5 +^3 - Revision history

Charles Eaton: /* Blocks with defect group 5_+^{1+2} */

2023-10-04T09:36:21Z

‎Blocks with defect group 5_+^{1+2}

Charles Eaton at 15:47, 3 October 2023

2023-10-03T15:47:45Z

Charles Eaton: Created page with "== Blocks with defect group $5_+^{1+2}$ == left The Morita equivalence classes of blocks with this defect group are classifi..."

2023-10-03T14:00:03Z

Created page with "== Blocks with defect group <math>5_+^{1+2}</math> == 50px|left The Morita equivalence classes of blocks with this defect group are classifi..."

New page

== Blocks with defect group <math>5_+^{1+2}</math> ==

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

The Morita equivalence classes of blocks with this defect group are classified in [[References#A|[AE23]]]. Blocks in the same Morita equivalence class have the same fusion, and so the same inertial quotient, and the same Külshammer-Puig class.

{| class="wikitable"
|-
! scope="col"| Class
! scope="col"| Representative
! scope="col"| [[Glossary|# lifts / <math>\mathcal{O}</math>]]
! scope="col"| <math>k_0(B)</math>
! scope="col"| <math>k_1(B)</math>
! scope="col"| <math>l(B)</math>
! scope="col"| Inertial quotient
! scope="col"| <math>{\rm Pic}_\mathcal{O}(B)</math>
! scope="col"| <math>{\rm Pic}_k(B)</math>
! scope="col"| Notes

