Difference between revisions of "M(8,3,1)"

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'''Quiver:''' a:<1,1>, b:<1,1>
'''Quiver:''' a:<1,1>, b:<1,1>
'''Relations w.r.t. <math>k</math>:''' a^b=b^2=(ab)^2+(ba)^2=0
'''Relations w.r.t. <math>k</math>:''' a^2=b^2=(ab)^2+(ba)^2=0
== Other notatable representatives ==
== Other notatable representatives ==

Latest revision as of 21:47, 5 October 2018

M(8,3,1) - [math]kD_8[/math]
Representative: [math]kD_8[/math]
Defect groups: [math]D_8[/math]
Inertial quotients: [math]1[/math]
[math]k(B)=[/math] 5
[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}D_8[/math]
Decomposition matrices: [math]\left( \begin{array}{c} 2 \\ 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] {{{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: {{{pcoveringblocks}}}

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)^2+(ba)^2=0

Other notatable representatives

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 \\ \end{array} [/math]

Irreducible characters

[math]k_0(B)=4, k_1(B)=1[/math]