Difference between revisions of "C2xC2xC2"

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[[M(8,5,2)]] and [[M(8,5,3)]] are derived equivalent over <math>\mathcal{O}</math>.
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[[M(8,5,2)]] and [[M(8,5,3)]] are derived equivalent over <math>\mathcal{O}</math>. This is a consequence of the derived equivalence between [[M(4,2,2)]] and [[M(4,2,3)]] (see [[References|[Ri96]]]).
  
[[M(8,5,4)]] and [[M(8,5,5)]] are derived equivalent over <math>\mathcal{O}</math>.
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[[M(8,5,4)]] and [[M(8,5,5)]] are derived equivalent over <math>\mathcal{O}</math>. See [[References|[Ro95]]].
  
[[M(8,5,6)]], [[M(8,5,7)]] and [[M(8,5,8)]] are derived equivalent over <math>\mathcal{O}</math>.
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[[M(8,5,6)]], [[M(8,5,7)]] and [[M(8,5,8)]] are derived equivalent over <math>\mathcal{O}</math>. See [[References|[Go97]]], [[References|[Ok97]]] and [[References|[CR13]]].

Revision as of 15:47, 5 October 2018

Blocks with defect group [math]C_2 \times C_2 \times C_2[/math]

These were classified in [Ea16] using the CFSG. Each of the eight [math]k[/math]-Morita equivalence classes lifts to an unique class over [math]\mathcal{O}[/math].

Class Representative # lifts / [math]\mathcal{O}[/math] [math]k(B)[/math] [math]l(B)[/math] Inertial quotients [math]{\rm Pic}_\mathcal{O}(B)[/math] [math]{\rm Pic}_k(B)[/math] [math]{\rm mf_\mathcal{O}(B)}[/math] [math]{\rm mf_k(B)}[/math] Notes
M(8,5,1) [math]k(C_2 \times C_2 \times C_2)[/math] 1 8 1 [math]1[/math] 1 1
M(8,5,2) [math]B_0(k(A_5 \times C_2))[/math] 1 8 3 [math]C_3[/math] 1 1
M(8,5,3) [math]k(A_4 \times C_2)[/math] 1 8 3 [math]C_3[/math] 1 1
M(8,5,4) [math]k((C_2 \times C_2 \times C_2):C_7)[/math] 1 8 7 [math]C_7[/math] 1 1
M(8,5,5) [math]B_0(kSL_2(8))[/math] 1 8 7 [math]C_7[/math] 1 1
M(8,5,6) [math]k((C_2 \times C_2 \times C_2):(C_7:C_3))[/math] 1 8 5 [math]C_7:C_3[/math] 1 1
M(8,5,7) [math]B_0(kJ_1)[/math] 1 8 5 [math]C_7:C_3[/math] 1 1
M(8,5,8) [math]B_0(k{\rm Aut}(SL_2(8)))[/math] 1 8 5 [math]C_7:C_3[/math] 1 1

M(8,5,2) and M(8,5,3) are derived equivalent over [math]\mathcal{O}[/math]. This is a consequence of the derived equivalence between M(4,2,2) and M(4,2,3) (see [Ri96]).

M(8,5,4) and M(8,5,5) are derived equivalent over [math]\mathcal{O}[/math]. See [Ro95].

M(8,5,6), M(8,5,7) and M(8,5,8) are derived equivalent over [math]\mathcal{O}[/math]. See [Go97], [Ok97] and [CR13].