Q8 - Revision history

Charles Eaton: Picard groups over O

2022-05-24T08:35:51Z

Picard groups over O

Charles Eaton at 22:44, 18 December 2018

2018-12-18T22:44:32Z

Charles Eaton at 21:43, 3 December 2018

2018-12-03T21:43:51Z

Charles Eaton at 15:49, 27 November 2018

2018-11-27T15:49:51Z

Charles Eaton: Removed under construction label

2018-10-06T18:11:09Z

Removed under construction label

Charles Eaton at 21:46, 26 September 2018

2018-09-26T21:46:38Z

Charles Eaton at 15:26, 26 September 2018

2018-09-26T15:26:18Z

Charles Eaton: Added lifting column and NOTITLE

2018-09-15T08:15:36Z

Added lifting column and __NOTITLE__

Charles Eaton at 20:14, 31 August 2018

2018-08-31T20:14:14Z

Charles Eaton: Created page with "== Blocks with defect group $Q_8$ == These are examples of tame blocks and were first classified over $k$ by Erdmann (see References|[..."

2018-08-31T16:43:22Z

Created page with "== Blocks with defect group <math>Q_8</math> == These are examples of tame blocks and were first classified over <math>k</math> by Erdmann (see References|[..."

New page

== Blocks with defect group <math>Q_8</math> ==

These are examples of [[Tame blocks|tame blocks]] and were first classified over <math>k</math> by Erdmann (see [[References|[Er87] ]]). The classification with respect to <math>\mathcal{O}</math> is still unknown in general, except for the cases [[M(8,4,1)]] and [[M(8,4,3)]], which both lift to unique Morita equivalence classes over <math>\mathcal{O}</math> by [[References|[HKL07] ]] and the theory of [[nilpotent blocks]].

{| class="wikitable"
|-
! scope="col"| Class
! scope="col"| Representative
! scope="col"| <math>k(B)</math>
! scope="col"| <math>l(B)</math>
! scope="col"| Inertial quotients
! scope="col"| <math>{\rm Pic}_\mathcal{O}(B)</math>
! scope="col"| <math>{\rm Pic}_k(B)</math>
! scope="col"| <math>{\rm mf_\mathcal{O}(B)}</math>
! scope="col"| <math>{\rm mf_k(B)}</math>
! scope="col"| Notes

|-
|[[M(8,4,1)]] || <math>k(Q_8)</math> ||5 ||1 ||<math>1</math> || || ||1 ||1 ||
|-
|[[M(8,4,2)]] || <math>B_0(kSL_2(5))</math> ||5 ||3 ||<math>1</math> || || || ||1 || <math>Q(3 {\cal A})_2</math>
|-
|[[M(8,4,3)]] || <math>kSL_2(3)</math> ||5 ||3 ||<math>1</math> || || ||1 ||1 || <math>D(3 {\cal K})</math>
|}

[[M(8,4,2)]] and [[M(8,4,3)]] are derived equivalent over <math>k</math> by [[References|[Ho97] ]].

@@ Line 20: / Line 20: @@
 |-
-|[[M(8,4,1)]] || <math>kQ_8</math> || 1 ||5 ||1 ||<math>1</math> || || ||1 ||1 ||
+|[[M(8,4,1)]] || <math>kQ_8</math> || 1 ||5 ||1 ||<math>1</math> || <math>S_4</math> || ||1 ||1 ||
 |-
-|[[M(8,4,2)]] || <math>B_0(kSL_2(5))</math> || 1 ||7 ||3 ||<math>C_3</math> || || || ||1 || <math>Q(3 {\cal A})_2</math>
+|[[M(8,4,2)]] || <math>B_0(kSL_2(5))</math> || 1 ||7 ||3 ||<math>C_3</math> || <math>C_2</math> || || ||1 || <math>Q(3 {\cal A})_2</math>
 |-
-|[[M(8,4,3)]] || <math>kSL_2(3)</math> || 1 ||7 ||3 ||<math>C_3</math> || || ||1 ||1 || <math>Q(3 {\cal K})</math>
+|[[M(8,4,3)]] || <math>kSL_2(3)</math> || 1 ||7 ||3 ||<math>C_3</math> || <math>S_3</math> || ||1 ||1 || <math>Q(3 {\cal K})</math>
 |}
 [[M(8,4,2)]] and [[M(8,4,3)]] are derived equivalent over <math>k</math> by [[References#H|[Ho97] ]].

