M(16,2,2) - Revision history

Charles Eaton at 09:57, 28 July 2019

2019-07-28T09:57:54Z

Charles Eaton: PI group

2019-01-22T14:00:12Z

PI group

Charles Eaton: Corrected Picard group

2018-11-14T13:21:53Z

Corrected Picard group

Charles Eaton at 19:51, 22 October 2018

2018-10-22T19:51:40Z

Charles Eaton at 19:34, 22 October 2018

2018-10-22T19:34:41Z

Charles Eaton: Added notes and ref for O-Picard group

2018-10-04T09:19:39Z

Added notes and ref for O-Picard group

Charles Eaton at 08:52, 4 October 2018

2018-10-04T08:52:43Z

Charles Eaton: /* Projective indecomposable modules */

2018-10-04T08:42:19Z

‎Projective indecomposable modules

Charles Eaton at 21:13, 3 October 2018

2018-10-03T21:13:29Z

Charles Eaton: Created page with "left {{blockbox |title = M(16,2,2) - $k((C_4 \times C_4):C_3)$ |image = M(4,2,3)quiver.png |representative = $k((C_4 \..."$

2018-10-03T20:26:20Z

Created page with "50px|left {{blockbox |title = M(16,2,2) - <math>k((C_4 \times C_4):C_3)</math> |image = M(4,2,3)quiver.png |representative = <math>k((C_4 \..."

New page

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

{{blockbox
|title = M(16,2,2) - <math>k((C_4 \times C_4):C_3)</math>
|image = M(4,2,3)quiver.png
|representative = <math>k((C_4 \times C_4):C_3)</math>
|defect = [[C2xC2|<math>C_4 \times C_4</math>]]
|inertialquotients = <math>C_3</math>
|k(B) = 8
|l(B) = 3
|k-morita-frob = 1
|Pic-k= <math></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.

[[C2xC2|Back to <math>C_2 \times C_2</math>]]

[[Category: Morita equivalence classes|4,2,3]]
[[Category: Blocks with defect group C2xC2]]
[[Category: Tame blocks|4,2,3]]

@@ Line 30: / Line 30: @@
 |O-morita-frob = 1
 |Pic-O = <math>C_2 \times S_3</math><ref>Proposition 4.3 of [[References|[BKL18]]]</ref>
-|PIgroup =
+|PIgroup = <math>S_3 \times D_8 \times C_2</math><ref>Using GAP, with code from [[References#R|[Ru11]]]</ref>
 |source? = No
 |sourcereps =

@@ Line 29: / Line 29: @@
 \end{array}\right)</math>
 |O-morita-frob = 1
-|Pic-O = <math>S_3</math><ref>Proposition 4.3 of [[References|[BKL18]]]</ref>
+|Pic-O = <math>C_2 \times S_3</math><ref>Proposition 4.3 of [[References|[BKL18]]]</ref>
 |PIgroup =
 |source? = No

@@ Line 1: / Line 1: @@
 {{blockbox
-|title = M(16,2,2) - <math>k((C_4 \times C_4):C_3)</math>
+|title = M(9,2,3) - <math>k((C_4 \times C_4):C_3)</math>
 |image = M(4,2,3)quiver.png
 |representative =  <math>k((C_4 \times C_4):C_3)</math>
@@ Line 30: / Line 30: @@
 |O-morita-frob = 1
 |Pic-O = <math>S_3</math><ref>Proposition 4.3 of [[References|[BKL18]]]</ref>
 |source? = No
 |sourcereps =
@@ Line 37: / Line 38: @@
 |coveringblocks =
 |coveredblocks = [[M(16,2,1)]]
 }}
@@ Line 87: / Line 89: @@
 All irreducible characters have height zero.
 [[C4xC4|Back to <math>C_4 \times C_4</math>]]

@@ Line 29: / Line 29: @@
 \end{array}\right)</math>
 |O-morita-frob = 1
-|Pic-O = <math>S_3</math>
+|Pic-O = <math>S_3</math><ref>Proposition 4.3 of [[References|[BKL18]]]</ref>
 |source? = No
 |sourcereps =
@@ Line 38: / Line 38: @@
 |coveredblocks = [[M(16,2,1)]]
 }}
 == Basic algebra ==
@@ Line 93: / Line 91: @@
 [[C4xC4|Back to <math>C_4 \times C_4</math>]]
-== References ==
+== Notes ==
-{{Reflist}}
+<references />

