On Kontsevich Automorphisms and Quiver Grassmannians Dylan Rupel - - PowerPoint PPT Presentation

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On Kontsevich Automorphisms and Quiver Grassmannians Dylan Rupel - - PowerPoint PPT Presentation

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Quiver Grass. from Non-Comm. Recursions On Kontsevich Automorphisms and Quiver Grassmannians Dylan Rupel University of Notre Dame November 20, 2017 Conference on Geometric Methods in Representation Theory University of Iowa D. Rupel (ND)

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Quiver Grass. from Non-Comm. Recursions

On Kontsevich Automorphisms and Quiver Grassmannians

Dylan Rupel

University of Notre Dame

November 20, 2017 Conference on Geometric Methods in Representation Theory University of Iowa

  • D. Rupel (ND)

Quiver Grass. from Non-Comm. Recursions November 20, 2017 1 / 12

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Quiver Grass. from Non-Comm. Recursions Polynomial Kontsevich Automorphisms Definition

k – field of characteristic zero K = k(X, Y ) – skew-field of formal rational expressions in non-commuting variables X and Y (Intuitively: W ∈ K is invertible if and only if its commutative specialization is non-zero) P(z) ∈ k[z] – any polynomial FP : K → K – algebra automorphism defined by FP :

  • X → XYX −1

Y → P(Y )X −1

  • D. Rupel (ND)

Quiver Grass. from Non-Comm. Recursions November 20, 2017 2 / 12

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SLIDE 3

Quiver Grass. from Non-Comm. Recursions Polynomial Kontsevich Automorphisms Definition

k – field of characteristic zero K = k(X, Y ) – skew-field of formal rational expressions in non-commuting variables X and Y (Intuitively: W ∈ K is invertible if and only if its commutative specialization is non-zero) P(z) ∈ k[z] – any polynomial FP : K → K – algebra automorphism defined by FP :

  • X → XYX −1

Y → P(Y )X −1

  • D. Rupel (ND)

Quiver Grass. from Non-Comm. Recursions November 20, 2017 2 / 12

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Quiver Grass. from Non-Comm. Recursions Polynomial Kontsevich Automorphisms Definition

k – field of characteristic zero K = k(X, Y ) – skew-field of formal rational expressions in non-commuting variables X and Y (Intuitively: W ∈ K is invertible if and only if its commutative specialization is non-zero) P(z) ∈ k[z] – any polynomial FP : K → K – algebra automorphism defined by FP :

  • X → XYX −1

Y → P(Y )X −1

  • D. Rupel (ND)

Quiver Grass. from Non-Comm. Recursions November 20, 2017 2 / 12

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SLIDE 5

Quiver Grass. from Non-Comm. Recursions Polynomial Kontsevich Automorphisms Definition

k – field of characteristic zero K = k(X, Y ) – skew-field of formal rational expressions in non-commuting variables X and Y (Intuitively: W ∈ K is invertible if and only if its commutative specialization is non-zero) P(z) ∈ k[z] – any polynomial FP : K → K – algebra automorphism defined by FP :

  • X → XYX −1

Y → P(Y )X −1

  • D. Rupel (ND)

Quiver Grass. from Non-Comm. Recursions November 20, 2017 2 / 12

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SLIDE 6

Quiver Grass. from Non-Comm. Recursions Polynomial Kontsevich Automorphisms Definition

k – field of characteristic zero K = k(X, Y ) – skew-field of formal rational expressions in non-commuting variables X and Y (Intuitively: W ∈ K is invertible if and only if its commutative specialization is non-zero) P(z) ∈ k[z] – any polynomial FP : K → K – algebra automorphism defined by FP :

  • X → XYX −1

Y → P(Y )X −1

  • D. Rupel (ND)

Quiver Grass. from Non-Comm. Recursions November 20, 2017 2 / 12

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Quiver Grass. from Non-Comm. Recursions Polynomial Kontsevich Automorphisms Setup

