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M(4,2,1) - [math]k(C_2 \times C_2)[/math]
Representative: [math]k(C_2 \times C_2)[/math]
Defect groups: [math]C_2 \times C_2[/math]
Inertial quotients: [math]1[/math]
[math]k(B)=[/math] 4
[math]l(B)=[/math] 1
[math]{\rm mf}_k(B)=[/math] 1
[math]{\rm Pic}_k(B)=[/math] [math](k \times k):GL_2(k)[/math]
Cartan matrix: [math]\left( \begin{array}{c} 4 \\ \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)[/math]
Decomposition matrices: [math]\left( \begin{array}{c} 1 \\ 1 \\ 1 \\ 1 \\ \end{array}\right)[/math]
[math]{\rm mf}_\mathcal{O}(B)=[/math] 1
[math]{\rm Pic}_{\mathcal{O}}(B)=[/math] [math]\mathcal{L}(B)=S_4[/math]
[math]PI(B)=[/math] [math]S_4 \times C_2[/math]
Source algebras known? Yes
Source algebra reps: [math]k(C_2 \times C_2)[/math]
[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: M(4,2,1), M(4,2,3) (complete)
[math]p'[/math]-index covered blocks: M(4,2,1), M(4,2,3)[1] (complete)
Index [math]p[/math] covering blocks: M(8,2,1), M(8,3,1), M(8,5,1) (complete)

These are nilpotent blocks.

Basic algebra

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

Relations w.r.t. [math]k[/math]: a^2=b^2=ab+ba=0

Other notatable representatives

Block number 2 of [math]k PGL_3(7)[/math] in the labelling used in [2]

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}{ccc} & S_1 & \\ S_1 & & S_1 \\ & S_1 & \\ \end{array} [/math]

Irreducible characters

All irreducible characters have height zero.

Back to [math]C_2 \times C_2[/math]


  1. For example consider block number 2 of [math]PSL_3(7) \triangleleft PGL_3(7)[/math] in the labelling used in [1].