Charles Eaton: Created page with "{{blockbox |title = M(8,4,3) - $B_0(kPSL_2(9))$ |image = M(4,2,2)quiver.png |representative = $B_0(kPSL_2(9))$ |defect = $D_8$ |inert..."

2018-10-04T16:51:42Z

Created page with "{{blockbox |title = M(8,4,3) - <math>B_0(kPSL_2(9))</math> |image = M(4,2,2)quiver.png |representative = <math>B_0(kPSL_2(9))</math> |defect = <math>D_8</math> |inert..."

New page

{{blockbox
|title = M(8,4,3) - <math>B_0(kPSL_2(9))</math>
|image = M(4,2,2)quiver.png
|representative = <math>B_0(kPSL_2(9))</math>
|defect = [[D8|<math>D_8</math>]]
|inertialquotients = <math>1</math>
|k(B) = 5
|l(B) = 3
|k-morita-frob = 1
|Pic-k=
|cartan = <math>\left( \begin{array}{ccc}
3 & 4 & 2 \\
4 & 8 & 4 \\
2 & 4 & 3 \\
\end{array} \right)</math>
|defect-morita-inv? = Yes
|inertial-morita-inv? = Yes
|O-morita? = Yes
|O-morita = <math>B_0(\mathcal{O}A_5)</math>
|decomp = <math>\left( \begin{array}{ccc}
0 & 1 & 0 \\
1 & 1 & 0 \\
0 & 1 & 1 \\
1 & 1 & 1 \\
1 & 2 & 1 \\
\end{array}\right)</math>
|O-morita-frob = 1
|Pic-O =
|source? = No
|sourcereps =
|k-derived-known? = Yes
|k-derived = [[M(8,3,5)]], [[M(8,3,6)]]
|O-derived-known? =
|coveringblocks =
|coveredblocks =
}}

These are [[Tame blocks|tame blocks]], and appear in the family <math>D(3 {\cal A})_1</math> in Erdmann's classification (see [[References|[Er87]]]). Derived equivalences over <math>k</math> are established in [[References|[Li94b]]].

== Basic algebra ==

'''Quiver:''' a:<1,2>, b:<2,3>, c:<3,2>, d:<2,1>

'''Relations w.r.t. <math>k</math>:''' <math>ad=cb=(bcda)^2+(dabc)^2=0</math>

== Other notatable representatives ==

== Projective indecomposable modules ==

Labelling the simple <math>B</math>-modules by <math>1,2,3</math>, the projective indecomposable modules have Loewy structure as follows:

<math>\begin{array}{ccc}
\begin{array}{c}
1 \\
2 \\
3 \\
2 \\
1 \\
2 \\
3 \\
2 \\
1 \\
\end{array},
&
\begin{array}{ccc}
& 2 & \\
\begin{array}{c} 1 \\ 2 \\ 3 \\ 2 \\ 1 \\ 2 \\ 3 \\ \end{array} & \oplus & \begin{array}{c} 3 \\ 2 \\ 1 \\ 2 \\ 3 \\ 2 \\ 1 \\ \end{array} \\
& 2 & \\
\end{array},
&
\begin{array}{c}
3 \\
2 \\
1 \\
2 \\
3 \\
2 \\
1 \\
2 \\
3 \\
\end{array}
\end{array}
</math>

== Irreducible characters ==

<math>k_0(B)=4, \ k_1(B)=1</math>

[[D8|Back to <math>D_8</math>]]

[[Category: Morita equivalence classes|8,3,4]]
[[Category: Blocks with defect group D8]]
[[Category: Tame blocks|8,3,4]]

M(8,3,4) - Revision history

Charles Eaton: Created page with "{{blockbox |title = M(8,4,3) - B_0(kPSL_2(9)) |image = M(4,2,2)quiver.png |representative = B_0(kPSL_2(9)) |defect = D_8 |inert..."

Charles Eaton: Created page with "{{blockbox |title = M(8,4,3) - $B_0(kPSL_2(9))$ |image = M(4,2,2)quiver.png |representative = $B_0(kPSL_2(9))$ |defect = $D_8$ |inert..."