Let P1, P2 ∈ k[z] be monic polynomials with Pi(0) = 1, say P1(z) = p1,0 + p1,1z + · · · + p1,d1−1zd1−1 + p1,d1zd1 P2(z) = p2,0 + p2,1z + · · · + p2,d2−1zd2−1 + p2,d2zd2 with p1,0 = p1,d1 = p2,0 = p2,d2 = 1. Take p1,i = 0 = p2,j for i, j < 0, i > d1, j > d2. Set A+ = Z≥0[p1,i, p2,j : 0 < i < d1, 0 < j < d2] and call this the pseudo-positive semiring associated to P1 and P2. For k ∈ Z, define Pk(z) =            zd2P2(z−1) if k ≡ 0 mod 4 P1(z) if k ≡ 1 mod 4 P2(z) if k ≡ 2 mod 4 zd1P1(z−1) if k ≡ 3 mod 4

  • D. Rupel (ND)

Quiver Grass. from Non-Comm. Recursions November 20, 2017 3 / 12

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Quiver Grass. from Non-Comm. Recursions Polynomial Kontsevich Automorphisms Setup

Let P1, P2 ∈ k[z] be monic polynomials with Pi(0) = 1, say P1(z) = p1,0 + p1,1z + · · · + p1,d1−1zd1−1 + p1,d1zd1 P2(z) = p2,0 + p2,1z + · · · + p2,d2−1zd2−1 + p2,d2zd2 with p1,0 = p1,d1 = p2,0 = p2,d2 = 1. Take p1,i = 0 = p2,j for i, j < 0, i > d1, j > d2. Set A+ = Z≥0[p1,i, p2,j : 0 < i < d1, 0 < j < d2] and call this the pseudo-positive semiring associated to P1 and P2. For k ∈ Z, define Pk(z) =            zd2P2(z−1) if k ≡ 0 mod 4 P1(z) if k ≡ 1 mod 4 P2(z) if k ≡ 2 mod 4 zd1P1(z−1) if k ≡ 3 mod 4

  • D. Rupel (ND)

Quiver Grass. from Non-Comm. Recursions November 20, 2017 3 / 12

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Quiver Grass. from Non-Comm. Recursions Polynomial Kontsevich Automorphisms Setup

Let P1, P2 ∈ k[z] be monic polynomials with Pi(0) = 1, say P1(z) = p1,0 + p1,1z + · · · + p1,d1−1zd1−1 + p1,d1zd1 P2(z) = p2,0 + p2,1z + · · · + p2,d2−1zd2−1 + p2,d2zd2 with p1,0 = p1,d1 = p2,0 = p2,d2 = 1. Take p1,i = 0 = p2,j for i, j < 0, i > d1, j > d2. Set A+ = Z≥0[p1,i, p2,j : 0 < i < d1, 0 < j < d2] and call this the pseudo-positive semiring associated to P1 and P2. For k ∈ Z, define Pk(z) =            zd2P2(z−1) if k ≡ 0 mod 4 P1(z) if k ≡ 1 mod 4 P2(z) if k ≡ 2 mod 4 zd1P1(z−1) if k ≡ 3 mod 4

  • D. Rupel (ND)

Quiver Grass. from Non-Comm. Recursions November 20, 2017 3 / 12

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Quiver Grass. from Non-Comm. Recursions Polynomial Kontsevich Automorphisms Setup

Let P1, P2 ∈ k[z] be monic polynomials with Pi(0) = 1, say P1(z) = p1,0 + p1,1z + · · · + p1,d1−1zd1−1 + p1,d1zd1 P2(z) = p2,0 + p2,1z + · · · + p2,d2−1zd2−1 + p2,d2zd2 with p1,0 = p1,d1 = p2,0 = p2,d2 = 1. Take p1,i = 0 = p2,j for i, j < 0, i > d1, j > d2. Set A+ = Z≥0[p1,i, p2,j : 0 < i < d1, 0 < j < d2] and call this the pseudo-positive semiring associated to P1 and P2. For k ∈ Z, define Pk(z) =            zd2P2(z−1) if k ≡ 0 mod 4 P1(z) if k ≡ 1 mod 4 P2(z) if k ≡ 2 mod 4 zd1P1(z−1) if k ≡ 3 mod 4

