Difference between revisions of "(C2)^4"
(Switched M(16,14,2) and M(16,14,3) to match C2xC2 and C3xC3xC3 labellings. Changed reference for k(B), l(B)) |
(Added some O-Picard groups) |
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== Blocks with defect group <math>(C_2)^4</math> == | == Blocks with defect group <math>(C_2)^4</math> == | ||
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These were classified in [[References|[Ea18]]] using the [[Glossary#CFSG|CFSG]]. Each of the sixteen <math>k</math>-Morita equivalence classes lifts to an unique class over <math>\mathcal{O}</math>. The possibilities for <math> k(B) \text{ and } l(B)</math> were computed in [[References|[KS13] and [EKKS14]]]. | These were classified in [[References|[Ea18]]] using the [[Glossary#CFSG|CFSG]]. Each of the sixteen <math>k</math>-Morita equivalence classes lifts to an unique class over <math>\mathcal{O}</math>. The possibilities for <math> k(B) \text{ and } l(B)</math> were computed in [[References|[KS13] and [EKKS14]]]. | ||
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| − | |[[M(16,14,1)]] || <math>k((C_2)^4)</math> || 1 ||16 ||1 ||<math>1</math> || || ||1 ||1 || | + | |[[M(16,14,1)]] || <math>k((C_2)^4)</math> || 1 ||16 ||1 ||<math>1</math> || <math>(C_2)^4:GL_4(2)</math> || ||1 ||1 || |
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| − | |[[M(16,14,2)]] || <math>B_0(k(C_2 \times C_2 \times A_5))</math> || 1 ||16 ||3 ||<math>C_3</math> || || ||1 ||1 || | + | |[[M(16,14,2)]] || <math>B_0(k(C_2 \times C_2 \times A_5))</math> || 1 ||16 ||3 ||<math>C_3</math> || <math>((C_2 \times C_2):S_3) \times C_2</math> || ||1 ||1 || |
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| − | |[[M(16,14,3)]] || <math>k(C_2 \times C_2 \times A_4)</math> || 1 ||16 ||3 ||<math>C_3</math> || || ||1 ||1 || | + | |[[M(16,14,3)]] || <math>k(C_2 \times C_2 \times A_4)</math> || 1 ||16 ||3 ||<math>C_3</math> || <math>((C_2 \times C_2):S_3) \times S_3</math> || ||1 ||1 || |
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| − | |[[M(16,14,4)]] || <math>k((C_2)^4 :C_3)</math> || 1 ||8 ||3 ||<math>C_3</math> || || ||1 ||1 || The action comes from the 5th power of a Singer cycle for <math>\mathbb{F}_{16}</math> | + | |[[M(16,14,4)]] || <math>k((C_2)^4 :C_3)</math> || 1 ||8 ||3 ||<math>C_3</math> || <math>C_3</math> || ||1 ||1 || The action comes from the 5th power of a Singer cycle for <math>\mathbb{F}_{16}</math> |
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| − | |[[M(16,14,5)]] || <math>k((C_2)^4 : C_5)</math> || 1 ||8 ||5 ||<math>C_5</math> || || ||1 ||1 || | + | |[[M(16,14,5)]] || <math>k((C_2)^4 : C_5)</math> || 1 ||8 ||5 ||<math>C_5</math> || <math>C_5</math> || ||1 ||1 || |
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|[[M(16,14,6)]] || <math>k(C_2 \times ((C_2)^3:C_7))</math> || 1 ||16 ||7 ||<math>C_7</math> || || ||1 ||1 || | |[[M(16,14,6)]] || <math>k(C_2 \times ((C_2)^3:C_7))</math> || 1 ||16 ||7 ||<math>C_7</math> || || ||1 ||1 || | ||
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|[[M(16,14,7)]] || <math>B_0(k(C_2 \times SL_2(8)))</math> || 1 ||16 ||7 ||<math>C_7</math> || || ||1 ||1 || | |[[M(16,14,7)]] || <math>B_0(k(C_2 \times SL_2(8)))</math> || 1 ||16 ||7 ||<math>C_7</math> || || ||1 ||1 || | ||
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| − | |[[M(16,14,8)]] || <math>k(A_4 \times A_4)</math> || 1 ||16 ||9 ||<math>C_3 \times C_3</math> || || ||1 ||1 || | + | |[[M(16,14,8)]] || <math>k(A_4 \times A_4)</math> || 1 ||16 ||9 ||<math>C_3 \times C_3</math> || <math>S_3 \wr C_2</math>|| ||1 ||1 || |
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| − | |[[M(16,14,9)]] || <math>B_0(k(A_4 \times A_5))</math> || 1 ||16 ||9 ||<math>C_3 \times C_3</math> || || ||1 ||1 || | + | |[[M(16,14,9)]] || <math>B_0(k(A_4 \times A_5))</math> || 1 ||16 ||9 ||<math>C_3 \times C_3</math> || <math>S_3 \times C_2</math> || ||1 ||1 || |
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|[[M(16,14,10)]] || <math>B_0(k(A_5 \times A_5))</math> || 1 ||16 ||9 ||<math>C_3 \times C_3</math> || || ||1 ||1 || | |[[M(16,14,10)]] || <math>B_0(k(A_5 \times A_5))</math> || 1 ||16 ||9 ||<math>C_3 \times C_3</math> || || ||1 ||1 || | ||
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| − | |[[M(16,14,11)]] || <math>k((C_2)^4 : C_{15})</math> || 1 ||16 ||15 ||<math>C_{15}</math> || || ||1 ||1 || | + | |[[M(16,14,11)]] || <math>k((C_2)^4 : C_{15})</math> || 1 ||16 ||15 ||<math>C_{15}</math> || <math>C_{15}</math> || ||1 ||1 || |
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| − | |[[M(16,14,12)]] || <math>B_0(kSL_2(16))</math> || 1 ||16 ||15 ||<math>C_{15}</math> || || ||1 ||1 || | + | |[[M(16,14,12)]] || <math>B_0(kSL_2(16))</math> || 1 ||16 ||15 ||<math>C_{15}</math> || <math>C_{15}:C_4</math> || ||1 ||1 || |
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|[[M(16,14,13)]] || <math>k(C_2 \times ((C_2)^3:(C_7:C_3)))</math> || 1 ||16 ||5 ||<math>C_7:C_3</math> || || ||1 ||1 || | |[[M(16,14,13)]] || <math>k(C_2 \times ((C_2)^3:(C_7:C_3)))</math> || 1 ||16 ||5 ||<math>C_7:C_3</math> || || ||1 ||1 || | ||
Revision as of 07:03, 26 October 2018
Blocks with defect group [math](C_2)^4[/math]
These were classified in [Ea18] using the CFSG. Each of the sixteen [math]k[/math]-Morita equivalence classes lifts to an unique class over [math]\mathcal{O}[/math]. The possibilities for [math] k(B) \text{ and } l(B)[/math] were computed in [KS13] and [EKKS14].
