Difference between revisions of "M(25,1,1)"
m |
(Pic_k, image, etc.) |
||
| Line 1: | Line 1: | ||
{{blockbox | {{blockbox | ||
|title = M(25,1,1) - <math>kC_{25}</math> | |title = M(25,1,1) - <math>kC_{25}</math> | ||
| − | |image = | + | |image = M(2,1,1)quiver.png |
|representative = <math>kC_{25}</math> | |representative = <math>kC_{25}</math> | ||
|defect = [[C25|<math>C_{25}</math>]] | |defect = [[C25|<math>C_{25}</math>]] | ||
| Line 8: | Line 8: | ||
|l(B) = 1 | |l(B) = 1 | ||
|k-morita-frob = 1 | |k-morita-frob = 1 | ||
| − | |Pic-k= | + | |Pic-k= <math>k^{23}:k^*</math> |
|cartan = <math>\left( \begin{array}{c} | |cartan = <math>\left( \begin{array}{c} | ||
25 \\ | 25 \\ | ||
| Line 29: | Line 29: | ||
|k-derived = Forms a derived equivalence class | |k-derived = Forms a derived equivalence class | ||
|O-derived-known? = Yes | |O-derived-known? = Yes | ||
| + | |coveringblocks = [[M(25,1,2)]], [[M(25,1,4)]] | ||
| + | |coveredblocks = | ||
}} | }} | ||
| Line 38: | Line 40: | ||
'''Quiver:''' a:<1,1> | '''Quiver:''' a:<1,1> | ||
| − | '''Relations w.r.t. <math>k</math>:''' a^{25}=0 | + | '''Relations w.r.t. <math>k</math>:''' <math>a^{25}=0</math> |
== Other notatable representatives == | == Other notatable representatives == | ||
Latest revision as of 15:24, 8 October 2018
quiver.png)
| Representative: | [math]kC_{25}[/math] |
|---|---|
| Defect groups: | [math]C_{25}[/math] |
| Inertial quotients: | [math]1[/math] |
| [math]k(B)=[/math] | 25 |
| [math]l(B)=[/math] | 1 |
| [math]{\rm mf}_k(B)=[/math] | 1 |
| [math]{\rm Pic}_k(B)=[/math] | [math]k^{23}:k^*[/math] |
| Cartan matrix: | [math]\left( \begin{array}{c} 25 \\ \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} C_{25}[/math] |
| Decomposition matrices: | [math]\left( \begin{array}{c} 1 \\ 1 \\ \vdots \\ 1 \\ \end{array}\right)[/math] |
| [math]{\rm mf}_\mathcal{O}(B)=[/math] | 1 |
| [math]{\rm Pic}_{\mathcal{O}}(B)=[/math] | [math]\mathcal{L}(B)=C_{25}:C_{20}[/math] |
| [math]PI(B)=[/math] | {{{PIgroup}}} |
| Source algebras known? | Yes |
| Source algebra reps: | [math]kC_{25}[/math] |
| [math]k[/math]-derived equiv. classes known? | Yes |
| [math]k[/math]-derived equivalent to: | Forms a derived equivalence class |
| [math]\mathcal{O}[/math]-derived equiv. classes known? | Yes |
| [math]p'[/math]-index covering blocks: | M(25,1,2), M(25,1,4) |
| [math]p'[/math]-index covered blocks: | |
| Index [math]p[/math] covering blocks: | {{{pcoveringblocks}}} |
These are nilpotent blocks.
Contents
Basic algebra
Quiver: a:<1,1>
Relations w.r.t. [math]k[/math]: [math]a^{25}=0[/math]
Other notatable representatives
Covering blocks and covered blocks
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].
If [math]b[/math] lies in M(25,1,1), then [math]B[/math] must lie in M(25,1,1), M(25,1,2) or M(25,1,4). For example consider the principal blocks of [math]C_{25} \triangleleft D_{50}, C_{25}:C_4[/math].
If [math]B[/math] lies in M(25,1,1), then [math]b[/math] must lie in M(25,1,1), M(25,1,2) or M(25,1,4). Example needed.
Projective indecomposable modules
Labelling the unique simple [math]B[/math]-module by [math]S_1[/math], the unique projective indecomposable module has Loewy structure as follows:
[math]\begin{array}{c} S_1 \\ S_1 \\ \vdots \\ S_1 \\ \end{array} [/math]
Irreducible characters
All irreducible characters have height zero.
