M(16,14,4)

M(16,14,4) - [math]k((C_2)^4 : C_3)[/math]

[[File: |250px]]

Representative:	[math]k((C_2)^4 : C_3)[/math]
Defect groups:	[math](C_2)^4[/math]
Inertial quotients:	[math]C_3[/math]
[math]k(B)=[/math]	8
[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} 6 & 5 & 5\\ 5 & 6 & 5 \\ 5 & 5 & 6 \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)^4 : C_3)[/math]
Decomposition matrices:	[math]\left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ 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:	Forms a derived equivalence class
[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:

The action of [math]C_3[/math] on the defect group, distinguished from the one in M(16,4,3), comes from the 5th power of a Singer cycle.

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(16,14,4), then [math]B[/math] is in M(16,14,1), M(16,14,4), M(16,14,8) or M(16,14,11).

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_2 S_2 S_3 S_3 \\ S_1 S_1 S_1 S_1 S_2 S_3 \\ S_2 S_2 S_3 S_3 \\ S_1 \\ \end{array} & \begin{array}{c} S_2 \\ S_1 S_1 S_3 S_3 \\ S_1 S_2 S_2 S_2 S_2 S_3 \\ S_1 S_1 S_3 S_3 \\ S_2 \\ \end{array} & \begin{array}{c} S_3 \\ S_1 S_1 S_2 S_2 \\ S_1 S_2 S_3 S_3 S_3 S_3 \\ S_1 S_1 S_2 S_2 \\ S_3 \\ \end{array} \end{array} [/math]

Irreducible characters

All irreducible characters have height zero.

Back to [math](C_2)^4[/math]