feat(RepresentationTheory): Adds Subrepresentations #286
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@YaelDillies there is one remaining sorry at |
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The proof should be |
kbuzzard
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Dec 17, 2024
kbuzzard
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Dec 17, 2024
kbuzzard
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Dec 18, 2024
| variable {W : Type*} [AddCommMonoid W] [Module k W] | ||
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| IsIrreducible predicates that a given Representation ρ is irreducible (also known as simple), |
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| IsIrreducible predicates that a given Representation ρ is irreducible (also known as simple), | |
| `IsIrreducible ρ` is the statement that a given representation ρ is irreducible (also known as simple), |
My instinct is to make things easier for mathematicians to read, and I don't know if they know what "predicates" means.
| meaning that any subrepresentation must be either the full one (⊤) or zero (⊥) | ||
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| This notion is only well behaved when the representation is over a field k. If it were defined over | ||
| a ring A with a nontrivial ideal J, the subrepresentation JW would be a non trivial subrepresentation, |
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| a ring A with a nontrivial ideal J, the subrepresentation JW would be a non trivial subrepresentation, | |
| a ring A with a nontrivial ideal J, the subrepresentation JW would often be a non trivial subrepresentation, |
If W was A/J then this trick doesn't work.
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| This notion is only well behaved when the representation is over a field k. If it were defined over | ||
| a ring A with a nontrivial ideal J, the subrepresentation JW would be a non trivial subrepresentation, | ||
| so ρ would never be irreducible. |
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| so ρ would never be irreducible. | |
| so ρ would rarely be irreducible. |
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| This notion is only well behaved when the representation is over a field k. If it were defined over | ||
| a ring A with a nontrivial ideal J, the subrepresentation JW would be a non trivial subrepresentation, | ||
| so ρ would never be absolutely irreducible. |
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Suggested change
| This notion is only well behaved when the representation is over a field k. If it were defined over | |
| a ring A with a nontrivial ideal J, the subrepresentation JW would be a non trivial subrepresentation, | |
| so ρ would never be absolutely irreducible. |
Sorry, this was my misunderstanding.
| irreducible : IsSimpleOrder (Subrepresentation ρ) | ||
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| IsAbsolutelyIrreducible predicates that a given Representation ρ over a field k |
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Suggested change
| IsAbsolutelyIrreducible predicates that a given Representation ρ over a field k | |
| `IsAbsolutelyIrreducible ρ` states that a given representation ρ over a field k |
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Oh sorry, you dealt with all my comments but didn't resolve the conversations so I only just noticed! It's also quite hard to work out what you just did because for some reason you force-pushed. |
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Co-authored-by: Yaël Dillies [email protected]