M(16,2,2)

From Block library
Jump to: navigation, search
M(16,2,2) - [math]k((C_4 \times C_4):C_3)[/math]
M(4,2,3)quiver.png
Representative: [math]k((C_4 \times C_4):C_3)[/math]
Defect groups: [math]C_4 \times C_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] [math][/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_4 \times C_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]C_2 \times S_3[/math][1]
[math]PI(B)=[/math] [math]S_3 \times D_8 \times C_2[/math][2]
Source algebras known? No
Source algebra reps:
[math]k[/math]-derived equiv. classes known? Yes
[math]k[/math]-derived equivalent to: Forms its own derived equivalence class
[math]\mathcal{O}[/math]-derived equiv. classes known? Yes
[math]p'[/math]-index covering blocks:
[math]p'[/math]-index covered blocks: M(16,2,1)
Index [math]p[/math] covering blocks:

Basic algebra

Quiver: a:<1,2>, b:<2,3>, c:<3,1>, d:<2,1>, e:<3,2>, f:<1,3>

Relations w.r.t. [math]k[/math]: abca=bcab=cabc=0, dfed=fedf=edfe=0, ad=fc, be=da, cf=eb

Other notatable representatives

Projective indecomposable modules

Labelling the simple [math]B[/math]-modules by [math]1,2,3[/math], the projective indecomposable modules have Loewy structure as follows:

[math]\begin{array}{ccc} \begin{array}{c} 1 \\ 2 \ 3 \\ 3 \ 1 \ 2 \\ 1 \ 2 \ 3 \ 1 \\ 3 \ 1 \ 2 \\ 2 \ 3 \\ 1 \\ \end{array}, & \begin{array}{c} 2 \\ 1 \ 3 \\ 3 \ 2 \ 1 \\ 2 \ 1 \ 3 \ 2 \\ 3 \ 2 \ 1 \\ 1 \ 3 \\ 2 \\ \end{array}, & \begin{array}{c} 3 \\ 1 \ 2 \\ 2 \ 3 \ 1 \\ 3 \ 1 \ 2 \ 3 \\ 2 \ 3 \ 1 \\ 1 \ 2 \\ 2 \\ \end{array} \end{array} [/math]

Irreducible characters

All irreducible characters have height zero.

Back to [math]C_4 \times C_4[/math]

Notes

  1. Proposition 4.3 of [BKL18]
  2. Using GAP, with code from [Ru11]