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M(8,5,1) - [math]k(C_2 \times C_2 \times C_2)[/math]
Representative: [math]k(C_2 \times C_2 \times C_2)[/math]
Defect groups: [math]C_2 \times C_2 \times C_2[/math]
Inertial quotients: [math]1[/math]
[math]k(B)=[/math] 8
[math]l(B)=[/math] 1
[math]{\rm mf}_k(B)=[/math] 1
[math]{\rm Pic}_k(B)=[/math]
Cartan matrix: [math]\left( \begin{array}{c} 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 \times C_2)[/math]
Decomposition matrices: [math]\left( \begin{array}{c} 1 \\ 1 \\ \vdots \\ 1 \\ \end{array}\right)[/math]
[math]{\rm mf}_\mathcal{O}(B)=[/math] 1
[math]{\rm Pic}_{\mathcal{O}}(B)=[/math] [math](C_2 \times C_2 \times C_2):GL_3(2)[/math]
[math]PI(B)=[/math] {{{PIgroup}}}
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:

These are nilpotent blocks.

Basic algebra

Quiver: a:<1,1>, b:<1,1>, c:<1,1>

Relations w.r.t. [math]k[/math]: a^2=b^2=b^c=0, ab+ba=ac+ca=bc+cb=0

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(8,5,1), then [math]B[/math] is in M(8,5,1), M(8,5,3), M(8,5,4) or M(8,5,6).

Projective indecomposable modules

Labelling the unique simple [math]B[/math]-module by [math]S_1[/math], the unique projective indecomposable module has Loewy structure as follows:

[math]\begin{array}{c} S_1 \\ S_1 S_1 S_1 S_1 \\ S_1 S_1 S_1 S_1 S_1 S_1 \\ S_1 S_1 S_1 S_1 \\ S_1 \\ \end{array} [/math]

Irreducible characters

All irreducible characters have height zero.

Back to [math](C_2[/math]