M(32,51,12) - Revision history

CesareGArdito at 15:55, 9 December 2019

2019-12-09T15:55:01Z

CesareGArdito at 15:53, 9 December 2019

2019-12-09T15:53:18Z

CesareGArdito at 15:53, 9 December 2019

2019-12-09T15:53:01Z

CesareGArdito at 15:07, 9 December 2019

2019-12-09T15:07:23Z

CesareGArdito: /* Covering blocks and covered blocks */

2019-12-06T13:36:39Z

‎Covering blocks and covered blocks

CesareGArdito: Created page with "{{blockbox |title = M(32,51,12) - $B_0(k(SL_2(16) \times C_2))$ |image = |representative = $B_0(k(SL_2(16) \times C_2))$ |defect = (C2)%5E5|<..."

2019-12-06T13:35:36Z

Created page with "{{blockbox |title = M(32,51,12) - <math>B_0(k(SL_2(16) \times C_2))</math> |image = |representative = <math>B_0(k(SL_2(16) \times C_2))</math> |defect = (C2)%5E5|<..."

New page

{{blockbox
|title = M(32,51,12) - <math>B_0(k(SL_2(16) \times C_2))</math>
|image =  
|representative = <math>B_0(k(SL_2(16) \times C_2))</math>
|defect = [[(C2)%5E5|<math>(C_2)^5</math>]]
|inertialquotients = <math>C_{15}</math>
|k(B) = 32
|l(B) = 15
|k-morita-frob = 1
|Pic-k=  
|cartan = See below.
|defect-morita-inv? = Yes
|inertial-morita-inv? = Yes
|O-morita? = Yes
|O-morita = <math>B_0(\mathcal{O}(SL_2(16) \times C_2))</math>
|decomp = See below.
|O-morita-frob = 1
|Pic-O =
|PIgroup =
|source? = No
|sourcereps =
|k-derived-known? = Yes
|k-derived = [[M(32,51,11)]]
|O-derived-known? = Yes
|coveringblocks =
|coveredblocks =
|pcoveringblocks =
}}

== Basic algebra ==

== Other notatable representatives ==

== Covering blocks and covered blocks ==

Let <math>N \triangleleft G</math> with prime <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> is in M(32,51,11), then <math>B</math> is in [[M(32,51,2)]], [[M(32,51,5)]], or M(32,51,11).

== Projective indecomposable modules ==

== Irreducible characters ==

All irreducible characters have height zero.

== Cartan matrix ==
<math>\left( \begin{array}{ccc}
32 & 16 & 16 & 16 & 16 & 8 & 8 & 8 & 8 & 8 & 8 & 4 & 4 & 4 & 4 \\
16 & 16 & 8 & 8 & 8 & 4 & 8 & 4 & 4 & 8 & 0 & 0 & 2 & 4 & 0 \\
16 & 8 & 16 & 8 & 8 & 8 & 4 & 0 & 4 & 4 & 8 & 0 & 0 & 2 & 4 \\
16 & 8 & 8 & 16 & 8 & 4 & 8 & 8 & 0 & 4 & 4 & 4 & 0 & 0 & 2 \\
16 & 8 & 8 & 8 & 16 & 8 & 4 & 4 & 8 & 0 & 4 & 2 & 4 & 0 & 0 \\
8 & 4 & 8 & 4 & 8 & 8 & 2 & 0 & 4 & 0 & 4 & 0 & 0 & 0 & 0 \\
8 & 8 & 4 & 8 & 4 & 2 & 8 & 4 & 0 & 4 & 0 & 0 & 0 & 0 & 0 \\
8 & 4 & 0 & 8 & 4 & 0 & 4 & 8 & 0 & 2 & 0 & 4 & 0 & 0 & 0 \\
8 & 4 & 4 & 0 & 8 & 4 & 0 & 0 & 8 & 0 & 2 & 0 & 4 & 0 & 0 \\
8 & 8 & 4 & 4 & 0 & 0 & 4 & 2 & 0 & 8 & 0 & 0 & 0 & 4 & 0 \\
8 & 0 & 8 & 4 & 4 & 4 & 0 & 0 & 2 & 0 & 8 & 0 & 0 & 0 & 4 \\
4 & 0 & 0 & 4 & 2 & 0 & 0 & 4 & 0 & 0 & 0 & 4 & 0 & 0 & 0 \\
4 & 2 & 0 & 0 & 4 & 0 & 0 & 0 & 4 & 0 & 0 & 0 & 4 & 0 & 0 \\
4 & 4 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 0 & 0 & 0 & 4 & 0 \\
4 & 0 & 4 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 0 & 0 & 0 & 4 \\
\end{array} \right)</math>

