M(16,2,2) - Revision history

Charles Eaton at 13:35, 14 September 2026

2026-09-14T13:35:15Z

2026-09-14T13:34:49Z

‎Projective indecomposable modules: Correcte typos

2019-07-28T09:57:54Z

2019-01-22T14:00:12Z

PI group

2018-11-14T13:21:53Z

Corrected Picard group

2018-10-22T19:51:40Z

2018-10-22T19:34:41Z

2018-10-04T09:19:39Z

Added notes and ref for O-Picard group

2018-10-04T08:52:43Z

2018-10-04T08:42:19Z

‎Projective indecomposable modules

← Older revision		Revision as of 13:35, 14 September 2026
Line 98:		Line 98:
	[[Category: Morita equivalence classes\|16,2,2]]		[[Category: Morita equivalence classes\|16,2,2]]
	[[Category: Blocks with defect group C4xC4]]		[[Category: Blocks with defect group C4xC4]]
−	~~[[Category: Tame blocks\|4,2,3]]~~

@@ Line 81: / Line 81: @@
 \ 3 \ 1 \\
 \ 2 \\
 \\
     \end{array}
 \end{array}

@@ 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 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(9,2,3) - <math>k((C_4 \times C_4):C_3)</math>
+|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>

@@ 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 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 ==