Difference between revisions of "M(8,3,5)"
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Latest revision as of 21:11, 4 October 2018
Representative: | [math]B_0(kA_7)[/math] |
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Defect groups: | [math]D_8[/math] |
Inertial quotients: | [math]1[/math] |
[math]k(B)=[/math] | 5 |
[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} 3 & 2 & 1 \\ 2 & 4 & 2 \\ 1 & 2 & 2 \\ \end{array} \right)[/math] |
Defect group Morita invariant? | Yes |
Inertial quotient Morita invariant? | Yes |
[math]\mathcal{O}[/math]-Morita classes known? | No |
[math]\mathcal{O}[/math]-Morita classes: | |
Decomposition matrices: | [math]\left( \begin{array}{ccc} 0 & 1 & 0 \\ 1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 0 & 0 \\ \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: | M(8,3,4), M(8,3,6) |
[math]\mathcal{O}[/math]-derived equiv. classes known? | No |
[math]p'[/math]-index covering blocks: | |
[math]p'[/math]-index covered blocks: | |
Index [math]p[/math] covering blocks: | {{{pcoveringblocks}}} |
These are tame blocks, and appear in the family [math]D(3 {\cal B})_1[/math] in Erdmann's classification (see [Er87]). Derived equivalences over [math]k[/math] are established in [Li94b].
Contents
Basic algebra
Quiver: a:<1,1>, b:<1,2>, c:<2,3>, d:<3,2>, e:<2,1>
Relations w.r.t. [math]k[/math]: [math]ab=ea=be=dc=0[/math], [math]cdeb=ebcd[/math], [math]a^2=bcde[/math]
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}{ccc} & 1 & \\ 1 & \oplus & \begin{array}{c} 2 \\ 3 \\ 2 \\ \end{array} \\ & 1 & \\ \end{array}, & \begin{array}{ccc} & 2 & \\ \begin{array}{c} 1 \\ 2 \\ 3 \\ \end{array} & \oplus & \begin{array}{c} 3 \\ 2 \\ 1 \\ \end{array} \\ & 2 & \\ \end{array}, & \begin{array}{c} 3 \\ 2 \\ 1 \\ 2 \\ 3 \\ \end{array} \end{array} [/math]
Irreducible characters
[math]k_0(B)=4, \ k_1(B)=1[/math]