Difference between revisions of "M(9,2,4)"
(Created page with "{{blockbox |title = M(9,2,4) - <math>k(S_3 \times S_3)</math> |image = M(9,2,4)quiver.png |representative = <math>k(S_3 \times S_3)</math> |defect = C3xC3|<math>C_3 \times...") |
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|representative = <math>k(S_3 \times S_3)</math> | |representative = <math>k(S_3 \times S_3)</math> | ||
|defect = [[C3xC3|<math>C_3 \times C_3</math>]] | |defect = [[C3xC3|<math>C_3 \times C_3</math>]] | ||
| − | |inertialquotients = <math>C_2</math> | + | |inertialquotients = <math>C_2 \times C_2</math> |
|k(B) = 9 | |k(B) = 9 | ||
|l(B) = 4 | |l(B) = 4 | ||
| Line 18: | Line 18: | ||
|inertial-morita-inv? = | |inertial-morita-inv? = | ||
|O-morita? = Yes | |O-morita? = Yes | ||
| − | |O-morita = <math>\mathcal{O} | + | |O-morita = <math>\mathcal{O} (S_3 \times S_3)</math> |
|decomp = <math>\left( \begin{array}{cccc} | |decomp = <math>\left( \begin{array}{cccc} | ||
1 & 0 & 0 & 0 \\ | 1 & 0 & 0 & 0 \\ | ||
| Line 24: | Line 24: | ||
0 & 0 & 1 & 0 \\ | 0 & 0 & 1 & 0 \\ | ||
0 & 0 & 0 & 1 \\ | 0 & 0 & 0 & 1 \\ | ||
| − | 1 0 & 0 & | + | 1 & 0 & 0 & 1 \\ |
0 & 1 & 1 & 0 \\ | 0 & 1 & 1 & 0 \\ | ||
1 & 1 & 0 & 0 \\ | 1 & 1 & 0 & 0 \\ | ||
| Line 36: | Line 36: | ||
|sourcereps = | |sourcereps = | ||
|k-derived-known? = No | |k-derived-known? = No | ||
| − | |k-derived = | + | |k-derived = |
|O-derived-known? = No | |O-derived-known? = No | ||
|coveringblocks = | |coveringblocks = | ||
Revision as of 08:40, 23 October 2018
| Representative: | [math]k(S_3 \times S_3)[/math] |
|---|---|
| Defect groups: | [math]C_3 \times C_3[/math] |
| Inertial quotients: | [math]C_2 \times C_2[/math] |
| [math]k(B)=[/math] | 9 |
| [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} 4 & 2 & 1 & 2 \\ 2 & 4 & 2 & 1 \\ 1 & 2 & 4 & 2 \\ 2 & 1 & 2 & 4 \\ \end{array} \right)[/math] |
| Defect group Morita invariant? | |
| Inertial quotient Morita invariant? | |
| [math]\mathcal{O}[/math]-Morita classes known? | Yes |
| [math]\mathcal{O}[/math]-Morita classes: | [math]\mathcal{O} (S_3 \times S_3)[/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 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 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]C_2 \wr C_2[/math][1] |
| [math]PI(B)=[/math] | |
| Source algebras known? | No |
| Source algebra reps: | |
| [math]k[/math]-derived equiv. classes known? | No |
| [math]k[/math]-derived equivalent to: | |
| [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: |
Contents
Basic algebra
Quiver: a:<1,2>, b:<2,3>, c:<3,4>, d:<4,1>, e:<1,4>, f:<4,3>, g:<3,2>, h:<2,1>
Relations w.r.t. [math]k[/math]: [math]ab=ef[/math], [math]bc=he[/math], [math]cd=gh[/math], [math]da=fg[/math], [math]aha=ede=0[/math], [math]bgb=hah=0[/math], [math]cfc=gbg=0[/math], [math]ded=fcf=0[/math]
Other notatable representatives
Projective indecomposable modules
Labelling the simple [math]B[/math]-modules by [math]1,2,3,4[/math], the projective indecomposable modules have Loewy structure as follows:
[math]\begin{array}{cccc} \begin{array}{ccccc} & & 1 & & \\ & 2 & & 4 & \\ 1 & & 3 & & 1 \\ & 4 & & 2 & \\ & & 1 & & \\ \end{array}, & \begin{array}{ccccc} & & 2 & & \\ & 1 & & 3 & \\ 2 & & 4 & & 2 \\ & 3 & & 1 & \\ & & 2 & & \\ \end{array}, & \begin{array}{ccccc} & & 3 & & \\ & 2 & & 4 & \\ 3 & & 1 & & 3 \\ & 4 & & 2 & \\ & & 3 & & \\ \end{array}, & \begin{array}{ccccc} & & 4 & & \\ & 1 & & 3 & \\ 4 & & 2 & & 4 \\ & 3 & & 1 & \\ & & 4 & & \\ \end{array} \\ \end{array} [/math]
Irreducible characters
All irreducible characters have height zero.
Back to [math]C_3 \times C_3[/math]
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