  • D. Rupel (ND)

Quiver Grass. from Non-Comm. Recursions November 20, 2017 3 / 12

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Quiver Grass. from Non-Comm. Recursions Polynomial Kontsevich Automorphisms Main Theorem

Theorem (R. 2017)

For k ≥ 1, the elements Xk := FP1FP2 · · · FPk(X) and Yk := FP1FP2 · · · FPk(Y ) are pseudo-positive non-commutative Laurent polynomials in X and Y , i.e. are contained in A+X ±1, Y ±1 ⊂ K. Prior results: Usnich 2009: Laurentness when Pk(z) = 1 + z2 Di Francesco-Kedem 2009: Laurentness and positivity when P1(z) = 1 + zd1 and P2(z) = 1 + zd2 with d1d2 = 4 Usnich 2010: Laurentness when Pk(z) is independent of k Berenstein-Retakh 2010: Laurentness when P1(z) = 1 + zd1 and P2(z) = 1 + zd2 Lee-Schiffler 2011: Laurentness and positivity when Pk(z) = 1 + zd

  • R. 2012: Laurentness and positivity when P1(z) = 1 + zd1 and

P2(z) = 1 + zd2

  • D. Rupel (ND)

Quiver Grass. from Non-Comm. Recursions November 20, 2017 4 / 12

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Quiver Grass. from Non-Comm. Recursions Polynomial Kontsevich Automorphisms Main Theorem

Theorem (R. 2017)

For k ≥ 1, the elements Xk := FP1FP2 · · · FPk(X) and Yk := FP1FP2 · · · FPk(Y ) are pseudo-positive non-commutative Laurent polynomials in X and Y , i.e. are contained in A+X ±1, Y ±1 ⊂ K. Prior results: Usnich 2009: Laurentness when Pk(z) = 1 + z2 Di Francesco-Kedem 2009: Laurentness and positivity when P1(z) = 1 + zd1 and P2(z) = 1 + zd2 with d1d2 = 4 Usnich 2010: Laurentness when Pk(z) is independent of k Berenstein-Retakh 2010: Laurentness when P1(z) = 1 + zd1 and P2(z) = 1 + zd2 Lee-Schiffler 2011: Laurentness and positivity when Pk(z) = 1 + zd

  • R. 2012: Laurentness and positivity when P1(z) = 1 + zd1 and

P2(z) = 1 + zd2

  • D. Rupel (ND)

Quiver Grass. from Non-Comm. Recursions November 20, 2017 4 / 12

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Quiver Grass. from Non-Comm. Recursions Polynomial Kontsevich Automorphisms Main Theorem

Theorem (R. 2017)

For k ≥ 1, the elements Xk := FP1FP2 · · · FPk(X) and Yk := FP1FP2 · · · FPk(Y ) are pseudo-positive non-commutative Laurent polynomials in X and Y , i.e. are contained in A+X ±1, Y ±1 ⊂ K. Prior results: Usnich 2009: Laurentness when Pk(z) = 1 + z2 Di Francesco-Kedem 2009: Laurentness and positivity when P1(z) = 1 + zd1 and P2(z) = 1 + zd2 with d1d2 = 4 Usnich 2010: Laurentness when Pk(z) is independent of k Berenstein-Retakh 2010: Laurentness when P1(z) = 1 + zd1 and P2(z) = 1 + zd2 Lee-Schiffler 2011: Laurentness and positivity when Pk(z) = 1 + zd

  • R. 2012: Laurentness and positivity when P1(z) = 1 + zd1 and

P2(z) = 1 + zd2

  • D. Rupel (ND)

