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

SLIDE 1

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

SLIDE 2

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

SLIDE 3

Quivers and Generalized Minors Quiver Representations

Q = (Q0, Q1, s, t) - acyclic quiver

vertices Q0 = {1, . . . , n} arrows Q1 s, t : Q1 → Q0 - source and target maps

k = k - algebraically closed field of characteristic zero M = (Mi, Ma) - k-representation of Q

Mi - k-vector space for i ∈ Q0 Ma : Ms(a) → Mt(a) - k-linear map for a ∈ Q1

repkQ - k-linear abelian category of representations with Grothendieck group K0(Q) - vector space of dimension vectors M ∈ repkQ is rigid if Ext1(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

SLIDE 4

Quivers and Generalized Minors Quiver Representations

Q = (Q0, Q1, s, t) - acyclic quiver

vertices Q0 = {1, . . . , n} arrows Q1 s, t : Q1 → Q0 - source and target maps

k = k - algebraically closed field of characteristic zero M = (Mi, Ma) - k-representation of Q

Mi - k-vector space for i ∈ Q0 Ma : Ms(a) → Mt(a) - k-linear map for a ∈ Q1

repkQ - k-linear abelian category of representations with Grothendieck group K0(Q) - vector space of dimension vectors M ∈ repkQ is rigid if Ext1(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

SLIDE 5

Quivers and Generalized Minors Quiver Representations

Q = (Q0, Q1, s, t) - acyclic quiver

vertices Q0 = {1, . . . , n} arrows Q1 s, t : Q1 → Q0 - source and target maps

k = k - algebraically closed field of characteristic zero M = (Mi, Ma) - k-representation of Q

Mi - k-vector space for i ∈ Q0 Ma : Ms(a) → Mt(a) - k-linear map for a ∈ Q1

repkQ - k-linear abelian category of representations with Grothendieck group K0(Q) - vector space of dimension vectors M ∈ repkQ is rigid if Ext1(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

SLIDE 6

Quivers and Generalized Minors Quiver Representations

Q = (Q0, Q1, s, t) - acyclic quiver

vertices Q0 = {1, . . . , n} arrows Q1 s, t : Q1 → Q0 - source and target maps

k = k - algebraically closed field of characteristic zero M = (Mi, Ma) - k-representation of Q

Mi - k-vector space for i ∈ Q0 Ma : Ms(a) → Mt(a) - k-linear map for a ∈ Q1

repkQ - k-linear abelian category of representations with Grothendieck group K0(Q) - vector space of dimension vectors M ∈ repkQ is rigid if Ext1(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

SLIDE 7

Quivers and Generalized Minors Quiver Representations

Q = (Q0, Q1, s, t) - acyclic quiver

vertices Q0 = {1, . . . , n} arrows Q1 s, t : Q1 → Q0 - source and target maps

k = k - algebraically closed field of characteristic zero M = (Mi, Ma) - k-representation of Q

Mi - k-vector space for i ∈ Q0 Ma : Ms(a) → Mt(a) - k-linear map for a ∈ Q1

repkQ - k-linear abelian category of representations with Grothendieck group K0(Q) - vector space of dimension vectors M ∈ repkQ is rigid if Ext1(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

SLIDE 8

Quivers and Generalized Minors Quiver Representations

Q = (Q0, Q1, s, t) - acyclic quiver

vertices Q0 = {1, . . . , n} arrows Q1 s, t : Q1 → Q0 - source and target maps

k = k - algebraically closed field of characteristic zero M = (Mi, Ma) - k-representation of Q

Mi - k-vector space for i ∈ Q0 Ma : Ms(a) → Mt(a) - k-linear map for a ∈ Q1

repkQ - k-linear abelian category of representations with Grothendieck group K0(Q) - vector space of dimension vectors M ∈ repkQ is rigid if Ext1(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

SLIDE 9

Quivers and Generalized Minors Quiver Representations

Q = (Q0, Q1, s, t) - acyclic quiver

vertices Q0 = {1, . . . , n} arrows Q1 s, t : Q1 → Q0 - source and target maps

k = k - algebraically closed field of characteristic zero M = (Mi, Ma) - k-representation of Q

Mi - k-vector space for i ∈ Q0 Ma : Ms(a) → Mt(a) - k-linear map for a ∈ Q1

repkQ - k-linear abelian category of representations with Grothendieck group K0(Q) - vector space of dimension vectors M ∈ repkQ is rigid if Ext1(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

SLIDE 10

Quivers and Generalized Minors Quiver Representations

Q = (Q0, Q1, s, t) - acyclic quiver

vertices Q0 = {1, . . . , n} arrows Q1 s, t : Q1 → Q0 - source and target maps

k = k - algebraically closed field of characteristic zero M = (Mi, Ma) - k-representation of Q

Mi - k-vector space for i ∈ Q0 Ma : Ms(a) → Mt(a) - k-linear map for a ∈ Q1

repkQ - k-linear abelian category of representations with Grothendieck group K0(Q) - vector space of dimension vectors M ∈ repkQ is rigid if Ext1(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

SLIDE 11

Quivers and Generalized Minors Quiver Representations

Q = (Q0, Q1, s, t) - acyclic quiver

vertices Q0 = {1, . . . , n} arrows Q1 s, t : Q1 → Q0 - source and target maps

k = k - algebraically closed field of characteristic zero M = (Mi, Ma) - k-representation of Q

Mi - k-vector space for i ∈ Q0 Ma : Ms(a) → Mt(a) - k-linear map for a ∈ Q1

repkQ - k-linear abelian category of representations with Grothendieck group K0(Q) - vector space of dimension vectors M ∈ repkQ is rigid if Ext1(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

