C7 - Revision history

Charles Eaton at 11:58, 22 November 2018

2018-11-22T11:58:17Z

Charles Eaton at 10:12, 14 October 2018

2018-10-14T10:12:31Z

Charles Eaton at 10:06, 14 October 2018

2018-10-14T10:06:20Z

Charles Eaton: M(7,1,14) example

2018-10-14T10:04:44Z

M(7,1,14) example

Charles Eaton at 06:55, 28 September 2018

2018-09-28T06:55:35Z

Charles Eaton: Picard groups

2018-09-28T06:52:03Z

Picard groups

Charles Eaton at 15:24, 26 September 2018

2018-09-26T15:24:01Z

Charles Eaton: Undo revision 379 by Mcbssce (talk)

2018-09-11T17:04:59Z

Undo revision 379 by Mcbssce (talk)

Charles Eaton at 17:02, 11 September 2018

2018-09-11T17:02:48Z

Charles Eaton at 09:32, 11 September 2018

2018-09-11T09:32:46Z

@@ Line 3: / Line 3: @@
 == Blocks with defect group <math>C_7</math> ==
-These are blocks with [[cyclic defect groups]] and so they are described by [[Brauer trees]]. There are candidate Brauer trees with no known block realising them.
+These are [[blocks with cyclic defect groups]] and so they are described by [[Brauer trees]]. There are candidate Brauer trees with no known block realising them.
 All <math>k</math>-Morita equivalence classes lift to <math>\mathcal{O}</math>-classes.

@@ Line 50: / Line 50: @@
 |[[M(7,1,13)]] || <math>B_0(k({}^2G_2(27)))</math> || 1 ||7 ||6 ||<math>C_6</math> || <math>1</math> || ||1 ||1 || [[Image:M(7,1,13)tree.png|45px]]
 |-
-|[[M(7,1,14)]] ||<math>B_0(k({}^2G_2(243)))</math><ref>Theorem 3.8 of [[References|[Du13]]] gives existence for <math>q=3^{2m+1}</math> such that <math>(3^{2m+1}-3^{m+1}+1)_7=7</math></ref> || 1 ||7 ||6 ||<math>C_6</math> ||<math>1</math> || ||1 ||1 || [[Image:M(7,1,14)tree.png|45px]]
+|[[M(7,1,14)]] ||<math>B_0(k({}^2G_2(243)))</math><ref>Theorem 3.8 of [[References|[Du14]]] gives existence for <math>q=3^{2m+1}</math> such that <math>(3^{2m+1}-3^{m+1}+1)_7=7</math></ref> || 1 ||7 ||6 ||<math>C_6</math> ||<math>1</math> || ||1 ||1 || [[Image:M(7,1,14)tree.png|45px]]
 |-
 | || || ||7 ||6 ||<math>C_6</math> |||| ||1 ||1 || [[Image:M(7,1,15)tree.png|45px]]

@@ Line 50: / Line 50: @@
 |[[M(7,1,13)]] || <math>B_0(k({}^2G_2(27)))</math> || 1 ||7 ||6 ||<math>C_6</math> || <math>1</math> || ||1 ||1 || [[Image:M(7,1,13)tree.png|45px]]
 |-
-|[[M(7,1,14)]] ||<math>B_0(k({}^2G_2(243)))</math><ref>Theorem 3.8 of [[References|[Du18]]] gives existence for <math>q=3^{2m+1}</math> such that <math>(3^{2m+1}-3^{m+1}+1)_7=7</math></ref> || 1 ||7 ||6 ||<math>C_6</math> ||<math>1</math> || ||1 ||1 || [[Image:M(7,1,14)tree.png|45px]]
+|[[M(7,1,14)]] ||<math>B_0(k({}^2G_2(243)))</math><ref>Theorem 3.8 of [[References|[Du13]]] gives existence for <math>q=3^{2m+1}</math> such that <math>(3^{2m+1}-3^{m+1}+1)_7=7</math></ref> || 1 ||7 ||6 ||<math>C_6</math> ||<math>1</math> || ||1 ||1 || [[Image:M(7,1,14)tree.png|45px]]
 |-
 | || || ||7 ||6 ||<math>C_6</math> |||| ||1 ||1 || [[Image:M(7,1,15)tree.png|45px]]

