M(8,5,1) - Revision history

CesareGArdito: fixed typo in blockbox

2019-11-27T13:25:42Z

fixed typo in blockbox

Charles Eaton at 21:19, 3 October 2018

2018-10-03T21:19:40Z

Charles Eaton at 16:00, 8 September 2018

2018-09-08T16:00:10Z

Charles Eaton: Edited following switcing of M(8,5,2) and M(8,5,3)

2018-09-08T15:59:41Z

Edited following switcing of M(8,5,2) and M(8,5,3)

Charles Eaton: Created page with "{{blockbox |title = M(8,5,1) - $k(C_2 \times C_2 \times C_2)$ |image = |representative = $k(C_2 \times C_2 \times C_2)$ |defect = C2xC2xC2| $C_2..."$

2018-09-08T15:26:38Z

Created page with "{{blockbox |title = M(8,5,1) - <math>k(C_2 \times C_2 \times C_2)</math> |image = |representative = <math>k(C_2 \times C_2 \times C_2)</math> |defect = C2xC2xC2|<math>C_2..."

New page

{{blockbox
|title = M(8,5,1) - <math>k(C_2 \times C_2 \times C_2)</math>
|image =
|representative = <math>k(C_2 \times C_2 \times C_2)</math>
|defect = [[C2xC2xC2|<math>C_2 \times C_2 \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_2 \times C_2 \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_2 \times C_2 \times C_2):GL_3(2)</math>
|source? = Yes
|sourcereps = <math>k(C_2 \times C_2 \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>, c:<1,1>

'''Relations w.r.t. <math>k</math>:''' a^2=b^2=b^c=0, ab+ba=ac+ca=bc+cb=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> is in M(8,5,1), then <math>B</math> is in M{8,5,1), [[M(8,5,2)]], [[M(8,5,4)]] or [[M(8,5,6)]].

== 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 S_1 S_1 \\
S_1 S_1 S_1 \\
S_1 \\

\end{array}
</math>

== Irreducible characters ==

All irreducible characters have height zero.

[[C2xC2xC2|Back to <math>C_2 \times C_2 \times C_2</math>]]

@@ Line 46: / Line 46: @@
 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> is in M(8,5,1), then <math>B</math> is in M{8,5,1), [[M(8,5,3)]], [[M(8,5,4)]] or [[M(8,5,6)]].
+If <math>b</math> is in M(8,5,1), then <math>B</math> is in M(8,5,1), [[M(8,5,3)]], [[M(8,5,4)]] or [[M(8,5,6)]].
 == Projective indecomposable modules ==

@@ Line 29: / Line 29: @@
 |k-derived = Forms a derived equivalence class
 |O-derived-known? = Yes
 }}
@@ Line 54: / Line 57: @@
 <math>\begin{array}{c}
        S_1 \\
-       S_1 S_1 S_1 \\
+       S_1 S_1 S_1 S_1 \\
-       S_1 S_1 S_1 \\
+       S_1 S_1 S_1 S_1 S_1 S_1 \\
        S_1 \\
     \end{array}
 </math>
@@ Line 65: / Line 68: @@
 All irreducible characters have height zero.
-[[C2xC2xC2|Back to <math>C_2 \times C_2 \times C_2</math>]]
+[[C2xC2xC2|Back to <math>(C_2</math>]]