C7

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Blocks with defect group [math]C_7[/math]

These are blocks with cyclic defect groups and so they are described by Brauer trees. There are candidate Brauer trees with no known block realising them.

All [math]k[/math]-Morita equivalence classes lift to [math]\mathcal{O}[/math]-classes.

CLASSIFICATION INCOMPLETE
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(7,1,1) [math]kC_7[/math] 1 7 1 [math]1[/math] [math]C_7:C_6[/math] 1 1 M(7,1,1)tree.png
M(7,1,2) [math]kD_{14}[/math] 1 5 2 [math]C_2[/math] [math]C_2 \times C_3[/math] 1 1 M(7,1,2)tree.png
M(7,1,3) [math]B_0(kPSL_2(13))[/math] 1 5 2 [math]C_2[/math] [math]C_3[/math] 1 1 M(7,1,3)tree.png
M(7,1,4) [math]k(C_7:C_3)[/math] 1 5 3 [math]C_3[/math] [math]C_3 \times C_2[/math] 1 1 M(7,1,4)tree.png
M(7,1,5) [math]B_9(k(3.A_7))[/math] 1 5 3 [math]C_3[/math] [math]C_2[/math] 1 1 M(7,1,5)tree.png
M(7,1,6) [math]B_0(kA_7)[/math] 1 5 3 [math]C_3[/math] [math]C_2[/math] 1 1 M(7,1,6)tree.png
M(7,1,7) [math]B_{15}(k(6.A_7))[/math] 1 5 3 [math]C_3[/math] [math]C_2[/math] 1 1 M(7,1,7)tree.png
M(7,1,8) [math]k(C_7:C_6)[/math] 1 7 6 [math]C_6[/math] [math]C_6[/math] 1 1 M(7,1,8)tree.png
M(7,1,9) [math]B_0(k(S_7))[/math] 1 7 6 [math]C_6[/math] [math]C_2[/math] 1 1 M(7,1,9)tree.png
M(7,1,10) [math]B_0(kPSL_3(29^3).3)[/math] 1 7 6 [math]C_6[/math] [math]C_3[/math] 1 1 M(7,1,10)tree.png
M(7,1,11) [math]B_{16}(k(2.J_2))[/math] 1 7 6 [math]C_6[/math]  ? 1 1 M(7,1,11)tree.png
M(7,1,12) [math]B_{29}(k(2.Ru))[/math] 1 7 6 [math]C_6[/math] [math]1[/math] 1 1 M(7,1,12)tree.png
M(7,1,13) [math]B_0(k({}^2G_2(27)))[/math] 1 7 6 [math]C_6[/math] [math]1[/math] 1 1 M(7,1,13)tree.png
M(7,1,14) [math]B_0(k({}^2G_2(243)))[/math][1] 1 7 6 [math]C_6[/math] [math]1[/math] 1 1 M(7,1,14)tree.png
7 6 [math]C_6[/math] 1 1 M(7,1,15)tree.png
7 6 [math]C_6[/math] 1 1 M(7,1,16)tree.png
7 6 [math]C_6[/math] 1 1 M(7,1,17)tree.png
7 6 [math]C_6[/math] 1 1 M(7,1,18)tree.png
7 6 [math]C_6[/math] 1 1 M(7,1,19)tree.png
7 6 [math]C_6[/math] 1 1 M(7,1,20)tree.png
7 6 [math]C_6[/math] 1 1 M(7,1,21)tree.png

To do:

  • Picard group for M(7,1,11) trivial or [math]C_2[/math]?

Notes

  1. Theorem 3.8 of [Du14] gives existence for [math]q=3^{2m+1}[/math] such that [math](3^{2m+1}-3^{m+1}+1)_7=7[/math]