@@ Line 13: / Line 13: @@
 ! scope="col"| Class
 ! scope="col"| Representative
 ! scope="col"| <math>k(B)</math>
 ! scope="col"| <math>l(B)</math>
@@ Line 23: / Line 24: @@
 |-
-|[[M(7,1,1)]] || <math>kC_7</math> ||7 ||1 ||<math>1</math> || <math>C_7:C_6</math> || ||1 ||1 || [[Image:M(7,1,1)tree.png|45px]]
+|[[M(7,1,1)]] || <math>kC_7</math> || 1 ||7 ||1 ||<math>1</math> || <math>C_7:C_6</math> || ||1 ||1 || [[Image:M(7,1,1)tree.png|45px]]
 |-
-|[[M(7,1,2)]] || <math>kD_{14}</math> ||5 ||2 ||<math>C_2</math> || <math>C_2 \times C_3</math> || ||1 ||1 || [[Image:M(7,1,2)tree.png|45px]]
+|[[M(7,1,2)]] || <math>kD_{14}</math> || 1 ||5 ||2 ||<math>C_2</math> || <math>C_2 \times C_3</math> || ||1 ||1 || [[Image:M(7,1,2)tree.png|45px]]
 |-
-|[[M(7,1,3)]] || <math>B_0(kPSL_2(13))</math> ||5 ||2 ||<math>C_2</math> || <math>C_3</math> || ||1 ||1 || [[Image:M(7,1,3)tree.png|45px]]
+|[[M(7,1,3)]] || <math>B_0(kPSL_2(13))</math> || 1 ||5 ||2 ||<math>C_2</math> || <math>C_3</math> || ||1 ||1 || [[Image:M(7,1,3)tree.png|45px]]
 |-
-|[[M(7,1,4)]] || <math>k(C_7:C_3)</math> ||5 ||3 ||<math>C_3</math> || <math>C_3 \times C_2</math> || ||1 ||1 || [[Image:M(7,1,4)tree.png|45px]]
+|[[M(7,1,4)]] || <math>k(C_7:C_3)</math> || 1 ||5 ||3 ||<math>C_3</math> || <math>C_3 \times C_2</math> || ||1 ||1 || [[Image:M(7,1,4)tree.png|45px]]
 |-
-|[[M(7,1,5)]] || <math>B_9(k(3.A_7))</math> ||5 ||3 ||<math>C_3</math> || <math>C_2</math> || ||1 ||1 || [[Image:M(7,1,5)tree.png|45px]]
+|[[M(7,1,5)]] || <math>B_9(k(3.A_7))</math> || 1 ||5 ||3 ||<math>C_3</math> || <math>C_2</math> || ||1 ||1 || [[Image:M(7,1,5)tree.png|45px]]
 |-
-|[[M(7,1,6)]] || <math>B_0(kA_7)</math> ||5 ||3 ||<math>C_3</math> ||<math>C_2</math> || ||1 ||1 || [[Image:M(7,1,6)tree.png|45px]]
+|[[M(7,1,6)]] || <math>B_0(kA_7)</math> || 1 ||5 ||3 ||<math>C_3</math> ||<math>C_2</math> || ||1 ||1 || [[Image:M(7,1,6)tree.png|45px]]
 |-
-|[[M(7,1,7)]] || <math>B_{15}(k(6.A_7))</math> ||5 ||3 ||<math>C_3</math> || <math>C_2</math> || ||1 ||1 || [[Image:M(7,1,7)tree.png|45px]]
+|[[M(7,1,7)]] || <math>B_{15}(k(6.A_7))</math> || 1 ||5 ||3 ||<math>C_3</math> || <math>C_2</math> || ||1 ||1 || [[Image:M(7,1,7)tree.png|45px]]
 |-
-|[[M(7,1,8)]] || <math>k(C_7:C_6)</math> ||7 ||6 ||<math>C_6</math> || <math>C_6</math> || ||1 ||1 || [[Image:M(7,1,8)tree.png|45px]]
+|[[M(7,1,8)]] || <math>k(C_7:C_6)</math> || 1 ||7 ||6 ||<math>C_6</math> || <math>C_6</math> || ||1 ||1 || [[Image:M(7,1,8)tree.png|45px]]
 |-,
-|[[M(7,1,9)]] || <math>B_0(k(S_7))</math> ||7 ||6 ||<math>C_6</math> || <math>C_2</math> || ||1 ||1 || [[Image:M(7,1,9)tree.png|45px]]
+|[[M(7,1,9)]] || <math>B_0(k(S_7))</math> || 1 ||7 ||6 ||<math>C_6</math> || <math>C_2</math> || ||1 ||1 || [[Image:M(7,1,9)tree.png|45px]]
 |-
