M(5,1,4)

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M(5,1,4) - [math]k(C_5:C_4)[/math]
M(5,1,4)quiver.png
Representative: [math]k(C_5:C_4)[/math]
Defect groups: [math]C_5[/math]
Inertial quotients: [math]C_4[/math]
[math]k(B)=[/math] 5
[math]l(B)=[/math] 4
[math]{\rm mf}_k(B)=[/math] 1
[math]{\rm Pic}_k(B)=[/math]
Cartan matrix: [math]\left( \begin{array}{cccc} 2 & 1 & 1 & 1 \\ 1 & 2 & 1 & 1 \\ 1 & 1 & 2 & 1 \\ 1 & 1 & 1 & 2 \\ \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_5:C_4)[/math]
Decomposition matrices: [math]\left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 1 & 1 & 1 \\ \end{array}\right)[/math]
[math]{\rm mf}_\mathcal{O}(B)=[/math] 1
[math]{\rm Pic}_{\mathcal{O}}(B)=[/math]
Source algebras known? Yes
Source algebra reps:
[math]k[/math]-derived equiv. classes known? Yes
[math]k[/math]-derived equivalent to M(5,1,5), M(5,1,6)
[math]\mathcal{O}[/math]-derived equiv. classes known? Yes
[math]p'[/math]-index covering blocks {{{coveringblocks}}}
[math]p'[/math]-index covered blocks {{{coveredblocks}}}

Basic algebra

Quiver: a:<1,2>, b:<2,3>, c:<3,4>, d:<4,1>

Relations w.r.t. [math]k[/math]: abcda=bcdab=cdabc=dabcd=0

Other notatable representatives

Covering blocks and covered blocks

Projective indecomposable modules

Labelling the simple [math]B[/math]-modules by [math]S_1, S_2, S_3, S_4[/math], the projective indecomposable modules have Loewy structure as follows:

[math]\begin{array}{cccc} \begin{array}{c} S_1 \\ S_2 \\ S_3 \\ S_4 \\ S_1 \\ \end{array}, & \begin{array}{c} S_2 \\ S_3 \\ S_4 \\ S_1 \\ S_2 \\ \end{array}, & \begin{array}{c} S_3 \\ S_4 \\ S_1 \\ S_2 \\ S_3 \\ \end{array}, & \begin{array}{c} S_4 \\ S_1 \\ S_2 \\ S_3 \\ S_4 \\ \end{array} \end{array} [/math]

Irreducible characters

All irreducible characters have height zero.

Back to [math]C_5[/math]