M(4,2,1) - Revision history

Charles Eaton at 13:30, 19 December 2018

2018-12-19T13:30:18Z

Charles Eaton at 08:35, 4 December 2018

2018-12-04T08:35:57Z

Charles Eaton at 19:42, 31 October 2018

2018-10-31T19:42:25Z

Charles Eaton at 17:53, 31 October 2018

2018-10-31T17:53:12Z

Charles Eaton at 17:00, 21 October 2018

2018-10-21T17:00:26Z

Charles Eaton: Covered and covering blocks

2018-09-09T20:48:33Z

Covered and covering blocks

Charles Eaton: Image added

2018-09-04T09:35:58Z

Image added

Charles Eaton at 13:39, 30 August 2018

2018-08-30T13:39:32Z

Charles Eaton at 11:59, 30 August 2018

2018-08-30T11:59:14Z

Charles Eaton: Created page with "{{blockbox |title = M(4,2,1) - $k(C_2 \times C_2)$ |image = |representative = $k(C_2 \times C_2)$ |defect = $C_2 \times C_2$ |inertialquoti..."

2018-08-30T11:57:27Z

Created page with "{{blockbox |title = M(4,2,1) - <math>k(C_2 \times C_2)</math> |image = |representative = <math>k(C_2 \times C_2)</math> |defect = <math>C_2 \times C_2</math> |inertialquoti..."

New page

{{blockbox
|title = M(4,2,1) - <math>k(C_2 \times C_2)</math>
|image =
|representative = <math>k(C_2 \times C_2)</math>
|defect = <math>C_2 \times C_2</math>
|inertialquotients = <math>1</math>
|k(B) = 4
|l(B) = 1
|k-morita-frob = 1
|Pic-k= <math>(k \times k):GL_2(k)</math>
|cartan = <math>\left( \begin{array}{c}
4 \\
\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)</math>
|decomp = <math>\left( \begin{array}{c}
1 \\
1 \\
1 \\
1 \\
\end{array}\right)</math>
|O-morita-frob = 1
|Pic-O = <math>\mathcal{L}(B)=S_4</math>
|source? = Yes
|sourcereps = <math>k(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>

'''Relations w.r.t. <math>k</math>:''' a^2=b^2=ab+ba=0

== Other notatable representatives ==

Block number 4 of <math>k PGL_3(7)</math> in the labelling used in [http://www.math.rwth-aachen.de/~MOC/decomposition/tex/L3(7)/]

== 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(4,2,1), then <math>B</math> must lie in M(4,2,1) or M(4,2,3). For example consider the principal blocks of <math>O_2(A_4) \triangleleft A_4</math>.

If <math>B</math> lies in M(4,2,1), then <math>b</math> must lie in M(4,2,1) or M(4,2,3). For example consider blocks of <math>PSL_3(7) \triangleleft PGL_3(7)</math>.

== 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}{ccc}
& S_1 & \\
S_1 & & S_1 \\
& S_1 & \\
\end{array}
</math>

== Irreducible characters ==

All irreducible characters have height zero.

@@ Line 31: / Line 31: @@
 |O-derived-known? = Yes
 |coveringblocks = M(4,2,1), [[M(4,2,3)]] (complete)
-|coveredblocks = M(4,2,1), [[M(4,2,3)]]<ref>For example consider block number 2 of <math>PSL_3(7) \triangleleft PGL_3(7)</math> in the labelling used in [1].</ref> (complete)
+|coveredblocks = M(4,2,1), [[M(4,2,3)]]<ref>For example consider block number 2 of <math>PSL_3(7) \triangleleft PGL_3(7)</math> in the labelling used in [http://www.math.rwth-aachen.de/~MOC/decomposition/tex/L3(7)/].</ref> (complete)
 |pcoveringblocks = [[M(8,2,1)]], [[M(8,3,1)]], [[M(8,5,1)]] (complete)
 }}