| 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(16,14,1) | [math]k((C_2)^4)[/math] | 1 | 16 | 1 | [math]1[/math] | [math](C_2)^4:GL_4(2)[/math] | 1 | 1 | ||
| M(16,14,2) | [math]B_0(k(C_2 \times C_2 \times A_5))[/math] | 1 | 16 | 3 | [math]C_3[/math] | [math]((C_2 \times C_2):S_3) \times C_2[/math] | 1 | 1 | ||
| M(16,14,3) | [math]k(C_2 \times C_2 \times A_4)[/math] | 1 | 16 | 3 | [math]C_3[/math] | [math]((C_2 \times C_2):S_3) \times S_3[/math] | 1 | 1 | ||
| M(16,14,4) | [math]k((C_2)^4 :C_3)[/math] | 1 | 8 | 3 | [math]C_3[/math] | [math]C_3[/math] | 1 | 1 | The action comes from the 5th power of a Singer cycle for [math]\mathbb{F}_{16}[/math] | |
| M(16,14,5) | [math]k((C_2)^4 : C_5)[/math] | 1 | 8 | 5 | [math]C_5[/math] | [math]C_5[/math] | 1 | 1 | ||
| M(16,14,6) | [math]k(C_2 \times ((C_2)^3:C_7))[/math] | 1 | 16 | 7 | [math]C_7[/math] | 1 | 1 | |||
| M(16,14,7) | [math]B_0(k(C_2 \times SL_2(8)))[/math] | 1 | 16 | 7 | [math]C_7[/math] | 1 | 1 | |||
| M(16,14,8) | [math]k(A_4 \times A_4)[/math] | 1 | 16 | 9 | [math]C_3 \times C_3[/math] | [math]S_3 \wr C_2[/math] | 1 | 1 | ||
| M(16,14,9) | [math]B_0(k(A_4 \times A_5))[/math] | 1 | 16 | 9 | [math]C_3 \times C_3[/math] | [math]S_3 \times C_2[/math] | 1 | 1 | ||
| M(16,14,10) | [math]B_0(k(A_5 \times A_5))[/math] | 1 | 16 | 9 | [math]C_3 \times C_3[/math] | 1 | 1 | |||
| M(16,14,11) | [math]k((C_2)^4 : C_{15})[/math] | 1 | 16 | 15 | [math]C_{15}[/math] | [math]C_{15}[/math] | 1 | 1 | ||
| M(16,14,12) | [math]B_0(kSL_2(16))[/math] | 1 | 16 | 15 | [math]C_{15}[/math] | [math]C_{15}:C_4[/math] | 1 | 1 | ||
| M(16,14,13) | [math]k(C_2 \times ((C_2)^3:(C_7:C_3)))[/math] | 1 | 16 | 5 | [math]C_7:C_3[/math] | 1 | 1 | |||
| M(16,14,14) | [math]B_0(k(C_2 \times J_1))[/math] | 1 | 16 | 5 | [math]C_7:C_3[/math] | 1 | 1 | |||
| M(16,14,15) | [math]B_0(k(C_2 \times{\rm Aut}(SL_2(8))))[/math] | 1 | 16 | 5 | [math]C_7:C_3[/math] | 1 | 1 | |||
| M(16,14,16) | [math]b(k((C_2)^4 : 3^{1+2}_{+}))[/math] | 1 | 8 | 1 | [math]C_3 \times C_3[/math] | 1 | 1 | Non-principal faithful block. The center acts trivially. |
Both non-principal faithful blocks of [math]k((C_2)^4 : 3^{1+2}_{+})[/math] and [math]k((C_2)^4 : 3^{1+2}_{-})[/math] are Morita equivalent.
Blocks are derived equivalent if and only if they have the same inertial quotient (with the same action on the defect group) and number of simple modules. All the derived equivalences here also occur over [math]\mathcal{O}[/math]. In particular:
M(16,14,2) and M(16,14,3) are derived equivalent over [math]\mathcal{O}[/math].
M(16,14,6) and M(16,14,7) are derived equivalent over [math]\mathcal{O}[/math].
M(16,14,8) M(16,14,9) and M(16,14,10) are derived equivalent over [math]\mathcal{O}[/math].
M(16,14,11) and M(16,14,12) are derived equivalent over [math]\mathcal{O}[/math].
M(16,14,13), M(16,14,14) and M(16,14,15) are derived equivalent over [math]\mathcal{O}[/math].
