M(8,5,5) - Revision history

Charles Eaton at 16:22, 21 November 2018

2018-11-21T16:22:51Z

Charles Eaton: Relations and references

2018-10-12T14:22:05Z

Relations and references

Charles Eaton: [Al79] reference

2018-09-24T14:05:03Z

[Al79] reference

Charles Eaton: Created page with "{{blockbox |title = M(8,5,5) - $B_0(kSL_2(8))$ |image = M(8,5,5)quiver.png |representative = $B_0(kSL_2(8))$ |defect = C2xC2xC2| $C_2 \times C_2..."$

2018-09-23T20:18:02Z

Created page with "{{blockbox |title = M(8,5,5) - <math>B_0(kSL_2(8))</math> |image = M(8,5,5)quiver.png |representative = <math>B_0(kSL_2(8))</math> |defect = C2xC2xC2|<math>C_2 \times C_2..."

New page

{{blockbox
|title = M(8,5,5) - <math>B_0(kSL_2(8))</math>
|image = M(8,5,5)quiver.png
|representative = <math>B_0(kSL_2(8))</math>
|defect = [[C2xC2xC2|<math>C_2 \times C_2 \times C_2</math>]]
|inertialquotients = <math>C_7</math>
|k(B) = 8
|l(B) = 7
|k-morita-frob = 1
|Pic-k=  
|cartan = <math>\left( \begin{array}{ccccccc}
8 & 4 & 4 & 4 & 2 & 2 & 2 \\
4 & 4 & 2 & 2 & 0 & 2 & 1 \\
4 & 2 & 4 & 2 & 1 & 0 & 2 \\
4 & 2 & 2 & 4 & 2 & 1 & 0 \\
2 & 0 & 1 & 2 & 2 & 0 & 0 \\
2 & 2 & 0 & 1 & 0 & 2 & 0 \\
2 & 1 & 2 & 0 & 0 & 0 & 2 \\
\end{array} \right)</math>
|defect-morita-inv? = Yes
|inertial-morita-inv? = Yes
|O-morita? = Yes
|O-morita = <math>B_0(\mathcal{O}SL_2(8))</math>
|decomp = <math>\left( \begin{array}{ccccccc}
1 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 1 & 1 & 0 & 0 & 0 \\
1 & 0 & 1 & 0 & 0 & 0 & 1 \\
1 & 0 & 0 & 1 & 1 & 0 & 0 \\
1 & 1 & 0 & 0 & 0 & 1 & 0 \\
1 & 1 & 1 & 0 & 0 & 0 & 1 \\
1 & 0 & 1 & 1 & 1 & 0 & 0 \\
1 & 1 & 0 & 1 & 0 & 1 & 0 \\
\end{array}\right)</math>
|O-morita-frob = 1
|Pic-O = <math>C_3</math>;
|source? = No
|sourcereps =  
|k-derived-known? = Yes
|k-derived = [[M(8,5,4)]]
|O-derived-known? = Yes
|coveringblocks =
|coveredblocks =
}}

== Basic algebra ==



== Irreducible characters ==

All irreducible characters have height zero.

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

@@ Line 33: / Line 33: @@
 \end{array}\right)</math><ref>Decomposition matrix taken from [http://www.math.rwth-aachen.de/~MOC/decomposition/]</ref>
 |O-morita-frob = 1
-|Pic-O = <math>C_3</math>;
+|Pic-O = <math>C_3</math><ref>See [[References|[EL18c]]]</ref>
 |source? = No
 |sourcereps = &nbsp;

