Difference between revisions of "(C2)^5"

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== Blocks with defect group <math>(C_2)^5</math> ==
 
== Blocks with defect group <math>(C_2)^5</math> ==
  
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These were classified in [[References#A|[Ar19]]] using the [[Glossary#CFSG|CFSG]]. Each of the 34 <math>k</math>-Morita equivalence classes lifts to an unique class over <math>\mathcal{O}</math>. The known Picard groups were computed in [[References#E|[EL18c]]] or using the main theorem of [[References#L|[Liv19]]]
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These were classified in [[References#A|[Ar19]]] using the [[Glossary#CFSG|CFSG]]. Each of the 34 <math>k</math>-Morita equivalence classes lifts to an unique class over <math>\mathcal{O}</math>. The known Picard groups were computed in [[References#E|[EL18c]]] or using the main theorem of [[References#L|[Liv19]]]. TO DO: PICARD GROUPS COMPUTED AND FILLED IN WHERE METHOD KNOWN
  
 
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When considering any non-principal block above, considering <math>3^{1+2}_{-}</math> instead of <math>3^{1+2}_{+}</math> gives Morita equivalent blocks.
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If a block of a finite group is Morita equivalent to another block with defect group <math>(C_2)^5</math>, then it also has defect group <math>(C_2)^5</math>.
  
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With the possible exception of M(32,51,5), M(32,51,11) and M(32,51,12), blocks in the same Morita equivalence class have the same inertial quotient (with the same action on the defect group).
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Blocks in the same Morita equivalence class have the same inertial quotient (with the same action on the defect group), as shown in [[References#A|[Ar19]]] and [[References#A|[AS20]]].
  
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With the possible exception of blocks in M(32,51,34), possible blocks in M(32,51,5) with inertial quotient <math>C_7:C_3</math> and possible blocks in M(32,51,11) or M(32,51,12) with inertial quotient <math>C_7:C_3 \times C_3</math>, blocks are splendid derived equivalent to their Brauer correspondent.
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All blocks with defect group  <math>(C_2)^5</math> are derived equivalent to their Brauer correspondent, and each derived equivalence class is determined by the number of simple modules and the inertial quotient (and its action on the defect group) [[References#A|[AS20]]].
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At the moment it is unknown whether the derived equivalence class of [[M(32,51,5)]] contains blocks with inertial quotient <math>C_7:C_3</math> (as in [[M(32,51,20)]]), and whether the derived equivalence class of [[M(32,51,11)]] contains blocks with inertial quotient <math>C_7:C_3 \times C_3</math>.
 
  
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All the derived equivalences above also occur over <math>\mathcal{O}</math>.
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It is unknown whether these derived equivalences are splendid, because we cannot say anything about the sources of the Morita equivalences determined in [[References#A|[Ar19]]].

Latest revision as of 09:18, 17 May 2022

Blocks with defect group [math](C_2)^5[/math]

These were classified in [Ar19] using the CFSG. Each of the 34 [math]k[/math]-Morita equivalence classes lifts to an unique class over [math]\mathcal{O}[/math]. The known Picard groups were computed in [EL18c] or using the main theorem of [Liv19]. TO DO: PICARD GROUPS COMPUTED AND FILLED IN WHERE METHOD KNOWN

