Difference between revisions of "M(4,2,3)"

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{{blockbox
 
{{blockbox
 
|title = M(4,2,3) - <math>kA_4</math>  
 
|title = M(4,2,3) - <math>kA_4</math>  
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|image =  
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|image = M(4,2,3)quiver.png
 
|representative =  <math>kA_4</math>
 
|representative =  <math>kA_4</math>
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|defect = <math>C_2 \times C_2</math>
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|defect = [[C2xC2|<math>C_2 \times C_2</math>]]
 
|inertialquotients = <math>C_3</math>
 
|inertialquotients = <math>C_3</math>
 
|k(B) = 4
 
|k(B) = 4
 
|l(B) = 3
 
|l(B) = 3
 
|k-morita-frob = 1  
 
|k-morita-frob = 1  
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|Pic-k= <math>(k^* \times k^* \times C_3):C_2</math>
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|Pic-k= <math>(k^* \times k^* \times C_3):C_2</math><ref>This is an elementary calculation.</ref>
 
|cartan = <math>\left( \begin{array}{ccc}
 
|cartan = <math>\left( \begin{array}{ccc}
 
2 & 1 & 1 \\
 
2 & 1 & 1 \\
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\end{array}\right)</math>
 
\end{array}\right)</math>
 
|O-morita-frob = 1
 
|O-morita-frob = 1
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|Pic-O = <math>\mathcal{T}(B)=S_3</math>
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|Pic-O = <math>\mathcal{T}(B)=S_3</math><ref>Every Morita equivalence is a source algebra equivalence by [[References|[CEKL13]]], so <math>{\rm Pic}(B)=\mathcal{T}(B)</math></ref>
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|PIgroup = <math>S_4 \times C_2</math><ref>See, for example, Proposition 2.8 of [[References|[EL18a]]]</ref>
 
|source? = Yes
 
|source? = Yes
 
|sourcereps = <math>\mathcal{O}A_4</math>
 
|sourcereps = <math>\mathcal{O}A_4</math>
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|k-derived = [[M(4,2,2)]]
 
|k-derived = [[M(4,2,2)]]
 
|O-derived-known? = Yes
 
|O-derived-known? = Yes
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|coveringblocks = [[M(4,2,1)]]<ref>For example consider 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)</math> is nilpotent.</ref>, M(4,2,3) (complete)
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|coveredblocks = [[M(4,2,1)]], M(4,2,2) (complete)
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|pcoveringblocks = [[M(8,3,3)]], [[M(8,5,3)]] (complete)<ref>A covering block cannot have defect group <math>C_4 \times C_2</math>.</ref>
 
}}
 
}}
  
 
== 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>
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'''Quiver:''' a:<1,2>, b:<2,3>, c:<3,1>, d:<2,1>, e:<3,2>, f:<1,3>
  
 
'''Relations w.r.t. <math>k</math>:''' ab=bc=ca=0, df=fe=ed=0, ad=fc, be=da, cf=eb
 
'''Relations w.r.t. <math>k</math>:''' ab=bc=ca=0, df=fe=ed=0, ad=fc, be=da, cf=eb
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Block number 2 of <math>k PSL_3(7)</math> in the labelling used in [http://www.math.rwth-aachen.de/~MOC/decomposition/tex/L3(7)/]
 
Block number 2 of <math>k PSL_3(7)</math> in the labelling used in [http://www.math.rwth-aachen.de/~MOC/decomposition/tex/L3(7)/]
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−
 
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== Covering blocks and covered blocks ==
 
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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>.
 
−
 
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If <math>b</math> lies in M(4,2,3), then <math>B</math> must lie in [[M(4,2,1)]] or M(4,2,3). For example consider blocks of <math>PSL_3(7) \triangleleft PGL_3(7)</math>.
 
−
 
−
If <math>B</math> lies in M(4,2,3), then <math>b</math> must lie in [[M(4,2,1)]] or M(4,2,3). For example consider the principal blocks of <math>O_2(A_4) \triangleleft A_4</math>.
 
  
 
== Projective indecomposable modules ==
 
== Projective indecomposable modules ==
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All irreducible characters have height zero.
 
All irreducible characters have height zero.
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 +
[[C2xC2|Back to <math>C_2 \times C_2</math>]]
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== Notes ==
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 +
<references />
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[[Category: Morita equivalence classes|4,2,3]]
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[[Category: Blocks with defect group C2xC2]]
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[[Category: Tame blocks|4,2,3]]

Latest revision as of 15:29, 22 November 2018

M(4,2,3) - [math]kA_4[/math]
M(4,2,3)quiver.png
Representative: [math]kA_4[/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^* \times k^* \times C_3):C_2[/math][1]
Cartan matrix: [math]\left( \begin{array}{ccc} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 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]\mathcal{O}A_4[/math]
Decomposition matrices: [math]\left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 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)=S_3[/math][2]
[math]PI(B)=[/math] [math]S_4 \times C_2[/math][3]
Source algebras known? Yes
Source algebra reps: [math]\mathcal{O}A_4[/math]
[math]k[/math]-derived equiv. classes known? Yes
[math]k[/math]-derived equivalent to: M(4,2,2)
[math]\mathcal{O}[/math]-derived equiv. classes known? Yes
[math]p'[/math]-index covering blocks: M(4,2,1)[4], M(4,2,3) (complete)
[math]p'[/math]-index covered blocks: M(4,2,1), M(4,2,2) (complete)
Index [math]p[/math] covering blocks: M(8,3,3), M(8,5,3) (complete)[5]

Basic algebra

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

Relations w.r.t. [math]k[/math]: ab=bc=ca=0, df=fe=ed=0, ad=fc, be=da, cf=eb

Other notatable representatives

Block number 2 of [math]k PSL_3(7)[/math] in the labelling used in [2]

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

Notes

  1. ↑ This is an elementary calculation.
  2. ↑ Every Morita equivalence is a source algebra equivalence by [CEKL13], so [math]{\rm Pic}(B)=\mathcal{T}(B)[/math]
  3. ↑ See, for example, Proposition 2.8 of [EL18a]
  4. ↑ For example consider block number 2 of [math]PSL_3(7) \triangleleft PGL_3(7)[/math] in the labelling used in [1]. The covering block of [math]PGL_3(7)[/math] is nilpotent.
  5. ↑ A covering block cannot have defect group [math]C_4 \times C_2[/math].
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