Difference between revisions of "M(16,14,13)"
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Revision as of 13:45, 28 November 2019
Representative: | [math]k(C_2 \times ((C_2)^3 : (C_7 : C_3)))[/math] |
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Defect groups: | [math](C_2)^4[/math] |
Inertial quotients: | [math]C_7:C_3[/math] |
[math]k(B)=[/math] | 16 |
[math]l(B)=[/math] | 5 |
[math]{\rm mf}_k(B)=[/math] | 1 |
[math]{\rm Pic}_k(B)=[/math] | |
Cartan matrix: | [math]\left( \begin{array}{ccccc} 4 & 0 & 0 & 2 & 2 \\ 0 & 4 & 0 & 2 & 2 \\ 0 & 0 & 4 & 2 & 2 \\ 2 & 2 & 2 & 8 & 6 \\ 2 & 2 & 2 & 6 & 8 \end{array}\right)[/math] |
Defect group Morita invariant? | Yes |
Inertial quotient Morita invariant? | Yes |
[math]\mathcal{O}[/math]-Morita classes known? | Yes |
[math]\mathcal{O}[/math]-Morita classes: | [math]\mathcal{O} (C_2 \times ((C_2)^3 : (C_7 : C_3)))[/math] |
Decomposition matrices: | [math]\left( \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 & 1 \\ 1 & 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 1 & 1 \\ 0 & 1 & 0 & 1 & 1 \\ 0 & 1 & 0 & 1 & 1 \\ 0 & 0 & 1 & 1 & 1 \end{array}\right)[/math] |
[math]{\rm mf}_\mathcal{O}(B)=[/math] | 1 |
[math]{\rm Pic}_{\mathcal{O}}(B)=[/math] | |
[math]PI(B)=[/math] | |
Source algebras known? | No |
Source algebra reps: | |
[math]k[/math]-derived equiv. classes known? | Yes |
[math]k[/math]-derived equivalent to: | M(16,14,14), M(16,14,15) |
[math]\mathcal{O}[/math]-derived equiv. classes known? | Yes |
[math]p'[/math]-index covering blocks: | |
[math]p'[/math]-index covered blocks: | |
Index [math]p[/math] covering blocks: |
Contents
Basic algebra
Other notatable representatives
Covering blocks and covered blocks
Let [math]N \triangleleft G[/math] with prime [math]p'[/math]-index and let [math]B[/math] be a block of [math]\mathcal{O} G[/math] covering a block [math]b[/math] of [math]\mathcal{O} N[/math].
If [math]b[/math] is in M(16,14,13), then [math]B[/math] is in M(16,14,3), M(16,14,6) or M(16,14,13).
Projective indecomposable modules
Projective indecomposable modules
Labelling the simple [math]B[/math]-modules by [math]S_1, S_2, S_3, S_4, S_5[/math], the projective indecomposable modules have Loewy structure as follows:
[math]\begin{array}{ccccc} \begin{array}{c} S_1 \\ S_1 S_5 \\ S_4 S_5 \\ S_1 S_4 \\ S_1 \\ \end{array} & \begin{array}{c} S_2 \\ S_2 S_5 \\ S_4 S_5 \\ S_2 S_4 \\ S_2 \\ \end{array} & \begin{array}{c} S_3 \\ S_3 S_5 \\ S_4 S_5 \\ S_3 S_4 \\ S_3 \\ \end{array} & \begin{array}{c} S_4 \\ S_1 S_2 S_3 S_4 S_4 S_5 \\ S_1 S_2 S_3 S_4 S_4 S_5 S_5 S_5 \\ S_4 S_4 S_5 S_5 \\ S_4 \\ \end{array} & \begin{array}{c} S_5 \\ S_4 S_4 S_5 S_5 \\ S_1 S_2 S_3 S_4 S_4 S_4 S_5 S_5 \\ S_1 S_2 S_3 S_4 S_5 S_5 \\ S_5 \\ \end{array} \end{array} [/math]
Irreducible characters
All irreducible characters have height zero.