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(32,51,1) [math]k((C_2)^5)[/math] 1 32 1 [math]1[/math] [math](C_2)^5:GL_5(2)[/math] 1 1
M(32,51,2) [math]k(A_4 \times (C_2)^3)[/math] 1 32 3 [math]C_3[/math] [math]((C_2)^3 : GL_3(2)) \times S_3[/math] 1 1
M(32,51,3) [math]B_0(k(A_5 \times (C_2)^3))[/math] 1 32 3 [math]C_3[/math] [math]((C_2)^3 : GL_3(2)) \times C_2[/math] 1 1
M(32,51,4) [math]k(((C_2)^4 :C_3) \times C_2)[/math] 1 16 3 [math]C_3[/math] 1 1 The action of [math]C_3[/math] comes from the 5th power of a Singer cycle for [math]\mathbb{F}_{16}[/math]
M(32,51,5) [math]k(((C_2)^4 :C_5) \times C_2)[/math] 1 16 5 [math]C_5[/math] 1 1
M(32,51,6) [math]k(((C_2)^3:C_7) \times (C_2)^2)[/math] 1 32 7 [math]C_7[/math] 1 1
M(32,51,7) [math]B_0(k(SL_2(8) \times (C_2)^2))[/math] 1 32 7 [math]C_7[/math] 1 1
M(32,51,8) [math]k(A_4 \times A_4 \times C_2)[/math] 1 32 9 [math]C_3 \times C_3[/math] 1 1
M(32,51,9) [math]B_0(k(A_4 \times A_5 \times C_2))[/math] 1 32 9 [math]C_3 \times C_3[/math] 1 1
M(32,51,10) [math]B_0(k(A_5 \times A_5 \times C_2))[/math] 1 32 9 [math]C_3 \times C_3[/math] 1 1
M(32,51,11) [math]k((C_2)^4 : C_{15} \times C_2)[/math] 1 32 15 [math]C_{15}[/math] 1 1
M(32,51,12) [math]B_0(k(SL_2(16) \times C_2))[/math] 1 32 15 [math]C_{15}[/math] 1 1
M(32,51,13) [math]k(((C_2)^3:C_7) \times A_4)[/math] 1 32 21 [math]C_{21}[/math] 1 1
M(32,51,14) [math]B_0(k(((C_2)^3:C_7) \times A_5))[/math] 1 32 21 [math]C_{21}[/math] 1 1
M(32,51,15) [math]B_0(k(SL_2(8) \times A_4))[/math] 1 32 21 [math]C_{21}[/math] 1 1
M(32,51,16) [math]B_0(k(SL_2(8) \times A_5))[/math] 1 32 21 [math]C_{21}[/math] 1 1
M(32,51,17) [math]k(((C_2)^3:(C_7:C_3)) \times (C_2)^2)[/math] 1 32 5 [math]C_7:C_3[/math] 1 1
M(32,51,18) [math]B_0(k(J_1 \times (C_2)^2))[/math] 1 32 5 [math]C_7:C_3[/math] 1 1
M(32,51,19) [math]B_0(k({\rm Aut}(SL_2(8)) \times (C_2)^2))[/math] 1 32 5 [math]C_7:C_3[/math] 1 1
M(32,51,20) [math]k((C_2)^5:(C_7:C_3))[/math] 1 16 5 [math]C_7:C_3[/math] 1 1 The action of the subgroup [math]C_3[/math] is as specified in M(32,51,4) above.
M(32,51,21) [math]B_0(k((SL_2(8) \times (C_2)^2):C_3)[/math] 1 16 5 [math]C_7:C_3[/math] 1 1 The action of the subgroup [math]C_3[/math] is as specified in M(32,51,4) above.
M(32,51,22) [math]k((C_2)^5:C_{31})[/math] 1 32 31 [math]C_{31}[/math] 1 1
M(32,51,23) [math]B_0(SL_2(32))[/math] 1 32 31 [math]C_{31}[/math] [math]C_{5}[/math] 1 1
M(32,51,24) [math]k(((C_2)^3:(C_7:C_3)) \times A_4)[/math] 1 32 15 [math](C_7:C_3) \times C_3[/math] 1 1
M(32,51,25) [math]B_0(k(((C_2)^3:(C_7:C_3)) \times A_5))[/math] 1 32 15 [math](C_7:C_3) \times C_3[/math] 1 1
M(32,51,26) [math]B_0(k(J_1 \times A_4))[/math] 1 32 15 [math](C_7:C_3) \times C_3[/math] 1 1
M(32,51,27) [math]B_0(k(J_1 \times A_5))[/math] 1 32 15 [math](C_7:C_3) \times C_3[/math] 1 1
M(32,51,28) [math]B_0(k({\rm Aut}(SL_2(8)) \times A_4))[/math] 1 32 15 [math](C_7:C_3) \times C_3[/math] 1 1
M(32,51,29) [math]B_0(k({\rm Aut}(SL_2(8)) \times A_5))[/math] 1 32 15 [math](C_7:C_3) \times C_3[/math] 1 1
M(32,51,30) [math]k((C_2)^5:(C_{31}:C_5))[/math] 1 16 11 [math]C_{31}:C_5[/math] 1 1
M(32,51,31) [math]B_0({\rm Aut}(SL_2(32)))[/math] 1 16 11 [math]C_{31}:C_5[/math] 1 1
M(32,51,32) [math]b_2(k((C_2)^4 : 3^{1+2}_{+}) \times C_2)[/math] 1 16 1 [math]C_3 \times C_3[/math] 1 1 Non-principal block. Cannot be Morita equivalent to a principal block of any finite group.
M(32,51,33) [math]b_2(k((C_2)^5 : (C_7 : 3^{1+2}_{+})))[/math] 1 16 7 [math](C_7:C_3) \times C_3[/math] 1 1 Non-principal block. Cannot be Morita equivalent to a principal block of any finite group.
M(32,51,34) [math]b_2(k((SL_2(8) \times (C_2)^2) : 3^{1+2}_{+}))[/math] 1 16 7 [math](C_7:C_3) \times C_3[/math] 1 1 Non-principal block. Cannot be Morita equivalent to a principal block of any finite group.

If a block of a finite group is Morita equivalent to another block with defect group [math](C_2)^5[/math], then it also has defect group [math](C_2)^5[/math].

Blocks in the same Morita equivalence class have the same inertial quotient (with the same action on the defect group), as shown in [Ar19] and [AS20].

All blocks with defect group [math](C_2)^5[/math] are derived equivalent to their Brauer correspondent, and each derived equivalence class is determined by the number of simple modules and the inertial quotient (and its action on the defect group) [AS20].

It is unknown whether these derived equivalences are splendid, because we cannot say anything about the sources of the Morita equivalences determined in [Ar19].

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