SLIDE 12

Quivers and Generalized Minors Quiver Representations

Q = (Q0, Q1, s, t) - acyclic quiver

vertices Q0 = {1, . . . , n} arrows Q1 s, t : Q1 → Q0 - source and target maps

k = k - algebraically closed field of characteristic zero M = (Mi, Ma) - k-representation of Q

Mi - k-vector space for i ∈ Q0 Ma : Ms(a) → Mt(a) - k-linear map for a ∈ Q1

repkQ - k-linear abelian category of representations with Grothendieck group K0(Q) - vector space of dimension vectors M ∈ repkQ is rigid if Ext1(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

SLIDE 13

Quivers and Generalized Minors Quiver Representations

Q = (Q0, Q1, s, t) - acyclic quiver

vertices Q0 = {1, . . . , n} arrows Q1 s, t : Q1 → Q0 - source and target maps

k = k - algebraically closed field of characteristic zero M = (Mi, Ma) - k-representation of Q

Mi - k-vector space for i ∈ Q0 Ma : Ms(a) → Mt(a) - k-linear map for a ∈ Q1

repkQ - k-linear abelian category of representations with Grothendieck group K0(Q) - vector space of dimension vectors M ∈ repkQ is rigid if Ext1(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

SLIDE 14

Quivers and Generalized Minors Quiver Representations Classification of Representations

τ : repkQ → repkQ - Auslander-Reiten translation functor for M ∈ repkQ, τ(M) is computed by 0 − → τ(M) − → DHomQ(P1, kQ) − → DHomQ(P0, kQ)

P1 − → P0 − → M − → 0 - projective presentation of M D = Homk(−, k) - standard k-linear duality kQ - 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 ∈ repkQ 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

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Quivers and Generalized Minors Quiver Representations Classification of Representations

τ : repkQ → repkQ - Auslander-Reiten translation functor for M ∈ repkQ, τ(M) is computed by 0 − → τ(M) − → DHomQ(P1, kQ) − → DHomQ(P0, kQ)

P1 − → P0 − → M − → 0 - projective presentation of M D = Homk(−, k) - standard k-linear duality kQ - 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 ∈ repkQ 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

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Quivers and Generalized Minors Quiver Representations Classification of Representations

τ : repkQ → repkQ - Auslander-Reiten translation functor for M ∈ repkQ, τ(M) is computed by 0 − → τ(M) − → DHomQ(P1, kQ) − → DHomQ(P0, kQ)

P1 − → P0 − → M − → 0 - projective presentation of M D = Homk(−, k) - standard k-linear duality kQ - 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 ∈ repkQ 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

SLIDE 17

Quivers and Generalized Minors Quiver Representations Classification of Representations

τ : repkQ → repkQ - Auslander-Reiten translation functor for M ∈ repkQ, τ(M) is computed by 0 − → τ(M) − → DHomQ(P1, kQ) − → DHomQ(P0, kQ)

P1 − → P0 − → M − → 0 - projective presentation of M D = Homk(−, k) - standard k-linear duality kQ - 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 ∈ repkQ 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

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Quivers and Generalized Minors Quiver Representations Classification of Representations

τ : repkQ → repkQ - Auslander-Reiten translation functor for M ∈ repkQ, τ(M) is computed by 0 − → τ(M) − → DHomQ(P1, kQ) − → DHomQ(P0, kQ)

P1 − → P0 − → M − → 0 - projective presentation of M D = Homk(−, k) - standard k-linear duality kQ - 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 ∈ repkQ 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

SLIDE 19

Quivers and Generalized Minors Quiver Representations Classification of Representations

τ : repkQ → repkQ - Auslander-Reiten translation functor for M ∈ repkQ, τ(M) is computed by 0 − → τ(M) − → DHomQ(P1, kQ) − → DHomQ(P0, kQ)

P1 − → P0 − → M − → 0 - projective presentation of M D = Homk(−, k) - standard k-linear duality kQ - 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 ∈ repkQ 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

SLIDE 20

Quivers and Generalized Minors Quiver Representations Classification of Representations

τ : repkQ → repkQ - Auslander-Reiten translation functor for M ∈ repkQ, τ(M) is computed by 0 − → τ(M) − → DHomQ(P1, kQ) − → DHomQ(P0, kQ)

P1 − → P0 − → M − → 0 - projective presentation of M D = Homk(−, k) - standard k-linear duality kQ - 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 ∈ repkQ 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

SLIDE 21

Quivers and Generalized Minors Quiver Representations Classification of Representations

τ : repkQ → repkQ - Auslander-Reiten translation functor for M ∈ repkQ, τ(M) is computed by 0 − → τ(M) − → DHomQ(P1, kQ) − → DHomQ(P0, kQ)

P1 − → P0 − → M − → 0 - projective presentation of M D = Homk(−, k) - standard k-linear duality kQ - 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 ∈ repkQ 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

SLIDE 22

Quivers and Generalized Minors Quiver Representations Classification of Representations

τ : repkQ → repkQ - Auslander-Reiten translation functor for M ∈ repkQ, τ(M) is computed by 0 − → τ(M) − → DHomQ(P1, kQ) − → DHomQ(P0, kQ)

P1 − → P0 − → M − → 0 - projective presentation of M D = Homk(−, k) - standard k-linear duality kQ - 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 ∈ repkQ 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

SLIDE 23

Quivers and Generalized Minors Quiver Representations Classification of Representations

τ : repkQ → repkQ - Auslander-Reiten translation functor for M ∈ repkQ, τ(M) is computed by 0 − → τ(M) − → DHomQ(P1, kQ) − → DHomQ(P0, kQ)

P1 − → P0 − → M − → 0 - projective presentation of M D = Homk(−, k) - standard k-linear duality kQ - 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 ∈ repkQ 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

SLIDE 24

Quivers and Generalized Minors Quiver Representations Classification of Representations

