Difference between revisions of "M(8,5,3)"

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|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 =
 
|source? = No
 
|source? = No
 
|sourcereps = &nbsp;
 
|sourcereps = &nbsp;
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|k-derived = [[M(8,5,2)]]
 
|k-derived = [[M(8,5,2)]]
 
|O-derived-known? = Yes
 
|O-derived-known? = Yes
 +
|coveringblocks =
 +
|coveredblocks =
 +
|pcoveringblocks = [[M(16,10,3)]], [[M(16,14,3)]]
 
}}
 
}}
 
 
  
 
== Basic algebra ==
 
== Basic algebra ==
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'''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 ==
 
== Covering blocks and covered blocks ==
 
  
 
== Projective indecomposable modules ==
 
== Projective indecomposable modules ==

Revision as of 20:32, 5 December 2018

M(8,5,3) - [math]k(A_4 \times C_2)[/math]
M(8,5,3)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]
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:
[math]p'[/math]-index covered blocks:
Index [math]p[/math] covering blocks: M(16,10,3), M(16,14,3)

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]

Notes

  1. See [EL18c]