M(4,2,3) - Revision history

Charles Eaton at 15:29, 22 November 2018

2018-11-22T15:29:16Z

Charles Eaton at 19:51, 31 October 2018

2018-10-31T19:51:33Z

Charles Eaton at 17:54, 31 October 2018

2018-10-31T17:54:43Z

Charles Eaton at 17:41, 31 October 2018

2018-10-31T17:41:52Z

Charles Eaton at 15:40, 8 September 2018

2018-09-08T15:40:17Z

Charles Eaton at 13:50, 31 August 2018

2018-08-31T13:50:13Z

Charles Eaton: Added image

2018-08-31T13:47:19Z

Added image

Charles Eaton: Created page with "{{blockbox |title = M(4,2,3) - $kA_4$ |image = |representative = $kA_4$ |defect = $C_2 \times C_2$ |inertialquotients = $C_3$ |k..."

2018-08-30T13:38:35Z

Created page with "{{blockbox |title = M(4,2,3) - <math>kA_4</math> |image = |representative = <math>kA_4</math> |defect = <math>C_2 \times C_2</math> |inertialquotients = <math>C_3</math> |k..."

New page

{{blockbox
|title = M(4,2,3) - <math>kA_4</math>
|image =
|representative = <math>kA_4</math>
|defect = <math>C_2 \times C_2</math>
|inertialquotients = <math>C_3</math>
|k(B) = 4
|l(B) = 3
|k-morita-frob = 1
|Pic-k= <math>(k^* \times k^* \times C_3):C_2</math>
|cartan = <math>\left( \begin{array}{ccc}
2 & 1 & 1 \\
1 & 2 & 1 \\
1 & 1 & 2 \\
\end{array} \right)</math>
|defect-morita-inv? = Yes
|inertial-morita-inv? = Yes
|O-morita? = Yes
|O-morita = <math>\mathcal{O}A_4</math>
|decomp = <math>\left( \begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
1 & 1 & 1 \\
\end{array}\right)</math>
|O-morita-frob = 1
|Pic-O = <math>\mathcal{T}(B)=S_3</math>
|source? = Yes
|sourcereps = <math>\mathcal{O}A_4</math>
|k-derived-known? = Yes
|k-derived = [[M(4,2,2)]]
|O-derived-known? = Yes
}}

== Basic algebra ==

'''Quiver:''' a: <1,2>, b: <2,3> , c: <3,1>, d: <2,1>, e: <3,2>, f: <1,3>

'''Relations w.r.t. <math>k</math>:''' ab=bc=ca=0, df=fe=ed=0, ad=fc, be=da, cf=eb

== Other notatable representatives ==

Block number 2 of <math>k PSL_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(4,2,3), 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(4,2,3), 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>.

== Projective indecomposable modules ==

Labelling the simple <math>B</math>-modules by <math>S_1, S_2, S_3</math>, the projective indecomposable modules have Loewy structure as follows:

<math>\begin{array}{ccc}
\begin{array}{ccc}
& S_1 & \\
S_2 & & S_3 \\
& S_1 & \\
\end{array},
&
\begin{array}{ccc}
& S_2 & \\
S_1 & & S_3 \\
& S_2 & \\
\end{array},
&
\begin{array}{ccc}
& S_3 & \\
S_1 & & S_2 \\
& S_3 & \\
\end{array}
\end{array}
</math>

== Irreducible characters ==

All irreducible characters have height zero.

@@ Line 34: / Line 34: @@
 |coveringblocks = [[M(4,2,1)]]<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)/]. The covering block of <math>PGL_3(7)</math> is nilpotent.</ref>, M(4,2,3) (complete)
 |coveredblocks = [[M(4,2,1)]], M(4,2,2) (complete)
-|pcoveringblocks = [[M(8,5,3)]] (complete)<ref>This follows from Theorem 2.15 of [[References|[EL18a]]] since a covering block cannot have defect group <math>C_4 \times C_2</math>.</ref>
+|pcoveringblocks = [[M(8,3,3)]], [[M(8,5,3)]] (complete)<ref>A covering block cannot have defect group <math>C_4 \times C_2</math>.</ref>
 }}

