M(8,2,1) - Revision history

Charles Eaton at 11:33, 1 November 2018

2018-11-01T11:33:36Z

Charles Eaton: /* Projective indecomposable modules */

2018-10-08T15:17:29Z

‎Projective indecomposable modules

Charles Eaton at 21:21, 3 October 2018

2018-10-03T21:21:18Z

Charles Eaton: Image added

2018-09-09T20:28:32Z

Image added

Charles Eaton: Created page with "{{blockbox |title = M(8,2,1) - $k(C_4 \times C_2)$ |image = |representative = $k(C_4 \times C_2)$ |defect = $C_4 \times C_2$ |ine..."

2018-09-08T15:12:31Z

Created page with "{{blockbox |title = M(8,2,1) - <math>k(C_4 \times C_2)</math> |image = |representative = <math>k(C_4 \times C_2)</math> |defect = <math>C_4 \times C_2</math> |ine..."

New page

{{blockbox
|title = M(8,2,1) - <math>k(C_4 \times C_2)</math>
|image =
|representative = <math>k(C_4 \times C_2)</math>
|defect = [[C4xC2|<math>C_4 \times C_2</math>]]
|inertialquotients = <math>1</math>
|k(B) = 8
|l(B) = 1
|k-morita-frob = 1
|Pic-k=
|cartan = <math>\left( \begin{array}{c}
8 \\
\end{array} \right)</math>
|defect-morita-inv? = Yes
|inertial-morita-inv? = Yes
|O-morita? = Yes
|O-morita = <math>\mathcal{O} (C_4 \times C_2)</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_4 \times C_2):(C_2 \times C_2 \times C_2)</math>
|source? = Yes
|sourcereps = <math>k(C_4 \times C_2)</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>, b:<1,1>

'''Relations w.r.t. <math>k</math>:''' a^4=b^2=ab+ba=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> or <math>b</math> is in M(8,2,1), then <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 \\
\vdots \\
S_1 \\

\end{array}
</math>

== Irreducible characters ==

All irreducible characters have height zero.

[[C4xC2|Back to <math>C_4 \times C_2</math>]]

@@ Line 24: / Line 24: @@
 |O-morita-frob = 1
 |Pic-O = <math>(C_4 \times C_2):(C_2 \times C_2 \times C_2)</math>
 |source? =
 |sourcereps =
@@ Line 29: / Line 30: @@
 |k-derived = Forms a derived equivalence class
 |O-derived-known? = Yes
 }}
@@ Line 41: / Line 47: @@
 == Other notatable representatives ==
 == Projective indecomposable modules ==

@@ Line 50: / Line 50: @@
 == 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:
+Labelling the unique simple <math>B</math>-module by <math>1</math>, the unique projective indecomposable module has Loewy structure as follows:
 <math>\begin{array}{c}
-       S_1 \\
+\\
-       S_1 \\
+\ 1 \\
-       \vdots \\
+\ 1 \\
-       S_1 \\
+\ 1 \\
+\\
     \end{array}
 </math>

@@ Line 23: / Line 23: @@
 \end{array}\right)</math>
 |O-morita-frob = 1
-|Pic-O = <math>\mathcal{L}(B)=(C_4 \times C_2):(C_2 \times C_2 \times C_2)</math>
+|Pic-O = <math>(C_4 \times C_2):(C_2 \times C_2 \times C_2)</math>
-|source? = Yes
+|source? =
-|sourcereps = <math>k(C_4 \times C_2)</math>
+|sourcereps =
 |k-derived-known? = Yes
 |k-derived = Forms a derived equivalence class