Charles Eaton: Created page with "NOTITLE == Blocks with defect group $D_{16}$ == These are examples of tame blocks and were first classified over $k$ by Erdmann (se..."

2018-10-05T09:07:32Z

Created page with "__NOTITLE__ == Blocks with defect group <math>D_{16}</math> == These are examples of tame blocks and were first classified over <math>k</math> by Erdmann (se..."

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== Blocks with defect group <math>D_{16}</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. Note that the class <math>D(3 {\cal B})_1</math> is only realised for defect groups <math>D_8</math> (see [[References|[Er87, Proposition 7.5.1]]]).

{| class="wikitable"
|-
! scope="col"| Class
! scope="col"| Representative
! scope="col"| # lifts / <math>\mathcal{O}</math>
! 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(16,7,1)]] || <math>k(D_{16})</math> || 1 ||7 ||1 ||<math>1</math> || || || ||1 ||
|-
|[[M(16,7,2)]] || <math>B_0(kPGL_2(9))</math> || ? ||7 ||2 ||<math>1</math> || || || ||1 || <math>D(2 {\cal A})</math>
|-
|[[M(16,7,3)]] || <math>B_0(kPGL_2(7))</math> || ? ||7 ||2 ||<math>1</math> || || || ||1 || <math>D(2 {\cal B})</math>
|-
|[[M(16,7,4)]] || <math>B_0(kPSL_2(17))</math> || ? ||7 ||3 ||<math>1</math> || || || ||1 || <math>D(3 {\cal A})_1</math>
|-
|[[M(16,7,5)]] || <math>B_0(kPSL_2(47))</math> || ? ||7 ||3 ||<math>1</math> || || || ||1 || <math>D(3 {\cal K})</math>
|}

[[M(16,7,2)]] and [[M(16,7,3)]] are derived equivalent over <math>k</math> by [[References|[Ho97] ]].

[[M(16,7,4)]] and [[M(17,7,5)]] are derived equivalent over <math>k</math> by [[References|[Li94b] ]].

D16 - Revision history

Charles Eaton: Created page with "__NOTITLE__ == Blocks with defect group D_{16} == These are examples of tame blocks and were first classified over k by Erdmann (se..."

Charles Eaton: Created page with "NOTITLE == Blocks with defect group $D_{16}$ == These are examples of tame blocks and were first classified over $k$ by Erdmann (se..."