@@ Line 3: / Line 3: @@
 == Blocks with defect group <math>Q_8</math> ==
-These are examples of [[Tame blocks|tame blocks]] and were first classified over <math>k</math> by Erdmann (see [[References|[Er87] ]]). The Morita equivalence classes lift to unique Morita equivalence classes over <math>\mathcal{O}</math> by [[References|[HKL07], [Ei16]]] and the theory of [[nilpotent blocks]].
+These are examples of [[Tame blocks|tame blocks]] and were first classified over <math>k</math> by Erdmann (see [[References#E|[Er87] ]]). The Morita equivalence classes lift to unique Morita equivalence classes over <math>\mathcal{O}</math> by [[References#H|[HKL07]]], [[References#E|[Ei16]]] and the theory of [[nilpotent blocks]].
 {| class="wikitable"
@@ Line 27: / Line 27: @@
 |}
-[[M(8,4,2)]] and [[M(8,4,3)]] are derived equivalent over <math>k</math> by [[References|[Ho97] ]].
+[[M(8,4,2)]] and [[M(8,4,3)]] are derived equivalent over <math>k</math> by [[References#H|[Ho97] ]].

@@ Line 22: / Line 22: @@
 |[[M(8,4,1)]] || <math>kQ_8</math> || 1 ||5 ||1 ||<math>1</math> || || ||1 ||1 ||
 |-
-|[[M(8,4,2)]] || <math>B_0(kSL_2(5))</math> || 1 ||5 ||3 ||<math>C_3</math> || || || ||1 || <math>Q(3 {\cal A})_2</math>
+|[[M(8,4,2)]] || <math>B_0(kSL_2(5))</math> || 1 ||7 ||3 ||<math>C_3</math> || || || ||1 || <math>Q(3 {\cal A})_2</math>
 |-
-|[[M(8,4,3)]] || <math>kSL_2(3)</math> || 1 ||5 ||3 ||<math>C_3</math> || || ||1 ||1 || <math>Q(3 {\cal K})</math>
+|[[M(8,4,3)]] || <math>kSL_2(3)</math> || 1 ||7 ||3 ||<math>C_3</math> || || ||1 ||1 || <math>Q(3 {\cal K})</math>
 |}
 [[M(8,4,2)]] and [[M(8,4,3)]] are derived equivalent over <math>k</math> by [[References|[Ho97] ]].

@@ Line 22: / Line 22: @@
 |[[M(8,4,1)]] || <math>kQ_8</math> || 1 ||5 ||1 ||<math>1</math> || || ||1 ||1 ||
 |-
-|[[M(8,4,2)]] || <math>B_0(kSL_2(5))</math> || 1 ||5 ||3 ||<math>1</math> || || || ||1 || <math>Q(3 {\cal A})_2</math>
+|[[M(8,4,2)]] || <math>B_0(kSL_2(5))</math> || 1 ||5 ||3 ||<math>C_3</math> || || || ||1 || <math>Q(3 {\cal A})_2</math>
 |-
-|[[M(8,4,3)]] || <math>kSL_2(3)</math> || 1 ||5 ||3 ||<math>1</math> || || ||1 ||1 || <math>Q(3 {\cal K})</math>
+|[[M(8,4,3)]] || <math>kSL_2(3)</math> || 1 ||5 ||3 ||<math>C_3</math> || || ||1 ||1 || <math>Q(3 {\cal K})</math>
 |}
 [[M(8,4,2)]] and [[M(8,4,3)]] are derived equivalent over <math>k</math> by [[References|[Ho97] ]].