Quiver Grass. from Non-Comm. Recursions November 20, 2017 4 / 12

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SLIDE 14

Quiver Grass. from Non-Comm. Recursions Polynomial Kontsevich Automorphisms Main Theorem

Theorem (R. 2017)

For k ≥ 1, the elements Xk := FP1FP2 · · · FPk(X) and Yk := FP1FP2 · · · FPk(Y ) are pseudo-positive non-commutative Laurent polynomials in X and Y , i.e. are contained in A+X ±1, Y ±1 ⊂ K. Prior results: Usnich 2009: Laurentness when Pk(z) = 1 + z2 Di Francesco-Kedem 2009: Laurentness and positivity when P1(z) = 1 + zd1 and P2(z) = 1 + zd2 with d1d2 = 4 Usnich 2010: Laurentness when Pk(z) is independent of k Berenstein-Retakh 2010: Laurentness when P1(z) = 1 + zd1 and P2(z) = 1 + zd2 Lee-Schiffler 2011: Laurentness and positivity when Pk(z) = 1 + zd

  • R. 2012: Laurentness and positivity when P1(z) = 1 + zd1 and

P2(z) = 1 + zd2

  • D. Rupel (ND)

Quiver Grass. from Non-Comm. Recursions November 20, 2017 4 / 12

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SLIDE 15

Quiver Grass. from Non-Comm. Recursions Polynomial Kontsevich Automorphisms Main Theorem

Theorem (R. 2017)

For k ≥ 1, the elements Xk := FP1FP2 · · · FPk(X) and Yk := FP1FP2 · · · FPk(Y ) are pseudo-positive non-commutative Laurent polynomials in X and Y , i.e. are contained in A+X ±1, Y ±1 ⊂ K. Prior results: Usnich 2009: Laurentness when Pk(z) = 1 + z2 Di Francesco-Kedem 2009: Laurentness and positivity when P1(z) = 1 + zd1 and P2(z) = 1 + zd2 with d1d2 = 4 Usnich 2010: Laurentness when Pk(z) is independent of k Berenstein-Retakh 2010: Laurentness when P1(z) = 1 + zd1 and P2(z) = 1 + zd2 Lee-Schiffler 2011: Laurentness and positivity when Pk(z) = 1 + zd

  • R. 2012: Laurentness and positivity when P1(z) = 1 + zd1 and

P2(z) = 1 + zd2

  • D. Rupel (ND)

Quiver Grass. from Non-Comm. Recursions November 20, 2017 4 / 12

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SLIDE 16

Quiver Grass. from Non-Comm. Recursions Polynomial Kontsevich Automorphisms Main Theorem

Theorem (R. 2017)

For k ≥ 1, the elements Xk := FP1FP2 · · · FPk(X) and Yk := FP1FP2 · · · FPk(Y ) are pseudo-positive non-commutative Laurent polynomials in X and Y , i.e. are contained in A+X ±1, Y ±1 ⊂ K. Prior results: Usnich 2009: Laurentness when Pk(z) = 1 + z2 Di Francesco-Kedem 2009: Laurentness and positivity when P1(z) = 1 + zd1 and P2(z) = 1 + zd2 with d1d2 = 4 Usnich 2010: Laurentness when Pk(z) is independent of k Berenstein-Retakh 2010: Laurentness when P1(z) = 1 + zd1 and P2(z) = 1 + zd2 Lee-Schiffler 2011: Laurentness and positivity when Pk(z) = 1 + zd

  • R. 2012: Laurentness and positivity when P1(z) = 1 + zd1 and

P2(z) = 1 + zd2

  • D. Rupel (ND)

Quiver Grass. from Non-Comm. Recursions November 20, 2017 4 / 12

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SLIDE 17

Quiver Grass. from Non-Comm. Recursions Polynomial Kontsevich Automorphisms Main Theorem

Theorem (R. 2017)

