On Generalized Minors and Quiver Representations Dylan Rupel Joint - PowerPoint PPT Presentation

Quivers and Generalized Minors On Generalized Minors and Quiver Representations Dylan Rupel Joint with: Salvatore Stella and Harold Williams University of Notre Dame October 19, 2016 Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 1 / 13

Quivers and Generalized Minors Plan for the talk 1 Quiver Representations 2 Cluster Algebras 3 Kac-Moody Groups Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 2 / 13

Quivers and Generalized Minors Quiver Representations Q = ( Q 0 , Q 1 , s , t ) - acyclic quiver vertices Q 0 = { 1 , . . . , n } arrows Q 1 s , t : Q 1 → Q 0 - source and target maps k = k - algebraically closed field of characteristic zero M = ( M i , M a ) - k -representation of Q M i - k -vector space for i ∈ Q 0 M a : M s ( a ) → M t ( a ) - k -linear map for a ∈ Q 1 rep k Q - k -linear abelian category of representations with Grothendieck group K 0 ( Q ) - vector space of dimension vectors M ∈ rep k Q is rigid if Ext 1 ( M , M ) = 0 Question How to get a handle on the representations of an arbitrary acyclic quiver? Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 3 / 13

Quivers and Generalized Minors Quiver Representations Q = ( Q 0 , Q 1 , s , t ) - acyclic quiver vertices Q 0 = { 1 , . . . , n } arrows Q 1 s , t : Q 1 → Q 0 - source and target maps k = k - algebraically closed field of characteristic zero M = ( M i , M a ) - k -representation of Q M i - k -vector space for i ∈ Q 0 M a : M s ( a ) → M t ( a ) - k -linear map for a ∈ Q 1 rep k Q - k -linear abelian category of representations with Grothendieck group K 0 ( Q ) - vector space of dimension vectors M ∈ rep k Q is rigid if Ext 1 ( M , M ) = 0 Question How to get a handle on the representations of an arbitrary acyclic quiver? Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 3 / 13

Quivers and Generalized Minors Quiver Representations Q = ( Q 0 , Q 1 , s , t ) - acyclic quiver vertices Q 0 = { 1 , . . . , n } arrows Q 1 s , t : Q 1 → Q 0 - source and target maps k = k - algebraically closed field of characteristic zero M = ( M i , M a ) - k -representation of Q M i - k -vector space for i ∈ Q 0 M a : M s ( a ) → M t ( a ) - k -linear map for a ∈ Q 1 rep k Q - k -linear abelian category of representations with Grothendieck group K 0 ( Q ) - vector space of dimension vectors M ∈ rep k Q is rigid if Ext 1 ( M , M ) = 0 Question How to get a handle on the representations of an arbitrary acyclic quiver? Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 3 / 13

Quivers and Generalized Minors Quiver Representations Q = ( Q 0 , Q 1 , s , t ) - acyclic quiver vertices Q 0 = { 1 , . . . , n } arrows Q 1 s , t : Q 1 → Q 0 - source and target maps k = k - algebraically closed field of characteristic zero M = ( M i , M a ) - k -representation of Q M i - k -vector space for i ∈ Q 0 M a : M s ( a ) → M t ( a ) - k -linear map for a ∈ Q 1 rep k Q - k -linear abelian category of representations with Grothendieck group K 0 ( Q ) - vector space of dimension vectors M ∈ rep k Q is rigid if Ext 1 ( M , M ) = 0 Question How to get a handle on the representations of an arbitrary acyclic quiver? Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 3 / 13

Quivers and Generalized Minors Quiver Representations Q = ( Q 0 , Q 1 , s , t ) - acyclic quiver vertices Q 0 = { 1 , . . . , n } arrows Q 1 s , t : Q 1 → Q 0 - source and target maps k = k - algebraically closed field of characteristic zero M = ( M i , M a ) - k -representation of Q M i - k -vector space for i ∈ Q 0 M a : M s ( a ) → M t ( a ) - k -linear map for a ∈ Q 1 rep k Q - k -linear abelian category of representations with Grothendieck group K 0 ( Q ) - vector space of dimension vectors M ∈ rep k Q is rigid if Ext 1 ( M , M ) = 0 Question How to get a handle on the representations of an arbitrary acyclic quiver? Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 3 / 13

Quivers and Generalized Minors Quiver Representations Q = ( Q 0 , Q 1 , s , t ) - acyclic quiver vertices Q 0 = { 1 , . . . , n } arrows Q 1 s , t : Q 1 → Q 0 - source and target maps k = k - algebraically closed field of characteristic zero M = ( M i , M a ) - k -representation of Q M i - k -vector space for i ∈ Q 0 M a : M s ( a ) → M t ( a ) - k -linear map for a ∈ Q 1 rep k Q - k -linear abelian category of representations with Grothendieck group K 0 ( Q ) - vector space of dimension vectors M ∈ rep k Q is rigid if Ext 1 ( M , M ) = 0 Question How to get a handle on the representations of an arbitrary acyclic quiver? Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 3 / 13