← Older revision		Revision as of 18:11, 6 October 2018
Line 2:		Line 2:

	== Blocks with defect group <math>Q_8</math> ==		== Blocks with defect group <math>Q_8</math> ==
−
−	~~[[Image:under-construction.png\|50px\|left]]~~

	These are examples of [[Tame blocks\|tame blocks]] and were first classified over <math>k</math> by Erdmann (see [[References\|[Er87] ]]). The Morita equivalence classes lift to unique Morita equivalence classes over <math>\mathcal{O}</math> by [[References\|[HKL07], [Ei16]]] and the theory of [[nilpotent blocks]].		These are examples of [[Tame blocks\|tame blocks]] and were first classified over <math>k</math> by Erdmann (see [[References\|[Er87] ]]). The Morita equivalence classes lift to unique Morita equivalence classes over <math>\mathcal{O}</math> by [[References\|[HKL07], [Ei16]]] and the theory of [[nilpotent blocks]].

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

	== Blocks with defect group <math>Q_8</math> ==		== Blocks with defect group <math>Q_8</math> ==
		+
		+	[[Image:under-construction.png\|50px\|left]]

	These are examples of [[Tame blocks\|tame blocks]] and were first classified over <math>k</math> by Erdmann (see [[References\|[Er87] ]]). The classification with respect to <math>\mathcal{O}</math> is still unknown in general, except for the cases [[M(8,4,1)]] and [[M(8,4,3)]], which both lift to unique Morita equivalence classes over <math>\mathcal{O}</math> by [[References\|[HKL07] ]] and the theory of [[nilpotent blocks]].		These are examples of [[Tame blocks\|tame blocks]] and were first classified over <math>k</math> by Erdmann (see [[References\|[Er87] ]]). The classification with respect to <math>\mathcal{O}</math> is still unknown in general, except for the cases [[M(8,4,1)]] and [[M(8,4,3)]], which both lift to unique Morita equivalence classes over <math>\mathcal{O}</math> by [[References\|[HKL07] ]] and the theory of [[nilpotent blocks]].

@@ Line 1: / Line 1: @@
 == Blocks with defect group <math>Q_8</math> ==
 These are examples of [[Tame blocks|tame blocks]] and were first classified over <math>k</math> by Erdmann (see [[References|[Er87] ]]). The classification with respect to <math>\mathcal{O}</math> is still unknown in general, except for the cases [[M(8,4,1)]] and [[M(8,4,3)]], which both lift to unique Morita equivalence classes over <math>\mathcal{O}</math> by [[References|[HKL07] ]] and the theory of [[nilpotent blocks]].
 {| class="wikitable"
@@ Line 8: / Line 9: @@
 ! scope="col"| Class
 ! scope="col"| Representative
 ! scope="col"| <math>k(B)</math>
 ! scope="col"| <math>l(B)</math>
@@ Line 18: / Line 20: @@
 |-
-|[[M(8,4,1)]] || <math>kQ_8</math> ||5 ||1 ||<math>1</math> || || ||1 ||1 ||
+|[[M(8,4,1)]] || <math>kQ_8</math> || 1 ||5 ||1 ||<math>1</math> || || ||1 ||1 ||
 |-
-|[[M(8,4,2)]] || <math>B_0(kSL_2(5))</math> ||5 ||3 ||<math>1</math> || || || ||1 || <math>Q(3 {\cal A})_2</math>
+|[[M(8,4,2)]] || <math>B_0(kSL_2(5))</math> || ? ||5 ||3 ||<math>1</math> || || || ||1 || <math>Q(3 {\cal A})_2</math>
 |-
-|[[M(8,4,3)]] || <math>kSL_2(3)</math> ||5 ||3 ||<math>1</math> || || ||1 ||1 || <math>Q(3 {\cal K})</math>
+|[[M(8,4,3)]] || <math>kSL_2(3)</math> || 1 ||5 ||3 ||<math>1</math> || || ||1 ||1 || <math>Q(3 {\cal K})</math>
 |}
 [[M(8,4,2)]] and [[M(8,4,3)]] are derived equivalent over <math>k</math> by [[References|[Ho97] ]].