-|[[M(7,1,10)]] || <math>B_0(kPSL_3(29^3).3)</math> ||7 ||6 ||<math>C_6</math> || <math>C_3</math> || ||1 ||1 || [[Image:M(7,1,10)tree.png|45px]]
+|[[M(7,1,10)]] || <math>B_0(kPSL_3(29^3).3)</math> || 1 ||7 ||6 ||<math>C_6</math> || <math>C_3</math> || ||1 ||1 || [[Image:M(7,1,10)tree.png|45px]]
 |-
-|[[M(7,1,11)]] || <math>B_{16}(k(2.J_2))</math> ||7 ||6 ||<math>C_6</math> || ? || ||1 ||1 || [[Image:M(7,1,11)tree.png|45px]]
+|[[M(7,1,11)]] || <math>B_{16}(k(2.J_2))</math> || 1 ||7 ||6 ||<math>C_6</math> || ? || ||1 ||1 || [[Image:M(7,1,11)tree.png|45px]]
 |-
-|[[M(7,1,12)]] || <math>B_{29}(k(2.Ru))</math> ||7 ||6 ||<math>C_6</math> || <math>1</math> || ||1 ||1 || [[Image:M(7,1,12)tree.png|45px]]
+|[[M(7,1,12)]] || <math>B_{29}(k(2.Ru))</math> || 1 ||7 ||6 ||<math>C_6</math> || <math>1</math> || ||1 ||1 || [[Image:M(7,1,12)tree.png|45px]]
 |-
-|[[M(7,1,13)]] || <math>B_0(k({}^2G_2(27)))</math> ||7 ||6 ||<math>C_6</math> || <math>1</math> || ||1 ||1 || [[Image:M(7,1,13)tree.png|45px]]
+|[[M(7,1,13)]] || <math>B_0(k({}^2G_2(27)))</math> || 1 ||7 ||6 ||<math>C_6</math> || <math>1</math> || ||1 ||1 || [[Image:M(7,1,13)tree.png|45px]]
 |-
-|[[M(7,1,14)]] || ||7 ||6 ||<math>C_6</math> |||| ||1 ||1 || [[Image:M(7,1,14)tree.png|45px]]
+|[[M(7,1,14)]] ||<math>B_0(k({}^2G_2(243)))</math><ref>Theorem 3.8 of [[References|[Du18]]] gives existence for <math>q=3^{2m+1}</math> such that <math>(3^{2m+1}-3^{m+1}+1)_7=7</math></ref> || 1 ||7 ||6 ||<math>C_6</math> ||<math>1</math> || ||1 ||1 || [[Image:M(7,1,14)tree.png|45px]]
 |-
-| || ||7 ||6 ||<math>C_6</math> |||| ||1 ||1 || [[Image:M(7,1,15)tree.png|45px]]
+| || || ||7 ||6 ||<math>C_6</math> |||| ||1 ||1 || [[Image:M(7,1,15)tree.png|45px]]
 |-
-| || ||7 ||6 ||<math>C_6</math> |||| ||1 ||1 || [[Image:M(7,1,16)tree.png|45px]]
+| || || ||7 ||6 ||<math>C_6</math> |||| ||1 ||1 || [[Image:M(7,1,16)tree.png|45px]]
 |-
-| || ||7 ||6 ||<math>C_6</math> |||| ||1 ||1 || [[Image:M(7,1,17)tree.png|45px]]
+| || || ||7 ||6 ||<math>C_6</math> |||| ||1 ||1 || [[Image:M(7,1,17)tree.png|45px]]
 |-
-| || ||7 ||6 ||<math>C_6</math> |||| ||1 ||1 || [[Image:M(7,1,18)tree.png|45px]]
+| || || ||7 ||6 ||<math>C_6</math> |||| ||1 ||1 || [[Image:M(7,1,18)tree.png|45px]]
 |-
-| || ||7 ||6 ||<math>C_6</math> |||| ||1 ||1 || [[Image:M(7,1,19)tree.png|45px]]
+| || || ||7 ||6 ||<math>C_6</math> |||| ||1 ||1 || [[Image:M(7,1,19)tree.png|45px]]
 |-
-| || ||7 ||6 ||<math>C_6</math> |||| ||1 ||1 || [[Image:M(7,1,20)tree.png|45px]]
+| || || ||7 ||6 ||<math>C_6</math> |||| ||1 ||1 || [[Image:M(7,1,20)tree.png|45px]]
 |-
-| || ||7 ||6 ||<math>C_6</math> |||| ||1 ||1 || [[Image:M(7,1,21)tree.png|45px]]
+| || || ||7 ||6 ||<math>C_6</math> |||| ||1 ||1 || [[Image:M(7,1,21)tree.png|45px]]
 |}
 To do:
 *Picard group for [[M(7,1,11)]] trivial or <math>C_2</math>?