@@ Line 32: / Line 32: @@
 |k-derived = [[M(4,2,2)]]
 |O-derived-known? = Yes
-|coveringblocks = [[M(4,2,1)]]<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)/]. The covering block of <math>PGL_3(7)</math> is nilpotent.</ref>, M(4,2,3)
+|coveringblocks = [[M(4,2,1)]]<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)/]. The covering block of <math>PGL_3(7)</math> is nilpotent.</ref>, M(4,2,3) (complete)
-|coveredblocks = [[M(4,2,1)]], M(4,2,2)
+|coveredblocks = [[M(4,2,1)]], M(4,2,2) (complete)
 |pcoveringblocks = [[M(8,5,3)]] (complete)<ref>This follows from Theorem 2.15 of [[References|[EL18a]]] since a covering block cannot have defect group <math>C_4 \times C_2</math>.</ref>
 }}

@@ Line 8: / Line 8: @@
 |l(B) = 3
 |k-morita-frob = 1
-|Pic-k= <math>(k^* \times k^* \times C_3):C_2</math>
+|Pic-k= <math>(k^* \times k^* \times C_3):C_2</math><ref>This is an elementary calculation.</ref>
 |cartan = <math>\left( \begin{array}{ccc}
 & 1 & 1 \\
@@ Line 26: / Line 26: @@
 |O-morita-frob = 1
 |Pic-O = <math>\mathcal{T}(B)=S_3</math><ref>Every Morita equivalence is a source algebra equivalence by [[References|[CEKL13]]], so <math>{\rm Pic}(B)=\mathcal{T}(B)</math></ref>
-|PIgroup = <math>S_4 \times C_2</math>
+|PIgroup = <math>S_4 \times C_2</math><ref>See, for example, Proposition 2.8 of [[References|[EL18a]]]</ref>
 |source? = Yes
 |sourcereps = <math>\mathcal{O}A_4</math>

@@ Line 25: / Line 25: @@
 \end{array}\right)</math>
 |O-morita-frob = 1
-|Pic-O = <math>\mathcal{T}(B)=S_3</math>
+|Pic-O = <math>\mathcal{T}(B)=S_3</math><ref>Every Morita equivalence is a source algebra equivalence by [[References|[CEKL13]]], so <math>{\rm Pic}(B)=\mathcal{T}(B)</math></ref>
 |source? = Yes
 |sourcereps = <math>\mathcal{O}A_4</math>
@@ Line 31: / Line 32: @@
 |k-derived = [[M(4,2,2)]]
 |O-derived-known? = Yes
 }}
@@ Line 42: / Line 46: @@
 Block number 2 of <math>k PSL_3(7)</math> in the labelling used in [http://www.math.rwth-aachen.de/~MOC/decomposition/tex/L3(7)/]
 == Projective indecomposable modules ==
@@ Line 81: / Line 76: @@
 All irreducible characters have height zero.
-[[C2xC2|Back to <math>C_2 \times C_2</math>]]
+<references />
 [[Category: Morita equivalence classes|4,2,3]]
 [[Category: Blocks with defect group C2xC2]]
 [[Category: Tame blocks|4,2,3]]

@@ Line 35: / Line 35: @@
 == Basic algebra ==
-'''Quiver:''' a: <1,2>, b: <2,3> , c: <3,1>, d: <2,1>, e: <3,2>, f: <1,3>
+'''Quiver:''' a:<1,2>, b:<2,3>, c:<3,1>, d:<2,1>, e:<3,2>, f:<1,3>
 '''Relations w.r.t. <math>k</math>:''' ab=bc=ca=0, df=fe=ed=0, ad=fc, be=da, cf=eb

@@ Line 3: / Line 3: @@
 |image = M(4,2,3)quiver.png
 |representative =  <math>kA_4</math>
-|defect = <math>C_2 \times C_2</math>
+|defect = [[C2xC2|<math>C_2 \times C_2</math>]]
 |inertialquotients = <math>C_3</math>
 |k(B) = 4
@@ Line 80: / Line 80: @@
 All irreducible characters have height zero.