<2,2>

'''Relations w.r.t. <math>k</math>:''' ac=cb=ba-c^4=0

== Other notatable representatives ==

== Covering blocks and covered blocks ==



== Projective indecomposable modules ==

Labelling the simple <math>B</math>-modules by <math>S_1, S_2</math>, the projective indecomposable modules have Loewy structure as follows:

<math>\begin{array}{cc}
\begin{array}{c}
S_1 \\
S_2 \\
S_1 \\
\end{array}, &
\begin{array}{ccc}
& S_2 & \\
S_1 & & \begin{array}{c} S_2 \\ S_2 \\ S_2 \\ \end{array} \\
& S_2 & \\
\end{array}
\end{array}
</math>

== Irreducible characters ==

All irreducible characters have height zero.

[[C9|Back to <math>C_9</math>]]

M(9,1,3) - Revision history

Charles Eaton: Created page with "{{blockbox |title = M(9,1,3) - B_0(kPSL_2(8)) |image = M(5,1,3)quiver.png |representative = B_0(kPSL_2(8)) |defect = C_9 |inert..."

Charles Eaton: Created page with "{{blockbox |title = M(9,1,3) - $B_0(kPSL_2(8))$ |image = M(5,1,3)quiver.png |representative = $B_0(kPSL_2(8))$ |defect = $C_9$ |inert..."