M(8,4,3)

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Under-construction.png
M(8,4,3) - [math]kSL_2(3)[/math]
M(4,2,3)quiver.png
Representative: [math]kSL_2(3)[/math]
Defect groups: [math]Q_8[/math]
Inertial quotients: [math]C_3[/math]
[math]k(B)=[/math] 7
[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} 4 & 2 & 2 \\ 2 & 4 & 2 \\ 2 & 2 & 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}SL_2(3)[/math]
Decomposition matrices: [math]\left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 1 \\ \end{array}\right)[/math]
[math]{\rm mf}_\mathcal{O}(B)=[/math] 1
[math]{\rm Pic}_{\mathcal{O}}(B)=[/math] [math]\mathcal{T}(B)=S_3[/math][1]
[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(8,4,2)
[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:

These are tame blocks, and appear in the family [math]Q(3 {\cal K})[/math] in Erdmann's classification (see [Er88a], [Er88b]). The class lifts to a unique [math]\mathcal{O}[/math]-Morita equivalence class by [HKL07]. A derived equivalence with M(8,4,2) over [math]k[/math] was established in [Ho97].

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]: ab=fcf, bc=dad, ca=ebe, fe=ada, df=beb, ed=cfc, dab=0=bed=cfe

Other notatable representatives

Projective indecomposable modules

Irreducible characters

[math]k_0(B)=4, k_1(B)=3[/math]

Back to [math]Q_8[/math]

Notes

  1. [math]{\rm Pic}(B)=\mathcal{L}(B)[/math] by [LiMa20b], so [math]{\rm Pic}(B)=\mathcal{T}(B)[/math]