M(8,4,2) - Revision history

Charles Eaton: Pic_O

2022-05-24T08:40:04Z

Pic_O

Charles Eaton: /* Basic algebra */

2021-06-03T15:41:01Z

‎Basic algebra

Charles Eaton at 21:49, 3 December 2018

2018-12-03T21:49:09Z

Charles Eaton at 21:44, 3 December 2018

2018-12-03T21:44:47Z

Charles Eaton at 17:54, 6 October 2018

2018-10-06T17:54:02Z

Charles Eaton at 17:51, 6 October 2018

2018-10-06T17:51:23Z

Charles Eaton: Created page with "{{blockbox |title = M(8,4,2) - $B_0(kSL_2(5))$ |image = M(4,2,2)quiver.png |representative = $B_0(kSL_2(5))$ |defect = $Q_8$ |inertia..."

2018-10-06T17:10:28Z

Created page with "{{blockbox |title = M(8,4,2) - <math>B_0(kSL_2(5))</math> |image = M(4,2,2)quiver.png |representative = <math>B_0(kSL_2(5))</math> |defect = <math>Q_8</math> |inertia..."

New page

{{blockbox
|title = M(8,4,2) - <math>B_0(kSL_2(5))</math>
|image = M(4,2,2)quiver.png
|representative = <math>B_0(kSL_2(5))</math>
|defect = [[Q8|<math>Q_8</math>]]
|inertialquotients = <math>C_3</math>
|k(B) = 5
|l(B) = 3
|k-morita-frob = 1
|Pic-k=
|cartan = <math>\left( \begin{array}{ccc}
4 & 4 & 2 \\
4 & 8 & 4 \\
2 & 4 & 4 \\
\end{array} \right)</math>
|defect-morita-inv? = Yes
|inertial-morita-inv? = Yes
|O-morita? = Yes
|O-morita = <math>B_0(\mathcal{O}SL_2(5))</math>
|decomp = <math>\left( \begin{array}{ccc}
0 & 1 & 0 \\
1 & 1 & 0 \\
0 & 1 & 1 \\
1 & 1 & 1 \\
1 & 0 & 0 \\
0 & 0 & 1 \\
1 & 2 & 1 \\
\end{array}\right)</math>
|O-morita-frob = 1
|Pic-O =
|source? = No
|sourcereps =
|k-derived-known? = Yes
|k-derived = [[M(8,2,3)]]
|O-derived-known? = No
|coveringblocks =
|coveredblocks =
}}

These are [[Tame blocks|tame blocks]], and appear in the family <math>D(3 {\cal A})_2</math> in Erdmann's classification (see [[References|[Er88a], [Er88b]]]). The class lifts to a unique <math>\mathcal{O}</math>-Morita equivalence class by [[References|[Ei16]]]. A derived equivalence with [[M(8,4,3)]] over <math>k</math> was established in [[References|[Ho97]]].

== Basic algebra ==

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

'''Relations w.r.t. <math>k</math>:''' ada=abcdabc, dad=bcdabcd, cbc=cdabcda, bcb=dabcdab, adab=cbcd

== Other notatable representatives ==

== Projective indecomposable modules ==



== Irreducible characters ==

<math>k_0(B)=4, k_1(B)=1</math>

[[Q8|Back to <math>Q_8</math>]]

[[Category: Morita equivalence classes|8,4,2]]
[[Category: Blocks with defect group Q8]]
[[Category: Tame blocks|8,4,2]]

@@ Line 28: / Line 28: @@
 \end{array}\right)</math>
 |O-morita-frob = 1
-|Pic-O =
+|Pic-O = <math>C_2</math><ref>[[References|[Mar]]]</ref>
 |source? = No
 |sourcereps =
@@ Line 90: / Line 90: @@
 [[Q8|Back to <math>Q_8</math>]]
 [[Category: Morita equivalence classes|8,4,2]]
 [[Category: Blocks with defect group Q8]]
 [[Category: Tame blocks|8,4,2]]

@@ Line 46: / Line 46: @@
 '''Quiver:''' a:<1,2>, b:<2,3>, c:<3,2>, d:<2,1>
-'''Relations w.r.t. <math>k</math>:''' ada=abcdabc, dad=bcdabcd, cbc=cdabcda, bcb=dabcdab, adab=cbcd
+'''Relations w.r.t. <math>k</math>:''' ada=abcdabc, dad=bcdabcd, cbc=cdabcda, bcb=dabcdab, adab=cbcd=0
 == Other notatable representatives ==

@@ Line 36: / Line 36: @@
 |coveringblocks =
 |coveredblocks =
 }}
@@ Line 86: / Line 87: @@
 == Irreducible characters ==
-<math>k_0(B)=4, k_1(B)=1</math>
+<math>k_0(B)=4, k_1(B)=3</math>
 [[Q8|Back to <math>Q_8</math>]]

@@ Line 5: / Line 5: @@
 |defect = [[Q8|<math>Q_8</math>]]
 |inertialquotients = <math>C_3</math>
-|k(B) = 5
+|k(B) = 7
 |l(B) = 3
 |k-morita-frob = 1

@@ Line 32: / Line 32: @@
 |sourcereps =
 |k-derived-known? = Yes
-|k-derived = [[M(8,2,3)]]
+|k-derived = [[M(8,4,3)]]
 |O-derived-known? = No
 |coveringblocks =

@@ Line 51: / Line 51: @@
 == Projective indecomposable modules ==
-<!--Labelling the simple <math>B</math>-modules by <math>1,2,3</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}
@@ Line 62: / Line 62: @@
     \end{array},
+&
-    \begin{array}{ccc}
+    \begin{array}{c}
-      & S_2 & \\
+\\
-      \begin{array}{c} S_1 \\ S_2 \\ S_3 \\ \end{array} & \oplus & \begin{array}{c} S_3 \\ S_2 \\ S_1 \\ \end{array} \\
+\ 3 \\
-      & S_2 & \\
+\ 2 \\
     \end{array},
+&
@@ Line 76: / Line 82: @@
     \end{array}
 \end{array}
-</math>-->
+</math>
 == Irreducible characters ==