Difference between revisions of "M(8,5,3)"
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{{blockbox | {{blockbox | ||
|title = M(8,5,3) - <math>k(A_4 \times C_2)</math> | |title = M(8,5,3) - <math>k(A_4 \times C_2)</math> | ||
| − | |image = | + | |image = M(8,5,3)quiver.png |
|representative = <math>k(A_4 \times C_2)</math> | |representative = <math>k(A_4 \times C_2)</math> | ||
|defect = [[C2xC2xC2|<math>C_2 \times C_2 \times C_2</math>]] | |defect = [[C2xC2xC2|<math>C_2 \times C_2 \times C_2</math>]] | ||
| − | |inertialquotients = <math> | + | |inertialquotients = <math>C_3</math> |
|k(B) = 8 | |k(B) = 8 | ||
|l(B) = 3 | |l(B) = 3 | ||
| Line 10: | Line 10: | ||
|Pic-k= | |Pic-k= | ||
|cartan = <math>\left( \begin{array}{ccc} | |cartan = <math>\left( \begin{array}{ccc} | ||
| − | 2 & | + | 4 & 2 & 2 \\ |
| − | + | 2 & 4 & 2 \\ | |
| − | + | 2 & 2 & 4 \\ | |
\end{array} \right)</math> | \end{array} \right)</math> | ||
|defect-morita-inv? = Yes | |defect-morita-inv? = Yes | ||
| Line 19: | Line 19: | ||
|O-morita = <math>\mathcal{O} (A_4 \times C_2)</math> | |O-morita = <math>\mathcal{O} (A_4 \times C_2)</math> | ||
|decomp = <math>\left( \begin{array}{ccc} | |decomp = <math>\left( \begin{array}{ccc} | ||
| + | 1 & 0 & 0 \\ | ||
1 & 0 & 0 \\ | 1 & 0 & 0 \\ | ||
| + | 0 & 1 & 0 \\ | ||
0 & 1 & 0 \\ | 0 & 1 & 0 \\ | ||
| + | 0 & 0 & 1 \\ | ||
0 & 0 & 1 \\ | 0 & 0 & 1 \\ | ||
| + | 1 & 1 & 1 \\ | ||
1 & 1 & 1 \\ | 1 & 1 & 1 \\ | ||
\end{array}\right)</math> | \end{array}\right)</math> | ||
|O-morita-frob = 1 | |O-morita-frob = 1 | ||
| − | |Pic-O = | + | |Pic-O = <math>S_3 \times C_2</math><ref>See [[References|[EL18c]]]</ref> |
| + | |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 31: | Line 36: | ||
|k-derived = [[M(8,5,2)]] | |k-derived = [[M(8,5,2)]] | ||
|O-derived-known? = Yes | |O-derived-known? = Yes | ||
| + | |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 = | ||
| + | |pcoveringblocks = [[M(16,10,3)]], [[M(16,14,3)]] | ||
}} | }} | ||
| − | |||
| − | |||
== Basic algebra == | == Basic algebra == | ||
| Line 39: | Line 45: | ||
'''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> | '''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> | |
| − | '''Relations w.r.t. <math>k</math>:''' ab=bc=ca=0, df=fe=ed=0, ad=fc, be=da, cf=eb, g^2=h^2=i^2=0, ah=ga, bi=hb, cg=ic, dg=hd, eh=ie, fi=gf | ||
== Other notatable representatives == | == Other notatable representatives == | ||
| − | |||
| − | |||
== Projective indecomposable modules == | == 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 == | == Irreducible characters == | ||
| Line 55: | Line 82: | ||
[[C2xC2xC2|Back to <math>C_2 \times C_2 \times C_2</math>]] | [[C2xC2xC2|Back to <math>C_2 \times C_2 \times C_2</math>]] | ||
| + | |||
| + | == Notes == | ||
| + | |||
| + | <references /> | ||
Latest revision as of 22:07, 5 December 2018
M(8,5,3) - [math]k(A_4 \times C_2)[/math]
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]