@@ Line 31: / Line 31: @@
 & 0 & 1 & 1 & 1 & 0 & 0 \\
 & 1 & 0 & 1 & 0 & 1 & 0 \\
-\end{array}\right)</math>
+\end{array}\right)</math><ref>Decomposition matrix taken from [http://www.math.rwth-aachen.de/~MOC/decomposition/]</ref>
 |O-morita-frob = 1
 |Pic-O = <math>C_3</math>;
@@ Line 38: / Line 38: @@
 |k-derived-known? = Yes
 |k-derived = [[M(8,5,4)]]
-|O-derived-known? = Yes
+|O-derived-known? = Yes<ref>A [[Glossary#Splendid equivalence|splendid]] Rickard equivalence is given in [[References|[Ro95, 2.3]]], which then lifts to <math>\mathcal{O}</math></ref>
-|coveringblocks =
+|coveringblocks = [[M(8,5,8)]]
-|coveredblocks =
+|coveredblocks = Potentially [[M(8,5,8)]]
 }}
-The projective indecomposable modules of the <math>2</math>-blocks of the groups <math>SL_2(2^n)</math> were computed by Alperin in [[References|[Al79]]].
+The projective indecomposable modules of the <math>2</math>-blocks of the groups <math>SL_2(2^n)</math> were computed by Alperin in [[References|[Al79]]] and the quiver and relations by Koshita in [[References|[Ko94]]]. A splendid derived equivalence with [[M(8,5,4)]] was constructed by Rouquier in [[References|[Ro95]]].
 == Basic algebra ==
-<!-- '''Quiver:''' <math>a_1:<1,2>, a_2:<2,3>, a_3:<3,4>, a_4:<4,5>, a_5:<5,6>, a_6:<6,7>, a_7:<7,1></math>,
+'''Quiver:''' a<1,2>, b:<2,3>, c:<3,2>, d:<2,1>, e:<1,4>, f:<4,5>, g:<5,4>, h:<4,1>, i:<1,6>, j:<6,7>, k:<7,6>, l:<6,1>
-'''Relations w.r.t. <math>k</math>:''' <math>a_1a_2=a_2a_3=a_3a_4=a_4a_5=a_5a_6=a_6a_7=a_7a_1=0</math>,
+'''Relations w.r.t. <math>k</math>:''' <math>da=he=li=0</math>,
-<math>b_1b_3=b_2b_4=b_3b_5=b_4b_6=b_5b_7=b_6b_1=b_7b_2=0</math>,
+<math>cb=gf=kj=0</math>,
-<math>c_1c_5=c_2c_6=c_3c_7=c_4c_1=c_5c_2=c_6c_3=c_7c_4=0</math>,
+<math>bcd=dil</math>, <math>fgh=had</math>, <math>jkl=leh</math>,
-<math>a_1b_2=b_1a_3</math>,
+<math>abc=ila</math>, <math>efg=ade</math>, <math>ijk=ehi</math>,
-<math>a_2b_3=b_2a_4</math>,
+<math>cdi=gha=kle</math>,
-<math>a_3b_4=b_3a_5</math>,
+<math>lab=def=hij</math>
 == Other notatable representatives ==
@@ Line 87: / Line 60: @@
 == 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:
+<!-- Labelling the simple <math>B</math>-modules by <math>1,2,3,4,5,6,7</math>, the projective indecomposable modules have Loewy structure as follows:
 <math>\begin{array}{ccccccc}
     \begin{array}{c}
-      S_1 \\
+\\
-     S_2 S_3 S_5 \\
+\ 4 \ 6 \\
-     S_4 S_6 S_7 \\
+\ 1 \ 5 \ 1 \ 7 \ 1 \\
       S_1 \\
     \end{array},
@@ Line 145: / Line 120: @@
 All irreducible characters have height zero.
 [[C2xC2xC2|Back to <math>C_2 \times C_2 \times C_2</math>]]

M(8,5,5) - Revision history

Charles Eaton at 16:22, 21 November 2018

Charles Eaton: Relations and references

Charles Eaton: [Al79] reference

Charles Eaton: Created page with "{{blockbox |title = M(8,5,5) - B_0(kSL_2(8)) |image = M(8,5,5)quiver.png |representative = B_0(kSL_2(8)) |defect = C2xC2xC2|C_2 \times C_2..."

Charles Eaton: Created page with "{{blockbox |title = M(8,5,5) - $B_0(kSL_2(8))$ |image = M(8,5,5)quiver.png |representative = $B_0(kSL_2(8))$ |defect = C2xC2xC2| $C_2 \times C_2..."$