M(8,3,6) - Revision history

Charles Eaton at 20:48, 4 October 2018

2018-10-04T20:48:59Z

Charles Eaton at 17:31, 4 October 2018

2018-10-04T17:31:09Z

Charles Eaton: Created page with "{{blockbox |title = M(8,3,6) - $B_0(kPSL_2(7))$ |image = M(4,2,3)quiver.png |representative = $B_0(kPSL_2(7))$ |defect = $D_8$ |inert..."

2018-10-04T17:01:55Z

Created page with "{{blockbox |title = M(8,3,6) - <math>B_0(kPSL_2(7))</math> |image = M(4,2,3)quiver.png |representative = <math>B_0(kPSL_2(7))</math> |defect = <math>D_8</math> |inert..."

New page

{{blockbox
|title = M(8,3,6) - <math>B_0(kPSL_2(7))</math>
|image = M(4,2,3)quiver.png
|representative = <math>B_0(kPSL_2(7))</math>
|defect = [[D8|<math>D_8</math>]]
|inertialquotients = <math>1</math>
|k(B) = 5
|l(B) = 3
|k-morita-frob = 1
|Pic-k=
|cartan = <math>\left( \begin{array}{ccc}
2 & 1 & 1 \\
1 & 3 & 2 \\
1 & 2 & 3 \\
\end{array} \right)</math>
|defect-morita-inv? = Yes
|inertial-morita-inv? = Yes
|O-morita? = No
|O-morita =
|decomp = <math>\left( \begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
1 & 1 & 1 \\
0 & 1 & 1 \\
\end{array}\right)</math>
|O-morita-frob = 1
|Pic-O =
|source? = No
|sourcereps =
|k-derived-known? = Yes
|k-derived = [[M(8,3,4)]], [[M(8,3,5)]]
|O-derived-known? = No
|coveringblocks =
|coveredblocks =
}}

== 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>:''' <math>ab=bc=ca=0</math>, <math>df=fe=ed=0</math>, <math>ad=fc</math>, <math>(be)^2=da</math>, <math>cf=(eb)^2</math>

== Other notatable representatives ==

== Projective indecomposable modules ==



== Irreducible characters ==

@@ Line 35: / Line 35: @@
 |coveredblocks =
 }}
 == Basic algebra ==

@@ Line 46: / Line 46: @@
 == 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</math>, the projective indecomposable modules have Loewy structure as follows:
 <math>\begin{array}{ccc}
     \begin{array}{ccc}
-      & S_1 & \\
+      & 1 & \\
-       S_2 & & S_3 \\
+& & 3 \\
-      & S_1 & \\
+      & 1 & \\
     \end{array},
+&
     \begin{array}{ccc}
-      & S_2 & \\
+      & 2 & \\
-       S_1 & & S_3 \\
+& \oplus & \begin{array}{c} 3 \\ 2 \\ 3 \\ \end{array} \\
-      & S_2 & \\
+      & 2 & \\
     \end{array},
+&
    \begin{array}{ccc}
-      & S_3 & \\
+      & 3 & \\
-       S_1 & & S_2 \\
+& \oplus & \begin{array}{c} 2 \\ 3 \\ 2 \\ \end{array} \\
-      & S_3 & \\
+      & 3 & \\
     \end{array}
 \end{array}
-</math>-->
+</math>
 == Irreducible characters ==
-<!--All irreducible characters have height zero.
+<math>k_0(B)=4, \ k_1(B)=1</math>
-[[C2xC2|Back to <math>C_2 \times C_2</math>]]
+[[D8|Back to <math>D_8</math>]]
-[[Category: Morita equivalence classes|4,2,3]]
+[[Category: Morita equivalence classes|8,3,6]]
-[[Category: Blocks with defect group C2xC2]]
+[[Category: Blocks with defect group D8]]
-[[Category: Tame blocks|4,2,3]]-->
+[[Category: Tame blocks|8,3,6]]

M(8,3,6) - Revision history

Charles Eaton at 20:48, 4 October 2018

Charles Eaton at 17:31, 4 October 2018

Charles Eaton: Created page with "{{blockbox |title = M(8,3,6) - B_0(kPSL_2(7)) |image = M(4,2,3)quiver.png |representative = B_0(kPSL_2(7)) |defect = D_8 |inert..."

Charles Eaton: Created page with "{{blockbox |title = M(8,3,6) - $B_0(kPSL_2(7))$ |image = M(4,2,3)quiver.png |representative = $B_0(kPSL_2(7))$ |defect = $D_8$ |inert..."