@@ Line 1: / Line 1: @@
 {{blockbox
 |title = M(16,2,2) - <math>k((C_4 \times C_4):C_3)</math>
@@ Line 40: / Line 38: @@
 |coveredblocks = [[M(16,2,1)]]
 }}
 == Basic algebra ==
@@ Line 91: / Line 91: @@
-[[C2xC2|Back to <math>C_2 \times C_2</math>]]
+[[C4xC4|Back to <math>C_4 \times C_4</math>]]
 [[Category: Morita equivalence classes|16,2,2]]
 [[Category: Blocks with defect group C4xC4]]
 [[Category: Tame blocks|4,2,3]]

@@ Line 3: / Line 3: @@
 |image = M(4,2,3)quiver.png
 |representative =  <math>k((C_4 \times C_4):C_3)</math>
-|defect = [[C2xC2|<math>C_4 \times C_4</math>]]
+|defect = [[C4xC4|<math>C_4 \times C_4</math>]]
 |inertialquotients = <math>C_3</math>
 |k(B) = 8

@@ Line 51: / Line 51: @@
 == 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}
+    \begin{array}{c}
-      & S_1 & \\
+\\
-      S_2 & & S_3 \\
+\ 3  \\
-      & S_1 & \\
+\ 1 \ 2 \\
     \end{array},
+&
-    \begin{array}{ccc}
+    \begin{array}{c}
-      & S_2 & \\
+\\
-      S_1 & & S_3 \\
+\ 3  \\
-      & S_2 & \\
+\ 2 \ 1 \\
     \end{array},
+&
-   \begin{array}{ccc}
+   \begin{array}{c}
-      & S_3 & \\
+\\
-      S_1 & & S_2 \\
+\ 2  \\
-      & S_3 & \\
+\ 3 \ 1 \\
-    \end{array}
+\ 1 \ 2 \ 3 \\
 \end{array}
-</math> -->
+</math>
 == Irreducible characters ==

@@ Line 12: / Line 12: @@
 |Pic-k= <math></math>
 |cartan = <math>\left( \begin{array}{ccc}
-& 1 & 1 \\
+& 5 & 5 \\
-& 2 & 1 \\
+& 6 & 5 \\
-& 1 & 2 \\
+& 5 & 6 \\
 \end{array} \right)</math>
 |defect-morita-inv? = Yes
 |inertial-morita-inv? = Yes
 |O-morita? = Yes
-|O-morita = <math>\mathcal{O}A_4</math>
+|O-morita = <math>\mathcal{O}((C_4 \times C_4):C_3)</math>
 |decomp = <math>\left( \begin{array}{ccc}
 & 0 & 0 \\
 & 1 & 0 \\
 & 0 & 1 \\
 & 1 & 1 \\
 \end{array}\right)</math>
 |O-morita-frob = 1
-|Pic-O = <math>\mathcal{T}(B)=S_3</math>
+|Pic-O = <math>S_3</math>
-|source? = Yes
+|source? = No
-|sourcereps = <math>\mathcal{O}A_4</math>
+|sourcereps =
 |k-derived-known? = Yes
-|k-derived = [[M(4,2,2)]]
+|k-derived = Forms its own derived equivalence class
 |O-derived-known? = Yes
 }}
@@ Line 39: / Line 45: @@
 '''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
+'''Relations w.r.t. <math>k</math>:''' abca=bcab=cabc=0, dfed=fedf=edfe=0, ad=fc, be=da, cf=eb
 == Other notatable representatives ==
 == 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>S_1, S_2, S_3</math>, the projective indecomposable modules have Loewy structure as follows:
 <math>\begin{array}{ccc}
@@ Line 77: / Line 72: @@
     \end{array}
 \end{array}
-</math>
+</math> -->
 == Irreducible characters ==
@@ Line 86: / Line 81: @@
 [[C2xC2|Back to <math>C_2 \times C_2</math>]]
-[[Category: Morita equivalence classes|4,2,3]]
+[[Category: Morita equivalence classes|16,2,2]]
-[[Category: Blocks with defect group C2xC2]]
+[[Category: Blocks with defect group C4xC4]]
 [[Category: Tame blocks|4,2,3]]