C7
Revision as of 10:06, 14 October 2018 by Charles Eaton (talk | contribs)
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(7,1,2) | [math]kD_{14}[/math] | 1 | 5 | 2 | [math]C_2[/math] | [math]C_2 \times C_3[/math] | 1 | 1 | ||
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(7,1,4) | [math]k(C_7:C_3)[/math] | 1 | 5 | 3 | [math]C_3[/math] | [math]C_3 \times C_2[/math] | 1 | 1 | ||
M(7,1,5) | [math]B_9(k(3.A_7))[/math] | 1 | 5 | 3 | [math]C_3[/math] | [math]C_2[/math] | 1 | 1 | ||
M(7,1,6) | [math]B_0(kA_7)[/math] | 1 | 5 | 3 | [math]C_3[/math] | [math]C_2[/math] | 1 | 1 | ||
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(7,1,8) | [math]k(C_7:C_6)[/math] | 1 | 7 | 6 | [math]C_6[/math] | [math]C_6[/math] | 1 | 1 | ||
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(7,1,10) | [math]B_0(kPSL_3(29^3).3)[/math] | 1 | 7 | 6 | [math]C_6[/math] | [math]C_3[/math] | 1 | 1 | ||
M(7,1,11) | [math]B_{16}(k(2.J_2))[/math] | 1 | 7 | 6 | [math]C_6[/math] | ? | 1 | 1 | ||
M(7,1,12) | [math]B_{29}(k(2.Ru))[/math] | 1 | 7 | 6 | [math]C_6[/math] | [math]1[/math] | 1 | 1 | ||
M(7,1,13) | [math]B_0(k({}^2G_2(27)))[/math] | 1 | 7 | 6 | [math]C_6[/math] | [math]1[/math] | 1 | 1 | ||
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 | ||
7 | 6 | [math]C_6[/math] | 1 | 1 | ||||||
7 | 6 | [math]C_6[/math] | 1 | 1 | ||||||
7 | 6 | [math]C_6[/math] | 1 | 1 | ||||||
7 | 6 | [math]C_6[/math] | 1 | 1 | ||||||
7 | 6 | [math]C_6[/math] | 1 | 1 | ||||||
7 | 6 | [math]C_6[/math] | 1 | 1 | ||||||
7 | 6 | [math]C_6[/math] | 1 | 1 |
To do:
- Picard group for M(7,1,11) trivial or [math]C_2[/math]?