Quivers and Generalized Minors Quiver Representations Q = ( Q 0 , Q 1 , s , t ) - acyclic quiver vertices Q 0 = { 1 , . . . , n } arrows Q 1 s , t : Q 1 → Q 0 - source and target maps k = k - algebraically closed field of characteristic zero M = ( M i , M a ) - k -representation of Q M i - k -vector space for i ∈ Q 0 M a : M s ( a ) → M t ( a ) - k -linear map for a ∈ Q 1 rep k Q - k -linear abelian category of representations with Grothendieck group K 0 ( Q ) - vector space of dimension vectors M ∈ rep k Q is rigid if Ext 1 ( M , M ) = 0 Question How to get a handle on the representations of an arbitrary acyclic quiver? Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 3 / 13

Quivers and Generalized Minors Quiver Representations Q = ( Q 0 , Q 1 , s , t ) - acyclic quiver vertices Q 0 = { 1 , . . . , n } arrows Q 1 s , t : Q 1 → Q 0 - source and target maps k = k - algebraically closed field of characteristic zero M = ( M i , M a ) - k -representation of Q M i - k -vector space for i ∈ Q 0 M a : M s ( a ) → M t ( a ) - k -linear map for a ∈ Q 1 rep k Q - k -linear abelian category of representations with Grothendieck group K 0 ( Q ) - vector space of dimension vectors M ∈ rep k Q is rigid if Ext 1 ( M , M ) = 0 Question How to get a handle on the representations of an arbitrary acyclic quiver? Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 3 / 13

Quivers and Generalized Minors Quiver Representations Q = ( Q 0 , Q 1 , s , t ) - acyclic quiver vertices Q 0 = { 1 , . . . , n } arrows Q 1 s , t : Q 1 → Q 0 - source and target maps k = k - algebraically closed field of characteristic zero M = ( M i , M a ) - k -representation of Q M i - k -vector space for i ∈ Q 0 M a : M s ( a ) → M t ( a ) - k -linear map for a ∈ Q 1 rep k Q - k -linear abelian category of representations with Grothendieck group K 0 ( Q ) - vector space of dimension vectors M ∈ rep k Q is rigid if Ext 1 ( M , M ) = 0 Question How to get a handle on the representations of an arbitrary acyclic quiver? Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 3 / 13

Quivers and Generalized Minors Quiver Representations Q = ( Q 0 , Q 1 , s , t ) - acyclic quiver vertices Q 0 = { 1 , . . . , n } arrows Q 1 s , t : Q 1 → Q 0 - source and target maps k = k - algebraically closed field of characteristic zero M = ( M i , M a ) - k -representation of Q M i - k -vector space for i ∈ Q 0 M a : M s ( a ) → M t ( a ) - k -linear map for a ∈ Q 1 rep k Q - k -linear abelian category of representations with Grothendieck group K 0 ( Q ) - vector space of dimension vectors M ∈ rep k Q is rigid if Ext 1 ( M , M ) = 0 Question How to get a handle on the representations of an arbitrary acyclic quiver? Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 3 / 13

Quivers and Generalized Minors Quiver Representations Q = ( Q 0 , Q 1 , s , t ) - acyclic quiver vertices Q 0 = { 1 , . . . , n } arrows Q 1 s , t : Q 1 → Q 0 - source and target maps k = k - algebraically closed field of characteristic zero M = ( M i , M a ) - k -representation of Q M i - k -vector space for i ∈ Q 0 M a : M s ( a ) → M t ( a ) - k -linear map for a ∈ Q 1 rep k Q - k -linear abelian category of representations with Grothendieck group K 0 ( Q ) - vector space of dimension vectors M ∈ rep k Q is rigid if Ext 1 ( M , M ) = 0 Question How to get a handle on the representations of an arbitrary acyclic quiver? Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 3 / 13

Quivers and Generalized Minors Quiver Representations Classification of Representations τ : rep k Q → rep k Q - Auslander-Reiten translation functor for M ∈ rep k Q , τ ( M ) is computed by 0 − → τ ( M ) − → D Hom Q ( P 1 , k Q ) − → D Hom Q ( P 0 , k Q ) P 1 − → P 0 − → M − → 0 - projective presentation of M D = Hom k ( − , k ) - standard k -linear duality k Q - path algebra of Q with basis paths in Q and multiplication given by concatenation of paths, thought of as a representation of Q Definition (Classification of Representations) P is projective ⇔ τ ( P ) = 0 M is preprojective ⇔ τ k ( M ) = 0 for some k ≥ 1 I is injective ⇔ I � = τ ( M ) for any M ∈ rep k Q M is postinjective ⇔ M = τ k ( I ) for some k ≥ 0 and some injective I M is regular otherwise Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 4 / 13