τ : repkQ → repkQ - Auslander-Reiten translation functor for M ∈ repkQ, τ(M) is computed by 0 − → τ(M) − → DHomQ(P1, kQ) − → DHomQ(P0, kQ)

P1 − → P0 − → M − → 0 - projective presentation of M D = Homk(−, k) - standard k-linear duality kQ - 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 ∈ repkQ 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

SLIDE 25

Quivers and Generalized Minors Quiver Representations Heuristic Picture

τ preprojective postinjective regular projective injective

Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 5 / 13

SLIDE 26

Quivers and Generalized Minors Cluster algebras

B - n × n signed adjacency matrix of Q F ⊃ k - purely transcendental field extension of degree 3n x = (x1, . . . , xn, xn+1, . . . , x2n, x2n+1, . . . , x3n) - transcendence basis of F over k (later it will be convenient to write yi = xn+i and yı = x2n+i) Σ = (x, ˜ B) - seed µkΣ = (x′, ˜ B′) - seed mutation in direction k (1 ≤ k ≤ n)

x′ = (x′

1, . . . , x′ 3n) where x′ i = xi for i = k

xkx′

k =

i:bik>0

xbik

i

+

i:bik<0

x−bik

i

˜ B′ = (b′

ij) where b′ ij =

−bij

if i = k or j = k bij + |bik|bkj+bik|bkj|

2

therwise

A(x, ˜ B) - cluster algebra

k-subalgebra of F generated by all transcendence bases from seeds

btained by iterated mutation from (x, ˜

B)

Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 6 / 13

SLIDE 27

Quivers and Generalized Minors Cluster algebras

B - n × n signed adjacency matrix of Q F ⊃ k - purely transcendental field extension of degree 3n x = (x1, . . . , xn, xn+1, . . . , x2n, x2n+1, . . . , x3n) - transcendence basis of F over k (later it will be convenient to write yi = xn+i and yı = x2n+i) Σ = (x, ˜ B) - seed µkΣ = (x′, ˜ B′) - seed mutation in direction k (1 ≤ k ≤ n)

x′ = (x′

1, . . . , x′ 3n) where x′ i = xi for i = k

xkx′

k =

i:bik>0

xbik

i

+

i:bik<0

x−bik

i

˜ B′ = (b′

ij) where b′ ij =

−bij

if i = k or j = k bij + |bik|bkj+bik|bkj|

2

therwise

A(x, ˜ B) - cluster algebra

k-subalgebra of F generated by all transcendence bases from seeds

btained by iterated mutation from (x, ˜

B)

Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 6 / 13

SLIDE 28

Quivers and Generalized Minors Cluster algebras

B - n × n signed adjacency matrix of Q F ⊃ k - purely transcendental field extension of degree 3n x = (x1, . . . , xn, xn+1, . . . , x2n, x2n+1, . . . , x3n) - transcendence basis of F over k (later it will be convenient to write yi = xn+i and yı = x2n+i) Σ = (x, ˜ B) - seed µkΣ = (x′, ˜ B′) - seed mutation in direction k (1 ≤ k ≤ n)

x′ = (x′

1, . . . , x′ 3n) where x′ i = xi for i = k

xkx′

k =

i:bik>0

xbik

i

+

i:bik<0

x−bik

i

˜ B′ = (b′

ij) where b′ ij =

−bij

if i = k or j = k bij + |bik|bkj+bik|bkj|

2

therwise

A(x, ˜ B) - cluster algebra

k-subalgebra of F generated by all transcendence bases from seeds

btained by iterated mutation from (x, ˜

B)

Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 6 / 13

SLIDE 29

Quivers and Generalized Minors Cluster algebras

B - n × n signed adjacency matrix of Q F ⊃ k - purely transcendental field extension of degree 3n x = (x1, . . . , xn, xn+1, . . . , x2n, x2n+1, . . . , x3n) - transcendence basis of F over k (later it will be convenient to write yi = xn+i and yı = x2n+i) Σ = (x, ˜ B) - seed µkΣ = (x′, ˜ B′) - seed mutation in direction k (1 ≤ k ≤ n)

x′ = (x′

1, . . . , x′ 3n) where x′ i = xi for i = k

xkx′

k =

i:bik>0

xbik

i

+

i:bik<0

x−bik

i

˜ B′ = (b′

ij) where b′ ij =

−bij

if i = k or j = k bij + |bik|bkj+bik|bkj|

2

therwise

A(x, ˜ B) - cluster algebra

k-subalgebra of F generated by all transcendence bases from seeds

btained by iterated mutation from (x, ˜

B)

Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 6 / 13

SLIDE 30

Quivers and Generalized Minors Cluster algebras

B - n × n signed adjacency matrix of Q F ⊃ k - purely transcendental field extension of degree 3n x = (x1, . . . , xn, xn+1, . . . , x2n, x2n+1, . . . , x3n) - transcendence basis of F over k (later it will be convenient to write yi = xn+i and yı = x2n+i) Σ = (x, ˜ B) - seed µkΣ = (x′, ˜ B′) - seed mutation in direction k (1 ≤ k ≤ n)

x′ = (x′

1, . . . , x′ 3n) where x′ i = xi for i = k

xkx′

k =

i:bik>0

xbik

i

+

i:bik<0

x−bik

i

˜ B′ = (b′

ij) where b′ ij =

−bij

if i = k or j = k bij + |bik|bkj+bik|bkj|

2

therwise

A(x, ˜ B) - cluster algebra

k-subalgebra of F generated by all transcendence bases from seeds

btained by iterated mutation from (x, ˜

B)

Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 6 / 13

SLIDE 31

Quivers and Generalized Minors Cluster algebras

B - n × n signed adjacency matrix of Q F ⊃ k - purely transcendental field extension of degree 3n x = (x1, . . . , xn, xn+1, . . . , x2n, x2n+1, . . . , x3n) - transcendence basis of F over k (later it will be convenient to write yi = xn+i and yı = x2n+i) Σ = (x, ˜ B) - seed µkΣ = (x′, ˜ B′) - seed mutation in direction k (1 ≤ k ≤ n)

x′ = (x′

1, . . . , x′ 3n) where x′ i = xi for i = k

xkx′

k =

i:bik>0

xbik

i

+

i:bik<0

x−bik

i

˜ B′ = (b′

ij) where b′ ij =

−bij

if i = k or j = k bij + |bik|bkj+bik|bkj|

2

therwise

A(x, ˜ B) - cluster algebra

k-subalgebra of F generated by all transcendence bases from seeds

btained by iterated mutation from (x, ˜

B)

Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 6 / 13

SLIDE 32

Quivers and Generalized Minors Cluster algebras

B - n × n signed adjacency matrix of Q F ⊃ k - purely transcendental field extension of degree 3n x = (x1, . . . , xn, xn+1, . . . , x2n, x2n+1, . . . , x3n) - transcendence basis of F over k (later it will be convenient to write yi = xn+i and yı = x2n+i) Σ = (x, ˜ B) - seed µkΣ = (x′, ˜ B′) - seed mutation in direction k (1 ≤ k ≤ n)

x′ = (x′

1, . . . , x′ 3n) where x′ i = xi for i = k

xkx′

k =

i:bik>0

xbik

i

+

i:bik<0

x−bik

i

˜ B′ = (b′

ij) where b′ ij =

−bij

if i = k or j = k bij + |bik|bkj+bik|bkj|

2

therwise

A(x, ˜ B) - cluster algebra

k-subalgebra of F generated by all transcendence bases from seeds

btained by iterated mutation from (x, ˜

B)

Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 6 / 13

SLIDE 33

Quivers and Generalized Minors Cluster algebras

B - n × n signed adjacency matrix of Q F ⊃ k - purely transcendental field extension of degree 3n x = (x1, . . . , xn, xn+1, . . . , x2n, x2n+1, . . . , x3n) - transcendence basis of F over k (later it will be convenient to write yi = xn+i and yı = x2n+i) Σ = (x, ˜ B) - seed µkΣ = (x′, ˜ B′) - seed mutation in direction k (1 ≤ k ≤ n)

x′ = (x′

1, . . . , x′ 3n) where x′ i = xi for i = k

xkx′

k =

i:bik>0

xbik

i

+

i:bik<0

x−bik

i

˜ B′ = (b′

ij) where b′ ij =

−bij

if i = k or j = k bij + |bik|bkj+bik|bkj|

2

therwise

A(x, ˜ B) - cluster algebra

k-subalgebra of F generated by all transcendence bases from seeds

btained by iterated mutation from (x, ˜

B)

Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 6 / 13

SLIDE 34

Quivers and Generalized Minors Cluster algebras

B - n × n signed adjacency matrix of Q F ⊃ k - purely transcendental field extension of degree 3n x = (x1, . . . , xn, xn+1, . . . , x2n, x2n+1, . . . , x3n) - transcendence basis of F over k (later it will be convenient to write yi = xn+i and yı = x2n+i) Σ = (x, ˜ B) - seed µkΣ = (x′, ˜ B′) - seed mutation in direction k (1 ≤ k ≤ n)

x′ = (x′

1, . . . , x′ 3n) where x′ i = xi for i = k

xkx′

k =

i:bik>0

xbik

i

+

i:bik<0

x−bik

i

˜ B′ = (b′

ij) where b′ ij =

−bij

if i = k or j = k bij + |bik|bkj+bik|bkj|

2

therwise

A(x, ˜ B) - cluster algebra

k-subalgebra of F generated by all transcendence bases from seeds

btained by iterated mutation from (x, ˜

B)

Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 6 / 13

SLIDE 35

Quivers and Generalized Minors Cluster algebras Categorification

Theorem (Caldero-Chapoton,Caldero-Keller)

There is a bijection between rigid indecomposable representations of Q and non-initial cluster variables of A(x, ˜ B) given by M → xM = xω(M)

e∈K0(Q)

χ

Gre(M)
ˆ

ye, ˆ yk =

3n

i=1

xbik

i

where Gre(M) - Grassmannian of subrepresentations with dimension vector e ω(M) - g-vector of the cluster variable ∼ coindex of M

Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 7 / 13

SLIDE 36

Quivers and Generalized Minors Kac-Moody Groups

A = (aij) - symmetric Cartan matrix, where aii = 2 and aij = −|bij| G - Kac-Moody group associated to A

generated by coroot subgroups ϕi : SL2 ֒ → G set si = ϕi

−1

1

and si = ϕi
1

−1

B± - opposite Borel subgroups

H = B+ ∩ B− - Cartan subgroup W = NG(H)/H - Weyl group of G G =

u,v∈W

G u,v

G u,v = B+uB+ ∩ B−vB− - double Bruhat cells

P = Hom(H, k×) - weight lattice P+ - dominant weights

Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 8 / 13

SLIDE 37

Quivers and Generalized Minors Kac-Moody Groups

A = (aij) - symmetric Cartan matrix, where aii = 2 and aij = −|bij| G - Kac-Moody group associated to A

generated by coroot subgroups ϕi : SL2 ֒ → G set si = ϕi

−1

1

and si = ϕi
1

−1

B± - opposite Borel subgroups

H = B+ ∩ B− - Cartan subgroup W = NG(H)/H - Weyl group of G G =

u,v∈W

G u,v

G u,v = B+uB+ ∩ B−vB− - double Bruhat cells

P = Hom(H, k×) - weight lattice P+ - dominant weights

Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 8 / 13

SLIDE 38

Quivers and Generalized Minors Kac-Moody Groups

A = (aij) - symmetric Cartan matrix, where aii = 2 and aij = −|bij| G - Kac-Moody group associated to A