@@ Line 2: / Line 2: @@
 == Blocks with defect group <math>C_7</math> ==
 These are blocks with [[cyclic defect groups]] and so they are described by [[Brauer trees]]. There are candidate Brauer trees with no known block realising them.
@@ Line 70: / Line 68: @@
 To do:
 *The tree labelling the potential class M(7,1,14) is believed to be realised, but an example has not been confirmed.

← Older revision		Revision as of 15:24, 26 September 2018
Line 2:		Line 2:

	== Blocks with defect group <math>C_7</math> ==		== Blocks with defect group <math>C_7</math> ==
		+
		+	[[Image:under-construction.png\|50px\|left]]

	These are blocks with [[cyclic defect groups]] and so they are described by [[Brauer trees]]. There are candidate Brauer trees with no known block realising them.		These are blocks with [[cyclic defect groups]] and so they are described by [[Brauer trees]]. There are candidate Brauer trees with no known block realising them.

@@ Line 29: / Line 29: @@
 |[[M(7,1,3)]] || <math>B_0(kPSL_2(13))</math> ||5 ||2 ||<math>C_2</math> || || ||1 ||1 || [[Image:M(7,1,3)tree.png|45px]]
 |-
-|[[M(7,1,4)]]
+|[[M(7,1,4)]] || <math>k(C_7:C_3)</math> ||5 ||3 ||<math>C_3</math> || || ||1 ||1 || [[Image:M(7,1,4)tree.png|45px]]

@@ Line 43: / Line 43: @@
 |[[M(7,1,10)]] || <math>B_0(kPSL_3(29^3).3)</math> ||7 ||6 ||<math>C_6</math> |||| ||1 ||1 || [[Image:M(7,1,10)tree.png|45px]]
 |-
-|[[M(7,1,11)]] || <math>B_{16}(k(2.J_2))</math> ||7 ||6 ||<math>C_6</math> |||| ||1 ||1 || [[Image:M(7,1,11)tree.png|45px]] CHECK
+|[[M(7,1,11)]] || <math>B_{16}(k(2.J_2))</math> ||7 ||6 ||<math>C_6</math> |||| ||1 ||1 || [[Image:M(7,1,11)tree.png|45px]]
 |-
 |[[M(7,1,12)]] || <math>B_{29}(k(2.Ru))</math> ||7 ||6 ||<math>C_6</math> |||| ||1 ||1 || [[Image:M(7,1,12)tree.png|45px]]
@@ Line 63: / Line 63: @@
 To do:
 *The tree labelling the potential class M(7,1,14) is believed to be realised, but an example has not been confirmed.