For k ≥ 1, the elements Xk := FP1FP2 · · · FPk(X) and Yk := FP1FP2 · · · FPk(Y ) are pseudo-positive non-commutative Laurent polynomials in X and Y , i.e. are contained in A+X ±1, Y ±1 ⊂ K. Prior results: Usnich 2009: Laurentness when Pk(z) = 1 + z2 Di Francesco-Kedem 2009: Laurentness and positivity when P1(z) = 1 + zd1 and P2(z) = 1 + zd2 with d1d2 = 4 Usnich 2010: Laurentness when Pk(z) is independent of k Berenstein-Retakh 2010: Laurentness when P1(z) = 1 + zd1 and P2(z) = 1 + zd2 Lee-Schiffler 2011: Laurentness and positivity when Pk(z) = 1 + zd

  • R. 2012: Laurentness and positivity when P1(z) = 1 + zd1 and

P2(z) = 1 + zd2

  • D. Rupel (ND)

Quiver Grass. from Non-Comm. Recursions November 20, 2017 4 / 12

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SLIDE 18

Quiver Grass. from Non-Comm. Recursions Polynomial Kontsevich Automorphisms Main Theorem

Theorem (R. 2017)

For k ≥ 1, the elements Xk := FP1FP2 · · · FPk(X) and Yk := FP1FP2 · · · FPk(Y ) are pseudo-positive non-commutative Laurent polynomials in X and Y , i.e. are contained in A+X ±1, Y ±1 ⊂ K. Prior results: Usnich 2009: Laurentness when Pk(z) = 1 + z2 Di Francesco-Kedem 2009: Laurentness and positivity when P1(z) = 1 + zd1 and P2(z) = 1 + zd2 with d1d2 = 4 Usnich 2010: Laurentness when Pk(z) is independent of k Berenstein-Retakh 2010: Laurentness when P1(z) = 1 + zd1 and P2(z) = 1 + zd2 Lee-Schiffler 2011: Laurentness and positivity when Pk(z) = 1 + zd

  • R. 2012: Laurentness and positivity when P1(z) = 1 + zd1 and

P2(z) = 1 + zd2

  • D. Rupel (ND)

Quiver Grass. from Non-Comm. Recursions November 20, 2017 4 / 12

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SLIDE 19

Quiver Grass. from Non-Comm. Recursions Proof - Graded Compatible Pairs Definition

For a = (a1, a2) ∈ Z2

≥0 with a = (0, 0), write D = Da for the maximal

Dyck path in the lattice rectangle [0, a1] × [0, a2]. E = {1, 2, . . . , a1 + a2} – set of edges of D H – set of horizontal edges of D V – set of vertical edges of D ee′ – subpath of D beginning with e and ending with e′

Definition

An edge grading ω : E → Z is compatible if for every h ∈ H and v ∈ V with h < v there exists e ∈ hv such that one of the following holds: e = v and |he ∩ V | =

  • h′∈he∩H

ω(h′)

  • r

e = h and |ev ∩ H| =

  • v′∈ev∩V

ω(v′).

  • D. Rupel (ND)

Quiver Grass. from Non-Comm. Recursions November 20, 2017 5 / 12

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SLIDE 20

Quiver Grass. from Non-Comm. Recursions Proof - Graded Compatible Pairs Definition

For a = (a1, a2) ∈ Z2

≥0 with a = (0, 0), write D = Da for the maximal

Dyck path in the lattice rectangle [0, a1] × [0, a2]. E = {1, 2, . . . , a1 + a2} – set of edges of D H – set of horizontal edges of D V – set of vertical edges of D ee′ – subpath of D beginning with e and ending with e′

Definition

An edge grading ω : E → Z is compatible if for every h ∈ H and v ∈ V with h < v there exists e ∈ hv such that one of the following holds: e = v and |he ∩ V | =

  • h′∈he∩H

ω(h′)

  • r

e = h and |ev ∩ H| =

  • v′∈ev∩V

ω(v′).

