Difference between revisions of "Morita invariants"
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| − | == Invariants preserved under Morita equivalence of blocks of <math>k</math>-algebras == | + | [[Image:under-construction.png|50px|left]] |
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| + | == Invariants preserved under Morita equivalence of blocks of f.d. <math>k</math>-algebras == | ||
| + | |||
| + | * Number of isomorphism classes of simple modules | ||
| + | * Isomorphism type of centre of algebra, and so dimension of centre (hence, in the case of <math>k</math>-blocks of finite groups, the number of irreducible characters associated to the corresponding <math>\mathcal{O}</math>-block) | ||
| + | * Loewy structure and socle structure | ||
== Invariants preserved under <math>k</math>-Morita equivalence of blocks of finite groups == | == Invariants preserved under <math>k</math>-Morita equivalence of blocks of finite groups == | ||
| + | |||
| + | * Invariants listed above | ||
| + | * Cartan matrix, up to rearrangement of rows and columns | ||
| + | * Order of defect group | ||
| + | * Exponent of defect group<ref>See (78) of [[References#K|[Ku91]]]</ref> | ||
| + | * <math>p</math>-rank of defect group<ref>See Corollary 4 of [[References#A|[AE81]]], attributed to Donovan</ref> | ||
== Invariants preserved under <math>\mathcal{O}</math>-Morita equivalence of blocks of finite groups == | == Invariants preserved under <math>\mathcal{O}</math>-Morita equivalence of blocks of finite groups == | ||
| + | |||
| + | * Invariants listed above | ||
| + | * Number <math>k_h(B)</math> of irreducible characters of a given [[Glossary#Height of an irreducible character|height]] <math>h</math> | ||
| + | * Decomposition matrix, up to rearrangement of rows and columns | ||
== Invariants preserved under basic Morita equivalence of blocks of finite groups<ref>Note that by [[References#K|[KL18]]], if <math>k</math>-blocks are basic Morita equivalent, then the corresponding <math>\mathcal{O}</math>-blocks are Morita equivalent</ref> == | == Invariants preserved under basic Morita equivalence of blocks of finite groups<ref>Note that by [[References#K|[KL18]]], if <math>k</math>-blocks are basic Morita equivalent, then the corresponding <math>\mathcal{O}</math>-blocks are Morita equivalent</ref> == | ||
| + | |||
| + | * Invariants listed above | ||
| + | * Isomorphism type of a defect group<ref>See Corollary 3.6 of [[References#P|[Pu99]]]</ref> | ||
| + | * [[Glossary#Fusion system|Fusion system]] of the block<ref>See Corollary 3.6 of [[References#P|[Pu99]]]</ref> | ||
== Invariants preserved under splendid Morita equivalence of blocks of finite groups == | == Invariants preserved under splendid Morita equivalence of blocks of finite groups == | ||
| + | |||
| + | * Invariants listed above | ||
| + | * [[Glossary#Source algebra|Source algebra]] | ||
== Notes == | == Notes == | ||
<references /> | <references /> | ||
Latest revision as of 11:40, 2 May 2024
Contents
- 1 Invariants preserved under Morita equivalence of blocks of f.d. [math]k[/math]-algebras
- 2 Invariants preserved under [math]k[/math]-Morita equivalence of blocks of finite groups
- 3 Invariants preserved under [math]\mathcal{O}[/math]-Morita equivalence of blocks of finite groups
- 4 Invariants preserved under basic Morita equivalence of blocks of finite groups[3]
- 5 Invariants preserved under splendid Morita equivalence of blocks of finite groups
- 6 Notes
Invariants preserved under Morita equivalence of blocks of f.d. [math]k[/math]-algebras
- Number of isomorphism classes of simple modules
- Isomorphism type of centre of algebra, and so dimension of centre (hence, in the case of [math]k[/math]-blocks of finite groups, the number of irreducible characters associated to the corresponding [math]\mathcal{O}[/math]-block)
- Loewy structure and socle structure
Invariants preserved under [math]k[/math]-Morita equivalence of blocks of finite groups
- Invariants listed above
- Cartan matrix, up to rearrangement of rows and columns
- Order of defect group
- Exponent of defect group[1]
- [math]p[/math]-rank of defect group[2]
Invariants preserved under [math]\mathcal{O}[/math]-Morita equivalence of blocks of finite groups
- Invariants listed above
- Number [math]k_h(B)[/math] of irreducible characters of a given height [math]h[/math]
- Decomposition matrix, up to rearrangement of rows and columns
Invariants preserved under basic Morita equivalence of blocks of finite groups[3]
- Invariants listed above
- Isomorphism type of a defect group[4]
- Fusion system of the block[5]
Invariants preserved under splendid Morita equivalence of blocks of finite groups
- Invariants listed above
- Source algebra