generated by coroot subgroups ϕi : SL2 ֒ → G set si = ϕi

−1

1

and si = ϕi
1

−1

B± - opposite Borel subgroups

H = B+ ∩ B− - Cartan subgroup W = NG(H)/H - Weyl group of G G =

u,v∈W

G u,v

G u,v = B+uB+ ∩ B−vB− - double Bruhat cells

P = Hom(H, k×) - weight lattice P+ - dominant weights

Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 8 / 13

SLIDE 39

Quivers and Generalized Minors Kac-Moody Groups

A = (aij) - symmetric Cartan matrix, where aii = 2 and aij = −|bij| G - Kac-Moody group associated to A

generated by coroot subgroups ϕi : SL2 ֒ → G set si = ϕi

−1

1

and si = ϕi
1

−1

B± - opposite Borel subgroups

H = B+ ∩ B− - Cartan subgroup W = NG(H)/H - Weyl group of G G =

u,v∈W

G u,v

G u,v = B+uB+ ∩ B−vB− - double Bruhat cells

P = Hom(H, k×) - weight lattice P+ - dominant weights

Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 8 / 13

SLIDE 40

Quivers and Generalized Minors Kac-Moody Groups

A = (aij) - symmetric Cartan matrix, where aii = 2 and aij = −|bij| G - Kac-Moody group associated to A

generated by coroot subgroups ϕi : SL2 ֒ → G set si = ϕi

−1

1

and si = ϕi
1

−1

B± - opposite Borel subgroups

H = B+ ∩ B− - Cartan subgroup W = NG(H)/H - Weyl group of G G =

u,v∈W

G u,v

G u,v = B+uB+ ∩ B−vB− - double Bruhat cells

P = Hom(H, k×) - weight lattice P+ - dominant weights

Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 8 / 13

SLIDE 41

Quivers and Generalized Minors Kac-Moody Groups

A = (aij) - symmetric Cartan matrix, where aii = 2 and aij = −|bij| G - Kac-Moody group associated to A

generated by coroot subgroups ϕi : SL2 ֒ → G set si = ϕi

−1

1

and si = ϕi
1

−1

B± - opposite Borel subgroups

H = B+ ∩ B− - Cartan subgroup W = NG(H)/H - Weyl group of G G =

u,v∈W

G u,v

G u,v = B+uB+ ∩ B−vB− - double Bruhat cells

P = Hom(H, k×) - weight lattice P+ - dominant weights

Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 8 / 13

SLIDE 42

Quivers and Generalized Minors Kac-Moody Groups

A = (aij) - symmetric Cartan matrix, where aii = 2 and aij = −|bij| G - Kac-Moody group associated to A

generated by coroot subgroups ϕi : SL2 ֒ → G set si = ϕi

−1

1

and si = ϕi
1

−1

B± - opposite Borel subgroups

H = B+ ∩ B− - Cartan subgroup W = NG(H)/H - Weyl group of G G =

u,v∈W

G u,v

G u,v = B+uB+ ∩ B−vB− - double Bruhat cells

P = Hom(H, k×) - weight lattice P+ - dominant weights

Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 8 / 13

SLIDE 43

Quivers and Generalized Minors Kac-Moody Groups

A = (aij) - symmetric Cartan matrix, where aii = 2 and aij = −|bij| G - Kac-Moody group associated to A

generated by coroot subgroups ϕi : SL2 ֒ → G set si = ϕi

−1

1

and si = ϕi
1

−1

B± - opposite Borel subgroups

H = B+ ∩ B− - Cartan subgroup W = NG(H)/H - Weyl group of G G =

u,v∈W

G u,v

G u,v = B+uB+ ∩ B−vB− - double Bruhat cells

P = Hom(H, k×) - weight lattice P+ - dominant weights

Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 8 / 13

SLIDE 44

Quivers and Generalized Minors Kac-Moody Groups

A = (aij) - symmetric Cartan matrix, where aii = 2 and aij = −|bij| G - Kac-Moody group associated to A

generated by coroot subgroups ϕi : SL2 ֒ → G set si = ϕi

−1

1

and si = ϕi
1

−1

B± - opposite Borel subgroups

H = B+ ∩ B− - Cartan subgroup W = NG(H)/H - Weyl group of G G =

u,v∈W

G u,v

G u,v = B+uB+ ∩ B−vB− - double Bruhat cells

P = Hom(H, k×) - weight lattice P+ - dominant weights

Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 8 / 13

SLIDE 45

Quivers and Generalized Minors Kac-Moody Groups

A = (aij) - symmetric Cartan matrix, where aii = 2 and aij = −|bij| G - Kac-Moody group associated to A

generated by coroot subgroups ϕi : SL2 ֒ → G set si = ϕi

−1

1

and si = ϕi
1

−1

B± - opposite Borel subgroups

H = B+ ∩ B− - Cartan subgroup W = NG(H)/H - Weyl group of G G =

u,v∈W

G u,v

G u,v = B+uB+ ∩ B−vB− - double Bruhat cells

P = Hom(H, k×) - weight lattice P+ - dominant weights

Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 8 / 13

SLIDE 46

Quivers and Generalized Minors Kac-Moody Groups

A = (aij) - symmetric Cartan matrix, where aii = 2 and aij = −|bij| G - Kac-Moody group associated to A

generated by coroot subgroups ϕi : SL2 ֒ → G set si = ϕi

−1

1

and si = ϕi
1

−1

B± - opposite Borel subgroups

H = B+ ∩ B− - Cartan subgroup W = NG(H)/H - Weyl group of G G =

u,v∈W

G u,v

G u,v = B+uB+ ∩ B−vB− - double Bruhat cells

P = Hom(H, k×) - weight lattice P+ - dominant weights

Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 8 / 13

SLIDE 47

Quivers and Generalized Minors Generalized Minors

Definition (Generalized Minors)