|-
|[[M(125,3,1)]] || <math>k5_+^{1+2}</math> || 1 || 20 || 4 ||1 || <math>1</math> || || ||
|-
|[[M(125,3,2)]] || <math>k5_+^{1+2}:C_2</math> || 1 || 20 || 2 ||2 || <math>C_2</math> || || || SmallGroup(250,5)
|-
|[[M(125,3,3)]] || <math>k5_+^{1+2}:C_2</math> || 1 || 14 || 8 ||2 || <math>C_2</math> || || || SmallGroup(250,8)
|-
|[[M(125,3,4)]] || <math>k5_+^{1+2}:C_3</math> || 1 || 11 || 12 ||3 || <math>C_3</math> || || || SmallGroup(375,2)
|-
|[[M(125,3,5)]] || <math>k5_+^{1+2}:C_4</math> || 1 || 25 || 1 ||4 || <math>C_4</math> || || || SmallGroup(500,17)
|-
|[[M(125,3,6)]] || <math>k5_+^{1+2}:C_4</math> || 1 || 13 || 1 ||4 || <math>C_4</math> || || || SmallGroup(500,21)
|-
|[[M(125,3,7)]] || <math>k(C_5 \times C_5):SL_2(5)</math> || 1 || 13 || 1 ||5 || <math>C_4</math> || || || Inertial quotient as in M(125,3,6)
|-
|[[M(125,3,8)]] || <math>k5_+^{1+2}:C_4</math> || 1 || 10 || 4 ||4 || <math>C_4</math> || || || SmallGroup(500,23)
|-
|[[M(125,3,9)]] || <math>k5_+^{1+2}:C_4</math> || 1 || 10 || 16 ||4 || <math>C_4</math> || || || SmallGroup(500,25)
|-
|[[M(125,3,10)]] || <math>k5_+^{1+2}:(C_2 \times C_2)</math> || 1 || 16 || 4 ||4 || <math>C_2 \times C_2</math> || || || SmallGroup(500,27)
|-
|[[M(125,3,11)]] || Faithful block of <math>k5_+^{1+2}:Q_8</math>, in which <math>Z(Q_8)</math> acts trivially || 1 || 13 || 4 ||1 || <math>C_2 \times C_2</math> || || || SmallGroup(1000,42)
|-
|[[M(125,3,12)]] || <math>k5_+^{1+2}:C_6</math> || 1 || 10 || 24 ||6 || <math>C_6</math> || || || SmallGroup(750,6)
|-
|[[M(125,3,13)]] || <math>k5_+^{1+2}:S_3</math> || 1 || 13 || 6 ||3 || <math>S_3</math> || || || SmallGroup(750,5)
|-
|[[M(125,3,14)]] || <math>k5_+^{1+2}:C_8</math> || 1 || 11 || 2 ||8 || <math>C_8</math> || || || SmallGroup(1000,86)
|-
|[[M(125,4,15)]] || <math>B_0(kSU_3(5))</math> || 1 || 11 || 2 || 8 || <math>C_8</math> || || ||
|-
|[[M(125,3,16)]] || Faithful block of maximal defect of <math>k(3.SU_3(5))</math> || 1 || 11 || 2 || 8 || <math>C_8</math> || || ||
|-
|[[M(125,3,17)]] || <math>k5_+^{1+2}:(C_4 \times C_2)</math> || 1 || 20 || 2 ||8 || <math>C_4 \times C_2</math> || || || SmallGroup(1000,89)
|-
|[[M(125,3,18)]] || Faithful block of <math>k5_+^{1+2}:M_4(2)</math>, in which <math>M_4(2)'</math> acts trivially || 1 || 14 || 2 ||2 || <math>C_2 \times C_2</math> || || || SmallGroup(2000,250)
|-
|-[[M(125,3,19)]] || <math>k((C_5 \times C_5):SL_2(5).2)</math> || 1 || 20 || 2 || 10 || <math>C_4 \times C_2</math> || Inertial quotient as in M(125,3,17)
|-
|[[M(125,3,20)]] || <math>k5_+^{1+2}:(C_4 \times C_2)</math> || 1 || 14 || 8 ||8 || <math>C_4 \times C_2</math> || || || SmallGroup(1000,91)
|-
|[[M(125,3,21)]] || Faithful block of <math>k5_+^{1+2}:M_4(2)</math>, in which <math>M_4(2)'</math> acts trivially || 1 || 8 || 8 ||2 || <math>C_2 \times C_2</math> || || || SmallGroup(2000,264)
|-
|[[M(125,3,22)]] || <math>k5_+^{1+2}:D_8</math> || 1 || 14 || 8 ||5 || <math>D_8</math> || || || SmallGroup(1000,92)
|-
|[[M(125,3,23)]] || Faithful block of <math>k5_+^{1+2}:D_{16}</math>, in which <math>Z(D_{16})</math> acts trivially || 1 || 11 || 8 ||2 || <math>D_8</math> || || ||
|-
|[[M(125,3,24)]] || <math>k5_+^{1+2}:Q_8</math> || 1 || 8 || 20 ||5 || <math>Q_8</math> || || || SmallGroup(1000,93)
|-
|[[M(125,3,25)]] || <math>k5_+^{1+2}:(C_3:C_4)</math> || 1 || 8 || 24 ||6 || <math>C_3:C_4</math> || || || SmallGroup(1500,35)
|-
|[[M(125,3,26)]] || <math>k5_+^{1+2}:C_{12}</math> || 1 || 14 || 12 ||12 || <math>C_{12}</math> || || || SmallGroup(1500,36)
|-
|[[M(125,3,27)]] || <math>k5_+^{1+2}:D_{12}</math> || 1 || 14 || 12 ||6 || <math>D_{12}</math> || || || SmallGroup(1000,37)
|-
|[[M(125,3,28)]] || Faithful block of <math>k5_+^{1+2}:D_{24}</math>, in which <math>Z(D_{24})</math> acts trivially || 1 || 10 || 4 ||1 || <math>D_{12}</math> || || ||
|-
|[[M(125,3,29)]] || <math>k5_+^{1+2}:(C_4 \times C_4)</math> || 1 || 25 || 4 ||16 || <math>C_4 \times C_4</math> || || || SmallGroup(2000,473)
|-
|[[M(125,3,30)]] || Faithful block of <math>k5_+^{1+2}:(C_2.(C_4 \times C_4))</math>, in which <math>(C_2.(C_4 \times C_4))'</math> acts trivially || 1 || 13 || 4 ||4 || <math>C_4 \times C_4</math> || || || <math>C_2.(C_4 \times C_4</math> is SmallGroup(32,2)
|-
|[[M(125,3,31)]] || Faithful block of <math>k5_+^{1+2}:((C_4 \times C_4):C_4)</math>, in which <math>Z((C_4 \times C_4):C_4)</math> acts trivially || 1 || 10 || 4 ||1 || <math>C_4 \times C_4</math> || || || <math>(C_4 \times C_4):C_4</math> is SmallGroup(64,18)
|-
|[[M(125,3,32)]] || <math>k(C_5 \times C_5):GL_2(5)</math> || 1 || 25 || 4 ||20 || <math>C_4 \times C_4</math> || || ||
|-
|[[M(125,3,33)]] || <math>B_0(kSL_3(5))</math> || 1 || 25 || 4 ||24 || <math>C_4 \times C_4</math> || || ||
|}

