Difference between revisions of "M(4,2,2)"

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|title = M(4,2,2) - <math>B_0(kA_5)</math>  
 
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|representative =  <math>B_0(kA_5)</math>
 
|representative =  <math>B_0(kA_5)</math>
 
|defect = <math>C_2 \times C_2</math>
 
|defect = <math>C_2 \times C_2</math>

Revision as of 16:38, 30 August 2018

M(4,2,2) - [math]B_0(kA_5)[/math]
[[File:|250px]]
Representative: [math]B_0(kA_5)[/math]
Defect groups: [math]C_2 \times C_2[/math]
Inertial quotients: [math]C_3[/math]
[math]k(B)=[/math] 4
[math]l(B)=[/math] 3
[math]{\rm mf}_k(B)=[/math] 1
[math]{\rm Pic}_k(B)=[/math] [math]k^* \wr C_2[/math]
Cartan matrix: [math]\left( \begin{array}{ccc} 4 & 2 & 2 \\ 2 & 2 & 1 \\ 2 & 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]B_0(\mathcal{O}A_5)[/math]
Decomposition matrices: [math]\left( \begin{array}{ccc} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \\ 1 & 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)=C_2[/math]
[math]PI(B)=[/math] {{{PIgroup}}}
Source algebras known? Yes
Source algebra reps: [math]B_0(\mathcal{O}A_5)[/math]
[math]k[/math]-derived equiv. classes known? Yes
[math]k[/math]-derived equivalent to: M(4,2,3)
[math]\mathcal{O}[/math]-derived equiv. classes known? Yes
[math]p'[/math]-index covering blocks: {{{coveringblocks}}}
[math]p'[/math]-index covered blocks: {{{coveredblocks}}}
Index [math]p[/math] covering blocks: {{{pcoveringblocks}}}


Basic algebra

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

Relations w.r.t. [math]k[/math]: ad=cb=bcda+dabc=0

Other notatable representatives

Covering blocks and covered blocks

Let [math]N \triangleleft G[/math] with [math]p'[/math]-index and let [math]B[/math] be a block of [math]\mathcal{O} G[/math] covering a block [math]b[/math] of [math]\mathcal{O} N[/math].

[math]B[/math] lies in M(4,2,2) iff [math]b[/math] does.

Projective indecomposable modules

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

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

Irreducible characters

All irreducible characters have height zero.