M(16,14,3)
Representative: | [math]k(C_2 \times C_2 \times A_4)[/math] |
---|---|
Defect groups: | [math](C_2)^4[/math] |
Inertial quotients: | [math]C_3[/math] |
[math]k(B)=[/math] | 16 |
[math]l(B)=[/math] | 3 |
[math]{\rm mf}_k(B)=[/math] | 1 |
[math]{\rm Pic}_k(B)=[/math] | |
Cartan matrix: | [math]\left( \begin{array}{ccc} 8 & 4 & 4\\ 4 & 8 & 4 \\ 4 & 4 & 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]B_0(\mathcal{O} (C_2 \times C_2 \times A_4)[/math] |
Decomposition matrices: | See below |
[math]{\rm mf}_\mathcal{O}(B)=[/math] | 1 |
[math]{\rm Pic}_{\mathcal{O}}(B)=[/math] | [math](C_2 \times C_2):S_3 \times S_3[/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(16,14,2) |
[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 [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(16,14,3), then [math]B[/math] is in M(16,14,1), M(16,14,3), M(16,14,8), M(16,14,16).
Projective indecomposable modules
Labelling the simple [math]B[/math]-modules by [math]S_1, S_2, S_3[/math], the projective indecomposable modules have Loewy structure as follows:
[math]\begin{array}{ccc} \begin{array}{c} S_1 \\ S_1 S_1 S_2 S_3 \\ S_1 S_1 S_2 S_2 S_3 S_3 \\ S_1 S_1 S_2 S_3 \\ S_1 \\ \end{array} & \begin{array}{c} S_2 \\ S_1 S_2 S_2 S_3 \\ S_1 S_1 S_2 S_2 S_3 S_3 \\ S_1 S_2 S_2 S_3 \\ S_2 \\ \end{array} & \begin{array}{c} S_3 \\ S_1 S_2 S_3 S_3 \\ S_1 S_1 S_2 S_2 S_3 S_3 \\ S_1 S_2 S_3 S_3 \\ S_3 \\ \end{array} \end{array} [/math]
Irreducible characters
All irreducible characters have height zero.
Decomposition matrix
[math]\left( \begin{array}{ccc} 1 &0 &0 \\ 1 &0 &0 \\ 1 &0 &0 \\ 1 &0 &0 \\ 0 &0 &1 \\ 0 &1 &0 \\ 0 &0 &1 \\ 0 &1 &0 \\ 0 &0 &1 \\ 0 &1 &0 \\ 0 &0 &1 \\ 0 &1 &0 \\ 1 &1 &1 \\ 1 &1 &1 \\ 1 &1 &1 \\ 1 &1 &1 \end{array}\right)[/math]