[Merged by Bors] - feat(RepresentationTheory/Coinvariants): more API about the coinvariants of a representation#21735
Closed
101damnations wants to merge 125 commits intomasterfrom
Closed
[Merged by Bors] - feat(RepresentationTheory/Coinvariants): more API about the coinvariants of a representation#21735101damnations wants to merge 125 commits intomasterfrom
101damnations wants to merge 125 commits intomasterfrom
Conversation
added 30 commits
January 22, 2025 14:37
added 2 commits
May 31, 2025 16:54
Member
|
LGTM now maintainer merge |
|
🚀 Pull request has been placed on the maintainer queue by kbuzzard. |
mathlib-bors bot
pushed a commit
that referenced
this pull request
Jun 2, 2025
…nts of a representation (#21735) In this PR we add more API for the coinvariants of a representation `(V, ρ)`: * `Representation.coinvariantsFinsuppLEquiv ρ α`: given a type `α`, this is the `k`-linear equivalence between `(α →₀ V)_G` and `α →₀ V_G`. * `Representation.coinvariantsTprodLeftRegularLEquiv ρ`: the `k`-linear equivalence between `(V ⊗[k] k[G])_G` and `V` sending `⟦v ⊗ single g r⟧ ↦ r • ρ(g⁻¹)(v)`. * `Rep.coinvariantsTensor k G`: the functor sending representations `A, B` to `(A ⊗[k] B)_G`. This is naturally isomorphic to the functor sending `A, B` to `A ⊗[k[G]] B`, where we give `A` the `k[G]ᵐᵒᵖ`-module structure defined by `g • a := A.ρ g⁻¹ a`. * `Rep.coinvariantsTensorFreeLEquiv A α`: given a representation `A` and a type `α`, this is the `k`-linear equivalence between `(A ⊗ (α →₀ k[G]))_G` and `α →₀ A` sending `⟦a ⊗ single x (single g r)⟧ ↦ single x (r • ρ(g⁻¹)(a))`. This is useful for group homology. Co-authored-by: 101damnations <al3717@ic.ac.uk>
Contributor
|
Pull request successfully merged into master. Build succeeded: |
joelriou
pushed a commit
that referenced
this pull request
Jun 8, 2025
…nts of a representation (#21735) In this PR we add more API for the coinvariants of a representation `(V, ρ)`: * `Representation.coinvariantsFinsuppLEquiv ρ α`: given a type `α`, this is the `k`-linear equivalence between `(α →₀ V)_G` and `α →₀ V_G`. * `Representation.coinvariantsTprodLeftRegularLEquiv ρ`: the `k`-linear equivalence between `(V ⊗[k] k[G])_G` and `V` sending `⟦v ⊗ single g r⟧ ↦ r • ρ(g⁻¹)(v)`. * `Rep.coinvariantsTensor k G`: the functor sending representations `A, B` to `(A ⊗[k] B)_G`. This is naturally isomorphic to the functor sending `A, B` to `A ⊗[k[G]] B`, where we give `A` the `k[G]ᵐᵒᵖ`-module structure defined by `g • a := A.ρ g⁻¹ a`. * `Rep.coinvariantsTensorFreeLEquiv A α`: given a representation `A` and a type `α`, this is the `k`-linear equivalence between `(A ⊗ (α →₀ k[G]))_G` and `α →₀ A` sending `⟦a ⊗ single x (single g r)⟧ ↦ single x (r • ρ(g⁻¹)(a))`. This is useful for group homology. Co-authored-by: 101damnations <al3717@ic.ac.uk>
TOMILO87
pushed a commit
that referenced
this pull request
Jun 8, 2025
…nts of a representation (#21735) In this PR we add more API for the coinvariants of a representation `(V, ρ)`: * `Representation.coinvariantsFinsuppLEquiv ρ α`: given a type `α`, this is the `k`-linear equivalence between `(α →₀ V)_G` and `α →₀ V_G`. * `Representation.coinvariantsTprodLeftRegularLEquiv ρ`: the `k`-linear equivalence between `(V ⊗[k] k[G])_G` and `V` sending `⟦v ⊗ single g r⟧ ↦ r • ρ(g⁻¹)(v)`. * `Rep.coinvariantsTensor k G`: the functor sending representations `A, B` to `(A ⊗[k] B)_G`. This is naturally isomorphic to the functor sending `A, B` to `A ⊗[k[G]] B`, where we give `A` the `k[G]ᵐᵒᵖ`-module structure defined by `g • a := A.ρ g⁻¹ a`. * `Rep.coinvariantsTensorFreeLEquiv A α`: given a representation `A` and a type `α`, this is the `k`-linear equivalence between `(A ⊗ (α →₀ k[G]))_G` and `α →₀ A` sending `⟦a ⊗ single x (single g r)⟧ ↦ single x (r • ρ(g⁻¹)(a))`. This is useful for group homology. Co-authored-by: 101damnations <al3717@ic.ac.uk>
