M(8,4,3) - Revision history

Charles Eaton: Pic_O

2021-06-03T14:51:20Z

Pic_O

Charles Eaton at 15:56, 16 August 2019

2019-08-16T15:56:47Z

Charles Eaton at 21:48, 3 December 2018

2018-12-03T21:48:01Z

Charles Eaton: Created page with "left {{blockbox |title = M(8,4,3) - $kSL_2(3)$ |image = M(4,2,3)quiver.png |representative = $kSL_2(3)$ |defect..."

2018-10-06T18:11:54Z

Created page with "50px|left {{blockbox |title = M(8,4,3) - <math>kSL_2(3)</math> |image = M(4,2,3)quiver.png |representative = <math>kSL_2(3)</math> |defect..."

New page

[[Image:under-construction.png|50px|left]]

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

These are [[Tame blocks|tame blocks]], and appear in the family <math>D(3 {\cal K})</math> in Erdmann's classification (see [[References|[Er88a], [Er88b]]]). The class lifts to a unique <math>\mathcal{O}</math>-Morita equivalence class by [[References|[HKL07]]]. A derived equivalence with [[M(8,4,2)]] over <math>k</math> was established in [[References|[Ho97]]].

== 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=fcf, bc=dad, ca=ebe, fe=ada, df=beb, ed=cfc, dab=0=bed=cfe

== 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|8,4,3]]
[[Category: Blocks with defect group Q8]]
[[Category: Tame blocks|8,4,3]]

@@ Line 30: / Line 30: @@
 \end{array}\right)</math>
 |O-morita-frob = 1
-|Pic-O =
+|Pic-O = <math>\mathcal{T}(B)=S_3</math><ref><math>{\rm Pic}(B)=\mathcal{L}(B)</math> by [[References|[LiMa20b]]], so <math>{\rm Pic}(B)=\mathcal{T}(B)</math></ref>
 |source? = No
 |sourcereps =
@@ Line 80: / Line 80: @@
 <math>k_0(B)=4, k_1(B)=3</math>
-[[Q8|Back to <math>Q_8</math>]]
+== Notes ==
 [[Category: Morita equivalence classes|8,4,3]]
 [[Category: Blocks with defect group Q8]]
 [[Category: Tame blocks|8,4,3]]

@@ Line 41: / Line 41: @@
 }}
-These are [[Tame blocks|tame blocks]], and appear in the family <math>D(3 {\cal K})</math> in Erdmann's classification (see [[References|[Er88a], [Er88b]]]). The class lifts to a unique <math>\mathcal{O}</math>-Morita equivalence class by [[References|[HKL07]]]. A derived equivalence with [[M(8,4,2)]] over <math>k</math> was established in [[References|[Ho97]]].
+These are [[Tame blocks|tame blocks]], and appear in the family <math>Q(3 {\cal K})</math> in Erdmann's classification (see [[References|[Er88a], [Er88b]]]). The class lifts to a unique <math>\mathcal{O}</math>-Morita equivalence class by [[References|[HKL07]]]. A derived equivalence with [[M(8,4,2)]] over <math>k</math> was established in [[References|[Ho97]]].
 == Basic algebra ==

@@ Line 7: / Line 7: @@
 |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 38: / Line 38: @@
 |coveringblocks =
 |coveredblocks =
 }}
@@ Line 77: / Line 78: @@
 == Irreducible characters ==
-<math>k_0(B)=4, k_1(B)=1</math>
+<math>k_0(B)=4, k_1(B)=3</math>

M(8,4,3) - Revision history

Charles Eaton: Pic_O

Charles Eaton at 15:56, 16 August 2019

Charles Eaton at 21:48, 3 December 2018

Charles Eaton: Created page with "left {{blockbox |title = M(8,4,3) - kSL_2(3) |image = M(4,2,3)quiver.png |representative = kSL_2(3) |defect..."

Charles Eaton: Created page with "left {{blockbox |title = M(8,4,3) - $kSL_2(3)$ |image = M(4,2,3)quiver.png |representative = $kSL_2(3)$ |defect..."