Difference between revisions of "M(8,5,3)"
| Line 30: | Line 30: | ||
|O-morita-frob = 1 | |O-morita-frob = 1 | ||
|Pic-O = <math>S_3 \times C_2</math><ref>See [[References|[EL18c]]]</ref> | |Pic-O = <math>S_3 \times C_2</math><ref>See [[References|[EL18c]]]</ref> | ||
| − | |PIgroup = | + | |PIgroup = <math>S_4 \times C_2 \times C_2</math><ref>By Theorem 3.7 of [[References|[EL18c]]].</ref> |
|source? = No | |source? = No | ||
|sourcereps = | |sourcereps = | ||
| Line 36: | Line 36: | ||
|k-derived = [[M(8,5,2)]] | |k-derived = [[M(8,5,2)]] | ||
|O-derived-known? = Yes | |O-derived-known? = Yes | ||
| − | |coveringblocks = | + | |coveringblocks = [[M(16,10,1)]]<ref>For example consider the block of <math>PSL_3(7) \times C_2</math> covering the block number 2 of <math>PSL_3(7) \triangleleft PGL_3(7)</math> in the labelling used in [http://www.math.rwth-aachen.de/~MOC/decomposition/tex/L3(7)/]. The covering block of <math>PGL_3(7) \times C_2</math> is nilpotent.</ref>, M(8,5,3) (complete) |
|coveredblocks = | |coveredblocks = | ||
|pcoveringblocks = [[M(16,10,3)]], [[M(16,14,3)]] | |pcoveringblocks = [[M(16,10,3)]], [[M(16,14,3)]] | ||
Latest revision as of 21:07, 5 December 2018
quiver.png)
| Representative: | [math]k(A_4 \times C_2)[/math] |
|---|---|
| Defect groups: | [math]C_2 \times C_2 \times C_2[/math] |
| Inertial quotients: | [math]C_3[/math] |
| [math]k(B)=[/math] | 8 |
| [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} (A_4 \times C_2)[/math] |
| Decomposition matrices: | [math]\left( \begin{array}{ccc} 1 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 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]S_3 \times C_2[/math][1] |
| [math]PI(B)=[/math] | [math]S_4 \times C_2 \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(8,5,2) |
| [math]\mathcal{O}[/math]-derived equiv. classes known? | Yes |
| [math]p'[/math]-index covering blocks: | M(16,10,1)[3], M(8,5,3) (complete) |
| [math]p'[/math]-index covered blocks: | |
| Index [math]p[/math] covering blocks: | M(16,10,3), M(16,14,3) |
Contents
Basic algebra
Quiver: a:<1,2>, b:<2,3>, c:<3,1>, d:<2,1>, e:<3,2>, f: <1,3>, g:<1,1>, h:<2,2>, i:<3,3>
Relations w.r.t. [math]k[/math]: [math]k[/math]: [math]ab=bc=ca=0[/math], [math]df=fe=ed=0[/math], [math]ad=fc[/math], [math]be=da[/math], [math]cf=eb[/math], [math]g^2=h^2=i^2=0[/math], [math]ah=ga[/math], [math]bi=hb[/math], [math]cg=ic[/math], [math]dg=hd[/math], [math]eh=ie[/math], [math]fi=gf[/math]
Other notatable representatives
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_1 & S_2 & S_3 \\ S_2 & S_3 & S_1 \\ & S_1 & \\ \end{array}, & \begin{array}{ccc} & S_2 & \\ S_1 & S_3 & S_2 \\ S_2 & S_1 & S_3 \\ & S_2 & \\ \end{array}, & \begin{array}{ccc} & S_3 & \\ S_1 & S_2 & S_3 \\ S_3 & S_1 & S_2 \\ & S_3 & \\ \end{array} \end{array} [/math]
Irreducible characters
All irreducible characters have height zero.
Back to [math]C_2 \times C_2 \times C_2[/math]
