M(16,10,2)

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M(16,10,2) - [math]B_0(k(C_4 \times A_5))[/math]
M(8,5,2)quiver.png
Representative: [math]B_0(k(C_4 \times A_5))[/math]
Defect groups: [math]C_4 \times C_2 \times C_2[/math]
Inertial quotients: [math]C_3[/math]
[math]k(B)=[/math] 16
[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} 8 & 8 & 4 \\ 8 & 16 & 8 \\ 4 & 8 & 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]B_0(\mathcal{O}(C_4\times A_5))[/math]
Decomposition matrices: [math]\left( \begin{array}{ccc} 0 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 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]D_8 \times C_2[/math][1]
[math]PI(B)=[/math] [math]D_8 \times S_4 \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: M(16,10,3)
[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:


Basic algebra

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

Relations w.r.t. [math]k[/math]: [math]ad=cb=0[/math], [math]bcda=dabc[/math], [math]e^4=f^4=g^4=0[/math], [math]af=ea[/math], [math]bg=fb[/math], [math]cf=gc[/math], [math]de=fd[/math]

Other notatable representatives

Projective indecomposable modules

Irreducible characters

All irreducible characters have height zero.

Back to [math]C_4 \times C_2 \times C_2[/math]

Notes

  1. [math]{\rm Pic}_{\mathcal{O}}(B_0(\mathcal{O}(C_4 \times A_5)))=\mathcal{L}(B_0(\mathcal{O}(C_4 \times A_5)))=\mathcal{L}(\mathcal{O}C_4) \times \mathcal{T}(B_0(\mathcal{O}A_5))[/math] by [EL18c], giving the isomorphism type of [math]{\rm Pic}_\mathcal{O}(B)[/math] in general.
  2. By Theorem 3.7 of [EL18c].