  • D. Rupel (ND)

Quiver Grass. from Non-Comm. Recursions November 20, 2017 5 / 12

slide-21
SLIDE 21

Quiver Grass. from Non-Comm. Recursions Proof - Graded Compatible Pairs Definition

For a = (a1, a2) ∈ Z2

≥0 with a = (0, 0), write D = Da for the maximal

Dyck path in the lattice rectangle [0, a1] × [0, a2]. E = {1, 2, . . . , a1 + a2} – set of edges of D H – set of horizontal edges of D V – set of vertical edges of D ee′ – subpath of D beginning with e and ending with e′

Definition

An edge grading ω : E → Z is compatible if for every h ∈ H and v ∈ V with h < v there exists e ∈ hv such that one of the following holds: e = v and |he ∩ V | =

  • h′∈he∩H

ω(h′)

  • r

e = h and |ev ∩ H| =

  • v′∈ev∩V

ω(v′).

  • D. Rupel (ND)

Quiver Grass. from Non-Comm. Recursions November 20, 2017 5 / 12

slide-22
SLIDE 22

Quiver Grass. from Non-Comm. Recursions Proof - Graded Compatible Pairs Definition

For a = (a1, a2) ∈ Z2

≥0 with a = (0, 0), write D = Da for the maximal

Dyck path in the lattice rectangle [0, a1] × [0, a2]. E = {1, 2, . . . , a1 + a2} – set of edges of D H – set of horizontal edges of D V – set of vertical edges of D ee′ – subpath of D beginning with e and ending with e′

Definition

An edge grading ω : E → Z is compatible if for every h ∈ H and v ∈ V with h < v there exists e ∈ hv such that one of the following holds: e = v and |he ∩ V | =

  • h′∈he∩H

ω(h′)

  • r

e = h and |ev ∩ H| =

  • v′∈ev∩V

ω(v′).

  • D. Rupel (ND)

Quiver Grass. from Non-Comm. Recursions November 20, 2017 5 / 12

slide-23
SLIDE 23

Quiver Grass. from Non-Comm. Recursions Proof - Graded Compatible Pairs Definition

For a = (a1, a2) ∈ Z2

≥0 with a = (0, 0), write D = Da for the maximal

Dyck path in the lattice rectangle [0, a1] × [0, a2]. E = {1, 2, . . . , a1 + a2} – set of edges of D H – set of horizontal edges of D V – set of vertical edges of D ee′ – subpath of D beginning with e and ending with e′

Definition

An edge grading ω : E → Z is compatible if for every h ∈ H and v ∈ V with h < v there exists e ∈ hv such that one of the following holds: e = v and |he ∩ V | =

  • h′∈he∩H

ω(h′)

  • r

e = h and |ev ∩ H| =

  • v′∈ev∩V

ω(v′).

  • D. Rupel (ND)

Quiver Grass. from Non-Comm. Recursions November 20, 2017 5 / 12

slide-24
SLIDE 24

Quiver Grass. from Non-Comm. Recursions Proof - Graded Compatible Pairs Definition

For a = (a1, a2) ∈ Z2

≥0 with a = (0, 0), write D = Da for the maximal

Dyck path in the lattice rectangle [0, a1] × [0, a2]. E = {1, 2, . . . , a1 + a2} – set of edges of D H – set of horizontal edges of D V – set of vertical edges of D ee′ – subpath of D beginning with e and ending with e′

Definition

An edge grading ω : E → Z is compatible if for every h ∈ H and v ∈ V with h < v there exists e ∈ hv such that one of the following holds: e = v and |he ∩ V | =

  • h′∈he∩H

ω(h′)

  • r

e = h and |ev ∩ H| =

  • v′∈ev∩V

ω(v′).