Let V =

µ∈P Vµ be a weight representation with dimkVλ = 1. Define

the principal generalized minor ∆V ,λ ∈ k[G] by g → πλ(gvλ)/vλ, where vλ ∈ Vλ is any nonzero vector; πλ : V → → Vλ is the orthogonal projection. For λ ∈ P+ ∩ −P+ and u, v ∈ W , define the generalized minor ∆uλ

vλ ∈ k[G] by

g →

∆V (λ),λ
u−1gv
if λ ∈ P+

∆V (λ),λ

u−1gv
if λ ∈ −P+

where V (λ) is the irreducible highest (or lowest) weight representation w = si1 · · · sik and w = si1 · · · sik for w = si1 · · · sik reduced decomp.

Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 9 / 13

SLIDE 48

Quivers and Generalized Minors Generalized Minors

Definition (Generalized Minors)

Let V =

µ∈P Vµ be a weight representation with dimkVλ = 1. Define

the principal generalized minor ∆V ,λ ∈ k[G] by g → πλ(gvλ)/vλ, where vλ ∈ Vλ is any nonzero vector; πλ : V → → Vλ is the orthogonal projection. For λ ∈ P+ ∩ −P+ and u, v ∈ W , define the generalized minor ∆uλ

vλ ∈ k[G] by

g →

∆V (λ),λ
u−1gv
if λ ∈ P+

∆V (λ),λ

u−1gv
if λ ∈ −P+

where V (λ) is the irreducible highest (or lowest) weight representation w = si1 · · · sik and w = si1 · · · sik for w = si1 · · · sik reduced decomp.

Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 9 / 13

SLIDE 49

Quivers and Generalized Minors Generalized Minors

Definition (Generalized Minors)

Let V =

µ∈P Vµ be a weight representation with dimkVλ = 1. Define

the principal generalized minor ∆V ,λ ∈ k[G] by g → πλ(gvλ)/vλ, where vλ ∈ Vλ is any nonzero vector; πλ : V → → Vλ is the orthogonal projection. For λ ∈ P+ ∩ −P+ and u, v ∈ W , define the generalized minor ∆uλ

vλ ∈ k[G] by

g →

∆V (λ),λ
u−1gv
if λ ∈ P+

∆V (λ),λ

u−1gv
if λ ∈ −P+

where V (λ) is the irreducible highest (or lowest) weight representation w = si1 · · · sik and w = si1 · · · sik for w = si1 · · · sik reduced decomp.

Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 9 / 13

SLIDE 50

Quivers and Generalized Minors Generalized Minors

Definition (Generalized Minors)

Let V =

µ∈P Vµ be a weight representation with dimkVλ = 1. Define

the principal generalized minor ∆V ,λ ∈ k[G] by g → πλ(gvλ)/vλ, where vλ ∈ Vλ is any nonzero vector; πλ : V → → Vλ is the orthogonal projection. For λ ∈ P+ ∩ −P+ and u, v ∈ W , define the generalized minor ∆uλ

vλ ∈ k[G] by

g →

∆V (λ),λ
u−1gv
if λ ∈ P+

∆V (λ),λ

u−1gv
if λ ∈ −P+

where V (λ) is the irreducible highest (or lowest) weight representation w = si1 · · · sik and w = si1 · · · sik for w = si1 · · · sik reduced decomp.

Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 9 / 13

SLIDE 51

Quivers and Generalized Minors Generalized Minors

Definition (Generalized Minors)

Let V =

µ∈P Vµ be a weight representation with dimkVλ = 1. Define

the principal generalized minor ∆V ,λ ∈ k[G] by g → πλ(gvλ)/vλ, where vλ ∈ Vλ is any nonzero vector; πλ : V → → Vλ is the orthogonal projection. For λ ∈ P+ ∩ −P+ and u, v ∈ W , define the generalized minor ∆uλ

vλ ∈ k[G] by

g →

∆V (λ),λ
u−1gv
if λ ∈ P+

∆V (λ),λ

u−1gv
if λ ∈ −P+

where V (λ) is the irreducible highest (or lowest) weight representation w = si1 · · · sik and w = si1 · · · sik for w = si1 · · · sik reduced decomp.

Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 9 / 13

SLIDE 52

Quivers and Generalized Minors Generalized Minors

Definition (Generalized Minors)

Let V =

µ∈P Vµ be a weight representation with dimkVλ = 1. Define

the principal generalized minor ∆V ,λ ∈ k[G] by g → πλ(gvλ)/vλ, where vλ ∈ Vλ is any nonzero vector; πλ : V → → Vλ is the orthogonal projection. For λ ∈ P+ ∩ −P+ and u, v ∈ W , define the generalized minor ∆uλ

vλ ∈ k[G] by

g →

∆V (λ),λ
u−1gv
if λ ∈ P+

∆V (λ),λ

u−1gv
if λ ∈ −P+

where V (λ) is the irreducible highest (or lowest) weight representation w = si1 · · · sik and w = si1 · · · sik for w = si1 · · · sik reduced decomp.

Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 9 / 13

SLIDE 53

Quivers and Generalized Minors Generalized Minors

Definition (Generalized Minors)

Let V =

µ∈P Vµ be a weight representation with dimkVλ = 1. Define

the principal generalized minor ∆V ,λ ∈ k[G] by g → πλ(gvλ)/vλ, where vλ ∈ Vλ is any nonzero vector; πλ : V → → Vλ is the orthogonal projection. For λ ∈ P+ ∩ −P+ and u, v ∈ W , define the generalized minor ∆uλ

vλ ∈ k[G] by

g →

∆V (λ),λ
u−1gv
if λ ∈ P+

∆V (λ),λ

u−1gv
if λ ∈ −P+

where V (λ) is the irreducible highest (or lowest) weight representation w = si1 · · · sik and w = si1 · · · sik for w = si1 · · · sik reduced decomp.

Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 9 / 13

SLIDE 54

Quivers and Generalized Minors Generalized Minors

Definition (Generalized Minors)

Let V =

µ∈P Vµ be a weight representation with dimkVλ = 1. Define

the principal generalized minor ∆V ,λ ∈ k[G] by g → πλ(gvλ)/vλ, where vλ ∈ Vλ is any nonzero vector; πλ : V → → Vλ is the orthogonal projection. For λ ∈ P+ ∩ −P+ and u, v ∈ W , define the generalized minor ∆uλ

vλ ∈ k[G] by

g →

∆V (λ),λ
u−1gv
if λ ∈ P+

∆V (λ),λ

u−1gv
if λ ∈ −P+

where V (λ) is the irreducible highest (or lowest) weight representation w = si1 · · · sik and w = si1 · · · sik for w = si1 · · · sik reduced decomp.

Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 9 / 13

SLIDE 55

Quivers and Generalized Minors Generalized Minors

Definition (Generalized Minors)

Let V =

µ∈P Vµ be a weight representation with dimkVλ = 1. Define

the principal generalized minor ∆V ,λ ∈ k[G] by g → πλ(gvλ)/vλ, where vλ ∈ Vλ is any nonzero vector; πλ : V → → Vλ is the orthogonal projection. For λ ∈ P+ ∩ −P+ and u, v ∈ W , define the generalized minor ∆uλ

vλ ∈ k[G] by

g →

∆V (λ),λ
u−1gv
if λ ∈ P+

∆V (λ),λ

u−1gv
if λ ∈ −P+

where V (λ) is the irreducible highest (or lowest) weight representation w = si1 · · · sik and w = si1 · · · sik for w = si1 · · · sik reduced decomp.

Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 9 / 13

SLIDE 56

Quivers and Generalized Minors Generalized Minors

Definition (Generalized Minors)

Let V =

µ∈P Vµ be a weight representation with dimkVλ = 1. Define

the principal generalized minor ∆V ,λ ∈ k[G] by g → πλ(gvλ)/vλ, where vλ ∈ Vλ is any nonzero vector; πλ : V → → Vλ is the orthogonal projection. For λ ∈ P+ ∩ −P+ and u, v ∈ W , define the generalized minor ∆uλ

vλ ∈ k[G] by

g →

∆V (λ),λ
u−1gv
if λ ∈ P+

∆V (λ),λ

u−1gv
if λ ∈ −P+

where V (λ) is the irreducible highest (or lowest) weight representation w = si1 · · · sik and w = si1 · · · sik for w = si1 · · · sik reduced decomp.

Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 9 / 13

SLIDE 57

Quivers and Generalized Minors Cluster Structures on Coexeter Double Bruhat Cells

σ - permutation so that bσi,σj ≥ 0 for 1 ≤ i < j ≤ n

Theorem (Berenstein-Fomin-Zelevinsky, Williams, R-Stella-Williams)

For c = sσ1 · · · sσn, we have k[G c,c−1] ∼ = A(x, ˜ B), where xi = ∆ωi

ωi,

yi = ∆ωi

cωi

j<i
∆ωj

cωj

aji, yi = ∆cωi

ωi

j<i
∆cωj

ωj

aji.

Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 10 / 13

SLIDE 58

Quivers and Generalized Minors Cluster Structures on Coexeter Double Bruhat Cells

σ - permutation so that bσi,σj ≥ 0 for 1 ≤ i < j ≤ n

Theorem (Berenstein-Fomin-Zelevinsky, Williams, R-Stella-Williams)

For c = sσ1 · · · sσn, we have k[G c,c−1] ∼ = A(x, ˜ B), where xi = ∆ωi

ωi,

yi = ∆ωi

cωi

j<i
∆ωj

cωj

aji, yi = ∆cωi

ωi

j<i
∆cωj

ωj

aji.

Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 10 / 13

SLIDE 59

Quivers and Generalized Minors Cluster Structures on Coexeter Double Bruhat Cells

σ - permutation so that bσi,σj ≥ 0 for 1 ≤ i < j ≤ n

Theorem (Berenstein-Fomin-Zelevinsky, Williams, R-Stella-Williams)

For c = sσ1 · · · sσn, we have k[G c,c−1] ∼ = A(x, ˜ B), where xi = ∆ωi

ωi,

yi = ∆ωi

cωi

j<i
∆ωj

cωj

aji, yi = ∆cωi

ωi

j<i
∆cωj

ωj

aji.

Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 10 / 13

SLIDE 60

Quivers and Generalized Minors Cluster Structures on Coexeter Double Bruhat Cells

σ - permutation so that bσi,σj ≥ 0 for 1 ≤ i < j ≤ n

Theorem (Berenstein-Fomin-Zelevinsky, Williams, R-Stella-Williams)

For c = sσ1 · · · sσn, we have k[G c,c−1] ∼ = A(x, ˜ B), where xi = ∆ωi

ωi,

yi = ∆ωi

cωi

j<i
∆ωj

cωj

aji, yi = ∆cωi

ωi

j<i
∆cωj

ωj

aji.

Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 10 / 13

SLIDE 61

Quivers and Generalized Minors Cluster Variables are Generalized Minors

Theorem (Yang-Zelevinsky, R-Stella-Williams)

1 All preprojective (resp. postinjective) cluster variables of k[G c,c−1] are

restrictions of principal generalized minors for irreducible highest-weight (resp. lowest-weight) representations.