@@ Line 1: / Line 1: @@
 == Blocks with defect group <math>5_+^{1+2}</math> ==
 The Morita equivalence classes of blocks with this defect group are classified in [[References#A|[AE23]]]. Blocks in the same Morita equivalence class have the same fusion, and so the same inertial quotient, and the same Külshammer-Puig class.
@@ Line 110: / Line 108: @@
 |-
 |[[M(125,3,46)]] || <math>k5_+^{1+2}:SL_2(3)</math> || 1 || 8 || 28 ||7 || <math>C_{24}</math> || || ||
 |}

@@ Line 49: / Line 49: @@
 |[[M(125,4,15)]] || <math>B_0(kSU_3(5))</math> || 1 || 11 || 2 || 8 || <math>C_8</math> || || ||
 |-
-|[[M(125,3,16)]] || Faithful block of maximal defect of <math>k(3.SU_3(5))</math> || 1 || 11 || 2 || 8 || <math>C_8</math> || || ||
+|[[M(125,3,16)]] || Faithful block of <math>k(3.SU_3(5))</math> || 1 || 11 || 2 || 8 || <math>C_8</math> || || ||
 |-
 |[[M(125,3,17)]] || <math>k5_+^{1+2}:(C_4 \times C_2)</math> || 1 || 20 || 2 ||8 || <math>C_4 \times C_2</math> || || || SmallGroup(1000,89)
@@ Line 77: / Line 77: @@
 |[[M(125,3,29)]] || <math>k5_+^{1+2}:(C_4 \times C_4)</math> || 1 || 25 || 4 ||16 || <math>C_4 \times C_4</math> || || || SmallGroup(2000,473)
 |-
-|[[M(125,3,30)]] || Faithful block of <math>k5_+^{1+2}:(C_2.(C_4 \times C_4))</math>, in which <math>(C_2.(C_4 \times C_4))'</math> acts trivially || 1 || 13 || 4 ||4 || <math>C_4 \times C_4</math> || || || <math>C_2.(C_4 \times C_4</math> is SmallGroup(32,2)
+|[[M(125,3,30)]] || Faithful block of <math>k5_+^{1+2}:(C_2.(C_4 \times C_4))</math>, in which <math>(C_2.(C_4 \times C_4))'</math> acts trivially || 1 || 13 || 4 ||4 || <math>C_4 \times C_4</math> || || || <math>C_2.(C_4 \times C_4)</math> is SmallGroup(32,2)
 |-
 |[[M(125,3,31)]] || Faithful block of <math>k5_+^{1+2}:((C_4 \times C_4):C_4)</math>, in which <math>Z((C_4 \times C_4):C_4)</math> acts trivially || 1 || 10 || 4 ||1 || <math>C_4 \times C_4</math> || || || <math>(C_4 \times C_4):C_4</math> is SmallGroup(64,18)
@@ Line 84: / Line 84: @@
 |-
 |[[M(125,3,33)]] || <math>B_0(kSL_3(5))</math> || 1 || 25 || 4 ||24 || <math>C_4 \times C_4</math> || || ||
 |}

5 +^3 - Revision history

Charles Eaton: /* Blocks with defect group 5_+^{1+2} */

Charles Eaton at 15:47, 3 October 2023

Charles Eaton: Created page with "== Blocks with defect group 5_+^{1+2} == left The Morita equivalence classes of blocks with this defect group are classifi..."

Charles Eaton: Created page with "== Blocks with defect group $5_+^{1+2}$ == left The Morita equivalence classes of blocks with this defect group are classifi..."