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

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|l(B) = 3
 
|l(B) = 3
 
|k-morita-frob = 1  
 
|k-morita-frob = 1  
|Pic-k= <math>k^* \wr C_2</math>
+
|Pic-k= <math>k^* \wr C_2</math><ref>This is an elementary calculation</ref>
 
|cartan = <math>\left( \begin{array}{ccc}
 
|cartan = <math>\left( \begin{array}{ccc}
 
2 & 2 & 1 \\
 
2 & 2 & 1 \\
Line 26: Line 26:
 
|O-morita-frob = 1
 
|O-morita-frob = 1
 
|Pic-O = <math>\mathcal{T}(B)=C_2</math><ref>Every Morita equivalence is a source algebra equivalence by [[References|[CEKL13]]], so <math>{\rm Pic}(B)=\mathcal{T}(B)</math></ref>
 
|Pic-O = <math>\mathcal{T}(B)=C_2</math><ref>Every Morita equivalence is a source algebra equivalence by [[References|[CEKL13]]], so <math>{\rm Pic}(B)=\mathcal{T}(B)</math></ref>
|PIgroup = <math>S_4 \times C_2</math>
+
|PIgroup = <math>S_4 \times C_2</math><ref>This follows from the result for [[M(4,2,3)]] since blocks in these classes are derived equivalent and so perfectly isometric</ref>
 
|source? = Yes
 
|source? = Yes
 
|sourcereps = <math>B_0(\mathcal{O}A_5)</math>
 
|sourcereps = <math>B_0(\mathcal{O}A_5)</math>

Revision as of 15:31, 22 November 2018

M(4,2,2) - [math]B_0(kA_5)[/math]
M(4,2,2)quiver.png
Representative: [math]B_0(kA_5)[/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^* \wr C_2[/math][1]
Cartan matrix: [math]\left( \begin{array}{ccc} 2 & 2 & 1 \\ 2 & 4 & 2 \\ 1 & 2 & 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]B_0(\mathcal{O}A_5)[/math]
Decomposition matrices: [math]\left( \begin{array}{ccc} 0 & 1 & 0 \\ 1 & 1 & 0 \\ 0 & 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]\mathcal{T}(B)=C_2[/math][2]
[math]PI(B)=[/math] [math]S_4 \times C_2[/math][3]
Source algebras known? Yes
Source algebra reps: [math]B_0(\mathcal{O}A_5)[/math]
[math]k[/math]-derived equiv. classes known? Yes
[math]k[/math]-derived equivalent to: M(4,2,3)
[math]\mathcal{O}[/math]-derived equiv. classes known? Yes
[math]p'[/math]-index covering blocks: M(4,2,2) (complete)
[math]p'[/math]-index covered blocks: M(4,2,2) (complete)
Index [math]p[/math] covering blocks: M(8,3,2), M(8,5,2) (complete)[4]


Basic algebra

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

Relations w.r.t. [math]k[/math]: ad=cb=bcda+dabc=0

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}{c} S_1 \\ S_2 \\ S_3 \\ S_2 \\ S_1 \\ \end{array}, & \begin{array}{ccc} & S_2 & \\ \begin{array}{c} S_1 \\ S_2 \\ S_3 \\ \end{array} & \oplus & \begin{array}{c} S_3 \\ S_2 \\ S_1 \\ \end{array} \\ & S_2 & \\ \end{array}, & \begin{array}{c} S_3 \\ S_2 \\ 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. This follows from the result for M(4,2,3) since blocks in these classes are derived equivalent and so perfectly isometric
  4. A covering block cannot have defect group [math]C_4 \times C_2[/math].