M(2,1,1) - Revision history

Charles Eaton: Image added

2018-09-09T12:18:58Z

Image added

Charles Eaton at 12:26, 31 August 2018

2018-08-31T12:26:09Z

Charles Eaton at 09:52, 31 August 2018

2018-08-31T09:52:29Z

Charles Eaton at 09:20, 31 August 2018

2018-08-31T09:20:37Z

Charles Eaton at 09:12, 31 August 2018

2018-08-31T09:12:56Z

Charles Eaton at 11:31, 30 August 2018

2018-08-30T11:31:08Z

Charles Eaton: Created page with "{{blockbox |title = M(2,1,1) - $kC_2$ |image = |representative = $kC_2$ |defect = $kC_2$ |inertialquotients = $1$ |k(B) = 2 |l(B..."

2018-08-30T11:25:46Z

Created page with "{{blockbox |title = M(2,1,1) - <math>kC_2</math> |image = |representative = <math>kC_2</math> |defect = <math>kC_2</math> |inertialquotients = <math>1</math> |k(B) = 2 |l(B..."

New page

{{blockbox
|title = M(2,1,1) - <math>kC_2</math>
|image =
|representative = <math>kC_2</math>
|defect = <math>kC_2</math>
|inertialquotients = <math>1</math>
|k(B) = 2
|l(B) = 1
|k-morita-frob = 1
|Pic-k=
|cartan = <math>\left( \begin{array}{c}
3 \\
\end{array} \right)</math>
|defect-morita-inv? = Yes
|inertial-morita-inv? = Yes
|O-morita? = Yes
|O-morita = <math>\mathcal{O} C_2</math>
|decomp = <math>\left( \begin{array}{c}
1 \\
1 \\
\end{array}\right)</math>
|O-morita-frob = 1
|Pic-O = <math>\mathcal{L}(B)=C_2</math>
|source? = Yes
|sourcereps = <math>kC_2</math>
|k-derived-known? = Yes
|k-derived = Forms a derived equivalence class
|O-derived-known? = Yes
}}

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

== Basic algebra ==

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

'''Relations w.r.t. <math>k</math>:''' a^2=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>.

<math>B</math> and <math>b</math> must be Morita equivalent.

== 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 \\
\end{array}
</math>

== Irreducible characters ==

All irreducible characters have height zero.

@@ Line 1: / Line 1: @@
 {{blockbox
 |title = M(2,1,1) - <math>kC_2</math>
 |image =
 |representative = <math>kC_2</math>
 |defect = [[C2|<math>C_2</math>]]
@@ Line 62: / Line 62: @@
 [[Category: Morita equivalence classes|2,1,1]]
-[[Category: Block with defect group C2]]
+[[Category: Blocks with defect group C2]]
-[[Category: Block with cyclic defect group|2,1]]
+[[Category: Blocks with cyclic defect group|2,1]]
-[[Category: Nilpotent block|2,1,1]]
+[[Category: Nilpotent blocks|2,1,1]]

@@ Line 61: / Line 61: @@
 [[C2|Back to <math>C_2</math>]]
-[[Category: Morita equivalence classes]]
+[[Category: Morita equivalence classes|2,1,1]]
 [[Category: Block with defect group C2]]
-[[Category: Block with cyclic defect group]]
+[[Category: Block with cyclic defect group|2,1]]
-[[Category: Nilpotent block]]
+[[Category: Nilpotent block|2,1,1]]

@@ Line 61: / Line 61: @@
 [[C2|Back to <math>C_2</math>]]
-[[Category: Morita equivalence class]]
+[[Category: Morita equivalence classes]]
 [[Category: Block with defect group C2]]
 [[Category: Block with cyclic defect group]]
 [[Category: Nilpotent block]]

@@ Line 2: / Line 2: @@
 |title = M(2,1,1) - <math>kC_2</math>
 |image =
-|representative =  <math>kC_2</math>
+|representative = <math>kC_2</math>
-|defect = <math>C_2</math>
+|defect = [[C2|<math>C_2</math>]]
 |inertialquotients = <math>1</math>
 |k(B) = 2
@@ Line 58: / Line 58: @@
 All irreducible characters have height zero.