2 If Q is an acyclic orientation of an n-cycle (i.e. G is the universal

central extension LSLn of the loop group of SLn), then all regular cluster variables of k[G c,c−1] are restrictions of principal generalized minors of level zero representations.

Remark

In each case, the g-vector of the cluster variable determines a

ne-dimensional weight space giving the principal generalized minor.

Conjecture

Part 2 holds for any quiver.

Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 11 / 13

SLIDE 62

Quivers and Generalized Minors Cluster Variables are Generalized Minors

Theorem (Yang-Zelevinsky, R-Stella-Williams)

1 All preprojective (resp. postinjective) cluster variables of k[G c,c−1] are

restrictions of principal generalized minors for irreducible highest-weight (resp. lowest-weight) representations.

2 If Q is an acyclic orientation of an n-cycle (i.e. G is the universal

central extension LSLn of the loop group of SLn), then all regular cluster variables of k[G c,c−1] are restrictions of principal generalized minors of level zero representations.

Remark

In each case, the g-vector of the cluster variable determines a

ne-dimensional weight space giving the principal generalized minor.

Conjecture

Part 2 holds for any quiver.

Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 11 / 13

SLIDE 63

Quivers and Generalized Minors Cluster Variables are Generalized Minors

Theorem (Yang-Zelevinsky, R-Stella-Williams)

1 All preprojective (resp. postinjective) cluster variables of k[G c,c−1] are

restrictions of principal generalized minors for irreducible highest-weight (resp. lowest-weight) representations.

2 If Q is an acyclic orientation of an n-cycle (i.e. G is the universal

central extension LSLn of the loop group of SLn), then all regular cluster variables of k[G c,c−1] are restrictions of principal generalized minors of level zero representations.

Remark

In each case, the g-vector of the cluster variable determines a

ne-dimensional weight space giving the principal generalized minor.

Conjecture

Part 2 holds for any quiver.

Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 11 / 13

SLIDE 64

Quivers and Generalized Minors Cluster Variables are Generalized Minors

Theorem (Yang-Zelevinsky, R-Stella-Williams)

1 All preprojective (resp. postinjective) cluster variables of k[G c,c−1] are

restrictions of principal generalized minors for irreducible highest-weight (resp. lowest-weight) representations.

2 If Q is an acyclic orientation of an n-cycle (i.e. G is the universal

central extension LSLn of the loop group of SLn), then all regular cluster variables of k[G c,c−1] are restrictions of principal generalized minors of level zero representations.

Remark

In each case, the g-vector of the cluster variable determines a

ne-dimensional weight space giving the principal generalized minor.

Conjecture

Part 2 holds for any quiver.

Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 11 / 13

SLIDE 65

Quivers and Generalized Minors Cluster Variables are Generalized Minors

Theorem (Yang-Zelevinsky, R-Stella-Williams)

1 All preprojective (resp. postinjective) cluster variables of k[G c,c−1] are

restrictions of principal generalized minors for irreducible highest-weight (resp. lowest-weight) representations.

2 If Q is an acyclic orientation of an n-cycle (i.e. G is the universal

central extension LSLn of the loop group of SLn), then all regular cluster variables of k[G c,c−1] are restrictions of principal generalized minors of level zero representations.

Remark

In each case, the g-vector of the cluster variable determines a

ne-dimensional weight space giving the principal generalized minor.

Conjecture

Part 2 holds for any quiver.

Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 11 / 13

SLIDE 66

Quivers and Generalized Minors Idea of Proof

Idea of Proof for Part 1

For u, v ∈ W with ℓ(u) < ℓ(usi) and ℓ(v) < ℓ(vsi), we have the generalized Jacobi-Desnanot identity: ∆uωi

vωi∆usiωi vsiωi =

1≤j≤n

j=i

(∆uωj

vωj)−aji + ∆uωi vsiωi∆usiωi vωi .

These basically give all necessary exchange relations to find the preprojective and postinjective cluster variables.

Remark

From part 2, we obtain new generalized minor identities by translating exchange relations involving regular cluster variables across the isomorphism k[G c,c−1] ∼ = A(x, ˜ B).

Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 12 / 13

SLIDE 67

Quivers and Generalized Minors Idea of Proof

Idea of Proof for Part 1

For u, v ∈ W with ℓ(u) < ℓ(usi) and ℓ(v) < ℓ(vsi), we have the generalized Jacobi-Desnanot identity: ∆uωi

vωi∆usiωi vsiωi =

1≤j≤n

j=i

(∆uωj

vωj)−aji + ∆uωi vsiωi∆usiωi vωi .

These basically give all necessary exchange relations to find the preprojective and postinjective cluster variables.

Remark

From part 2, we obtain new generalized minor identities by translating exchange relations involving regular cluster variables across the isomorphism k[G c,c−1] ∼ = A(x, ˜ B).

Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 12 / 13

SLIDE 68

Quivers and Generalized Minors Idea of Proof

Idea of Proof for Part 1

For u, v ∈ W with ℓ(u) < ℓ(usi) and ℓ(v) < ℓ(vsi), we have the generalized Jacobi-Desnanot identity: ∆uωi

vωi∆usiωi vsiωi =

1≤j≤n

j=i

(∆uωj

vωj)−aji + ∆uωi vsiωi∆usiωi vωi .

These basically give all necessary exchange relations to find the preprojective and postinjective cluster variables.

Remark

From part 2, we obtain new generalized minor identities by translating exchange relations involving regular cluster variables across the isomorphism k[G c,c−1] ∼ = A(x, ˜ B).

Dylan Rupel (ND) Quivers and Generalized Minors October 19, 2016 12 / 13

SLIDE 69

Quivers and Generalized Minors End Big Question