M(25,1,1) - Revision history

Charles Eaton: Pic_k, image, etc.

2018-10-08T15:24:37Z

Pic_k, image, etc.

Charles Eaton at 07:53, 31 August 2018

2018-08-31T07:53:23Z

Charles Eaton: Created page with "{{blockbox |title = M(25,1,1) - $kC_{25}$ |image = |representative = $kC_{25}$ |defect = $C_{25}$ |inertialquotients = $1$
2018-08-31T07:52:10Z

Created page with "{{blockbox |title = M(25,1,1) - <math>kC_{25}</math> |image = |representative = <math>kC_{25}</math> |defect = <math>C_{25}</math> |inertialquotients = <math>1</mat..."

New page
{{blockbox
|title = M(25,1,1) - <math>kC_{25}</math>
|image =
|representative = <math>kC_{25}</math>
|defect = [[C25|<math>C_{25}</math>]]
|inertialquotients = <math>1</math>
|k(B) = 25
|l(B) = 1
|k-morita-frob = 1
|Pic-k=
|cartan = <math>\left( \begin{array}{c}
25 \\
\end{array} \right)</math>
|defect-morita-inv? = Yes
|inertial-morita-inv? = Yes
|O-morita? = Yes
|O-morita = <math>\mathcal{O} C_{25}</math>
|decomp = <math>\left( \begin{array}{c}
1 \\
1 \\
\vdots \\
1 \\
\end{array}\right)</math>
|O-morita-frob = 1
|Pic-O = <math>\mathcal{L}(B)=C_{25}:C_{20}</math>
|source? = Yes
|sourcereps = <math>kC_{25}</math>
|k-derived-known? = Yes
|k-derived = Forms a derived equivalence class
|O-derived-known? = Yes
}}

These are [[Nilpotent blocks|nilpotent blocks]].

== Basic algebra ==

'''Quiver:''' a:<1,1>

'''Relations w.r.t. <math>k</math>:''' a^{25}=0

== Other notatable representatives ==

== Covering blocks and covered blocks ==

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>.

If <math>b</math> lies in M(25,1,1), then <math>B</math> must lie in M(25,1,1), [[M(25,1,2]] or [[M(25,1,4)]]. For example consider the principal blocks of <math>C_{25} \triangleleft D_{50}, C_{25}:C_4</math>.

If <math>B</math> lies in M(25,1,1), then <math>b</math> must lie in M(25,1,1), [[M(25,1,2]] or [[M(25,1,4)]]. <span style="color: red">Example needed.</span>

== Projective indecomposable modules ==

Labelling the unique simple <math>B</math>-module by <math>S_1</math>, the unique projective indecomposable module has Loewy structure as follows:

<math>\begin{array}{c}
S_1 \\
S_1 \\
\vdots \\
S_1 \\
\end{array}
</math>

== Irreducible characters ==

All irreducible characters have height zero.

[[C25|Back to <math>C_{25}</math>]]