== Decomposition matrix ==

<math>\left( \begin{array}{ccc}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\
1 & 1 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\
1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 \\
1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\
1 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
1 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\
1 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 \\
1 & 1 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 \\
1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\
1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\
1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\
1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 \\
1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\
1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0
\end{array}\right)</math>

[[(C2)%5E5|Back to <math>(C_2)^5</math>]]

@@ Line 28: / Line 28: @@
 }}
-A block with defect group [[(C2)%5E5|<math>(C_2)^5</math>]] and inertial quotient <math>C_{15}</math> is Morita equivalent to [[M(32,51,11)]] or M(32,51,12), and the latter are derived equivalent.
+A block with defect group [[(C2)%5E5|<math>(C_2)^5</math>]] and inertial quotient <math>C_{15}</math> is in this Morita equivalence class or in [[M(32,51,11)]], which is derived equivalent to this class.
 It is unknown whether this Morita equivalence class contains blocks with inertial quotient <math>C_7:C_3 \times C_3</math> (with action as in [[M(32,51,24)]]).

@@ Line 28: / Line 28: @@
 }}
-A block with defect group [[(C2)%5E5|<math>(C_2)^5</math>]] and inertial quotient <math>C_5</math> is Morita equivalent to [[M(32,51,11)]] or M(32,51,12), and the latter are derived equivalent.
+A block with defect group [[(C2)%5E5|<math>(C_2)^5</math>]] and inertial quotient <math>C_{15}</math> is Morita equivalent to [[M(32,51,11)]] or M(32,51,12), and the latter are derived equivalent.
 It is unknown whether this Morita equivalence class contains blocks with inertial quotient <math>C_7:C_3 \times C_3</math> (with action as in [[M(32,51,24)]]).

← Older revision		Revision as of 15:07, 9 December 2019
Line 28:		Line 28:
	}}		}}

		+	A block with defect group [[(C2)%5E5\|<math>(C_2)^5</math>]] and inertial quotient <math>C_5</math> is Morita equivalent to [[M(32,51,11)]] or M(32,51,12), and the latter are derived equivalent.
		+
		+	It is unknown whether this Morita equivalence class contains blocks with inertial quotient <math>C_7:C_3 \times C_3</math> (with action as in [[M(32,51,24)]]).

	== Basic algebra ==		== Basic algebra ==

@@ Line 4: / Line 4: @@
 |representative =  <math>B_0(k(SL_2(16) \times C_2))</math>
 |defect = [[(C2)%5E5|<math>(C_2)^5</math>]]
-|inertialquotients = <math>C_{15}</math>
+|inertialquotients = <math>C_{15}</math>, ?
 |k(B) = 32
 |l(B) = 15

@@ Line 37: / Line 37: @@
 Let <math>N \triangleleft G</math> with prime <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> is in M(32,51,11), then <math>B</math> is in [[M(32,51,2)]], [[M(32,51,5)]], or M(32,51,11).
+If <math>b</math> is in M(32,51,12), then <math>B</math> is also in M(32,51,12).
 == Projective indecomposable modules ==

M(32,51,12) - Revision history

CesareGArdito at 15:55, 9 December 2019

CesareGArdito at 15:53, 9 December 2019

CesareGArdito at 15:53, 9 December 2019

CesareGArdito at 15:07, 9 December 2019

CesareGArdito: /* Covering blocks and covered blocks */

CesareGArdito: Created page with "{{blockbox |title = M(32,51,12) - B_0(k(SL_2(16) \times C_2)) |image = |representative = B_0(k(SL_2(16) \times C_2)) |defect = (C2)%5E5|<..."

CesareGArdito: Created page with "{{blockbox |title = M(32,51,12) - $B_0(k(SL_2(16) \times C_2))$ |image = |representative = $B_0(k(SL_2(16) \times C_2))$ |defect = (C2)%5E5|<..."