@@ Line 32: / Line 32: @@
 |coveringblocks = M(4,2,1), [[M(4,2,3)]] (complete)
 |coveredblocks = M(4,2,1), [[M(4,2,3)]]<ref>For example consider block number 2 of <math>PSL_3(7) \triangleleft PGL_3(7)</math> in the labelling used in [1].</ref> (complete)
-|pcoveringblocks = [[M(8,2,1)]], [[M(8,5,1)]] (complete)
+|pcoveringblocks = [[M(8,2,1)]], [[M(8,3,1)]], [[M(8,5,1)]] (complete)
 }}

@@ Line 30: / Line 30: @@
 |k-derived = Forms a derived equivalence class
 |O-derived-known? = Yes
-|coveringblocks = [[M(4,2,3)]] (complete)
+|coveringblocks = M(4,2,1), [[M(4,2,3)]] (complete)
-|coveredblocks = [[M(4,2,3)]] (complete)
+|coveredblocks = M(4,2,1), [[M(4,2,3)]]<ref>For example consider block number 2 of <math>PSL_3(7) \triangleleft PGL_3(7)</math> in the labelling used in [1].</ref> (complete)
-|pcoveringblocks = [[M(8,2,1)]], [[M(8,5,1)]]
+|pcoveringblocks = [[M(8,2,1)]], [[M(8,5,1)]] (complete)
 }}
@@ Line 45: / Line 45: @@
 == Other notatable representatives ==
-Block number 4 of <math>k PGL_3(7)</math> in the labelling used in [http://www.math.rwth-aachen.de/~MOC/decomposition/tex/L3(7)/]
+Block number 2 of <math>k PGL_3(7)</math> in the labelling used in [http://www.math.rwth-aachen.de/~MOC/decomposition/tex/L3(7)/]
 == Projective indecomposable modules ==
@@ Line 69: / Line 61: @@
 All irreducible characters have height zero.

← Older revision		Revision as of 11:59, 30 August 2018
Line 50:		Line 50:

	If <math>B</math> lies in M(4,2,1), then <math>b</math> must lie in M(4,2,1) or M(4,2,3). For example consider blocks of <math>PSL_3(7) \triangleleft PGL_3(7)</math>.		If <math>B</math> lies in M(4,2,1), then <math>b</math> must lie in M(4,2,1) or M(4,2,3). For example consider blocks of <math>PSL_3(7) \triangleleft PGL_3(7)</math>.
−

	== Projective indecomposable modules ==		== Projective indecomposable modules ==

@@ Line 24: / Line 24: @@
 |O-morita-frob = 1
 |Pic-O = <math>\mathcal{L}(B)=S_4</math>
-|PIgroup =
+|PIgroup = <math>S_4 \times C_2</math>
 |source? = Yes
 |sourcereps = <math>k(C_2 \times C_2)</math>

@@ Line 1: / Line 1: @@
 {{blockbox
 |title = M(4,2,1) - <math>k(C_2 \times C_2)</math>
-|image =
+|image = M(4,2,1)quiver.png
 |representative =  <math>k(C_2 \times C_2)</math>
 |defect = <math>C_2 \times C_2</math>

@@ Line 47: / Line 47: @@
 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(4,2,1), then <math>B</math> must lie in M(4,2,1) or M(4,2,3). For example consider the principal blocks of <math>O_2(A_4) \triangleleft A_4</math>.
+If <math>b</math> lies in M(4,2,1), then <math>B</math> must lie in M(4,2,1) or [[M(4,2,3)]]. For example consider the principal blocks of <math>O_2(A_4) \triangleleft A_4</math>.
-If <math>B</math> lies in M(4,2,1), then <math>b</math> must lie in M(4,2,1) or M(4,2,3). For example consider blocks of <math>PSL_3(7) \triangleleft PGL_3(7)</math>.
+If <math>B</math> lies in M(4,2,1), then <math>b</math> must lie in M(4,2,1) or [[M(4,2,3)]]. For example consider blocks of <math>PSL_3(7) \triangleleft PGL_3(7)</math>.
 == Projective indecomposable modules ==