M(8,3,3) - Revision history

Charles Eaton at 22:53, 18 December 2018

2018-12-18T22:53:46Z

Charles Eaton: Created page with "{{blockbox |title = M(8,3,3) - $kS_4$ |image = M(8,3,3)quiver.png |representative = $kS_4$ |defect = $D_8$ |inertialquotients =
2018-10-04T14:33:03Z

Created page with "{{blockbox |title = M(8,3,3) - <math>kS_4</math> |image = M(8,3,3)quiver.png |representative = <math>kS_4</math> |defect = <math>D_8</math> |inertialquotients = <math..."

New page
{{blockbox
|title = M(8,3,3) - <math>kS_4</math>
|image = M(8,3,3)quiver.png
|representative = <math>kS_4</math>
|defect = [[D8|<math>D_8</math>]]
|inertialquotients = <math>1</math>
|k(B) = 5
|l(B) = 2
|k-morita-frob = 1
|Pic-k=
|cartan = <math>\left( \begin{array}{cc}
4 & 2 \\
2 & 3 \\
\end{array} \right)</math>
|defect-morita-inv? = Yes
|inertial-morita-inv? = Yes
|O-morita? = No
|O-morita =
|decomp = <math>\left( \begin{array}{c}
1 & 0 \\
1 & 0 \\
0 & 1 \\
1 & 1 \\
1 & 1 \\
\end{array}\right)</math>
|O-morita-frob =
|Pic-O =
|source? = No
|sourcereps =
|k-derived-known? = Yes
|k-derived = [[M(8,3,2)]]
|O-derived-known? =
|coveringblocks =
|coveredblocks =
}}

These are [[Tame blocks|tame blocks]], and appear in the family <math>D(2 {\cal B})</math> in Erdmann's classification (see [[References|[Er87] ]]). Derived equivalences over <math>k</math> are established in [[References|[Ho97]]].

== Basic algebra ==

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

'''Relations w.r.t. <math>k</math>:''' <math>a^2=bd=dc=cb=0</math>, <math>abc=bca</math>, <math>cab=d^2</math>

== Other notatable representatives ==

== Projective indecomposable modules ==

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

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

== Irreducible characters ==

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