This was referenced Jun 14, 2025
[Merged by Bors] - feat(RepresentationTheory/Homological/GroupHomology): long exact sequences
#25943
Closed
callesonne
pushed a commit
to callesonne/mathlib4
that referenced
this pull request
Jul 24, 2025
…nts of a representation (leanprover-community#21735) In this PR we add more API for the coinvariants of a representation `(V, ρ)`: * `Representation.coinvariantsFinsuppLEquiv ρ α`: given a type `α`, this is the `k`-linear equivalence between `(α →₀ V)_G` and `α →₀ V_G`. * `Representation.coinvariantsTprodLeftRegularLEquiv ρ`: the `k`-linear equivalence between `(V ⊗[k] k[G])_G` and `V` sending `⟦v ⊗ single g r⟧ ↦ r • ρ(g⁻¹)(v)`. * `Rep.coinvariantsTensor k G`: the functor sending representations `A, B` to `(A ⊗[k] B)_G`. This is naturally isomorphic to the functor sending `A, B` to `A ⊗[k[G]] B`, where we give `A` the `k[G]ᵐᵒᵖ`-module structure defined by `g • a := A.ρ g⁻¹ a`. * `Rep.coinvariantsTensorFreeLEquiv A α`: given a representation `A` and a type `α`, this is the `k`-linear equivalence between `(A ⊗ (α →₀ k[G]))_G` and `α →₀ A` sending `⟦a ⊗ single x (single g r)⟧ ↦ single x (r • ρ(g⁻¹)(a))`. This is useful for group homology. Co-authored-by: 101damnations <al3717@ic.ac.uk>
This file contains hidden or bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
Sign up for free
to join this conversation on GitHub.
Already have an account?
Sign in to comment
6 participants
Add this suggestion to a batch that can be applied as a single commit.This suggestion is invalid because no changes were made to the code.Suggestions cannot be applied while the pull request is closed.Suggestions cannot be applied while viewing a subset of changes.Only one suggestion per line can be applied in a batch.Add this suggestion to a batch that can be applied as a single commit.Applying suggestions on deleted lines is not supported.You must change the existing code in this line in order to create a valid suggestion.Outdated suggestions cannot be applied.This suggestion has been applied or marked resolved.Suggestions cannot be applied from pending reviews.Suggestions cannot be applied on multi-line comments.Suggestions cannot be applied while the pull request is queued to merge.Suggestion cannot be applied right now. Please check back later.
In this PR we add more API for the coinvariants of a representation
(V, ρ):Representation.coinvariantsFinsuppLEquiv ρ α: given a typeα, this is thek-linearequivalence between
(α →₀ V)_Gandα →₀ V_G.Representation.coinvariantsTprodLeftRegularLEquiv ρ: thek-linear equivalence between(V ⊗[k] k[G])_GandVsending⟦v ⊗ single g r⟧ ↦ r • ρ(g⁻¹)(v).Rep.coinvariantsTensor k G: the functor sending representationsA, Bto(A ⊗[k] B)_G. Thisis naturally isomorphic to the functor sending
A, BtoA ⊗[k[G]] B, where we giveAthek[G]ᵐᵒᵖ-module structure defined byg • a := A.ρ g⁻¹ a.Rep.coinvariantsTensorFreeLEquiv A α: given a representationAand a typeα, this is thek-linear equivalence between(A ⊗ (α →₀ k[G]))_Gandα →₀ Asending⟦a ⊗ single x (single g r)⟧ ↦ single x (r • ρ(g⁻¹)(a)). This is useful for group homology.Action.rhoaMonoidHominstead of a morphism inMonCat#21652Finsupp#21732