  • D. Rupel (ND)

Quiver Grass. from Non-Comm. Recursions November 20, 2017 5 / 12

slide-25
SLIDE 25

Quiver Grass. from Non-Comm. Recursions Proof - Graded Compatible Pairs Definition

For a = (a1, a2) ∈ Z2

≥0 with a = (0, 0), write D = Da for the maximal

Dyck path in the lattice rectangle [0, a1] × [0, a2]. E = {1, 2, . . . , a1 + a2} – set of edges of D H – set of horizontal edges of D V – set of vertical edges of D ee′ – subpath of D beginning with e and ending with e′

Definition

An edge grading ω : E → Z is compatible if for every h ∈ H and v ∈ V with h < v there exists e ∈ hv such that one of the following holds: e = v and |he ∩ V | =

  • h′∈he∩H

ω(h′)

  • r

e = h and |ev ∩ H| =

  • v′∈ev∩V

ω(v′).

  • D. Rupel (ND)

Quiver Grass. from Non-Comm. Recursions November 20, 2017 5 / 12

slide-26
SLIDE 26

Quiver Grass. from Non-Comm. Recursions Proof - Graded Compatible Pairs Definition

For a = (a1, a2) ∈ Z2

≥0 with a = (0, 0), write D = Da for the maximal

Dyck path in the lattice rectangle [0, a1] × [0, a2]. E = {1, 2, . . . , a1 + a2} – set of edges of D H – set of horizontal edges of D V – set of vertical edges of D ee′ – subpath of D beginning with e and ending with e′

Definition

An edge grading ω : E → Z is compatible if for every h ∈ H and v ∈ V with h < v there exists e ∈ hv such that one of the following holds: e = v and |he ∩ V | =

  • h′∈he∩H

ω(h′)

  • r

e = h and |ev ∩ H| =

  • v′∈ev∩V

ω(v′).

  • D. Rupel (ND)

Quiver Grass. from Non-Comm. Recursions November 20, 2017 5 / 12

slide-27
SLIDE 27

Quiver Grass. from Non-Comm. Recursions Proof - Graded Compatible Pairs Definition

For a = (a1, a2) ∈ Z2

≥0 with a = (0, 0), write D = Da for the maximal

Dyck path in the lattice rectangle [0, a1] × [0, a2]. E = {1, 2, . . . , a1 + a2} – set of edges of D H – set of horizontal edges of D V – set of vertical edges of D ee′ – subpath of D beginning with e and ending with e′

Definition

An edge grading ω : E → Z is compatible if for every h ∈ H and v ∈ V with h < v there exists e ∈ hv such that one of the following holds: e = v and |he ∩ V | =

  • h′∈he∩H

ω(h′)

  • r

e = h and |ev ∩ H| =

  • v′∈ev∩V

ω(v′).

  • D. Rupel (ND)

Quiver Grass. from Non-Comm. Recursions November 20, 2017 5 / 12

slide-28
SLIDE 28

Quiver Grass. from Non-Comm. Recursions Proof - Graded Compatible Pairs Non-Commutative Weights

For ω : E → Z, define a non-commutative weight wtω(e) ∈ K associated to each edge e of D as follows: wtω(e) =

  • p1,ω(e)Y ω(e)X −1

if e ∈ H p2,d2−ω(e)X ω(e)+1Y −1X −1 if e ∈ V Define YD =

  • ω compatible

YD(ω) for YD(ω) = wtω(1)wtω(2) · · · wtω(a1 + a2) Set a0 = (0, 1), a1 = (−1, 0), a2 = (0, −1), a3 = (1, 0) and define ak ∈ Z2

≥0 for k ∈ Z \ {0, 1, 2, 3} recursively by

ak−1 + ak+1 =

  • d2ak

if k is odd d1ak if k is even

Theorem

For k ≥ 1, Yk = YDak .

  • D. Rupel (ND)

Quiver Grass. from Non-Comm. Recursions November 20, 2017 6 / 12

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SLIDE 29

Quiver Grass. from Non-Comm. Recursions Proof - Graded Compatible Pairs Non-Commutative Weights

For ω : E → Z, define a non-commutative weight wtω(e) ∈ K associated to each edge e of D as follows: wtω(e) =

  • p1,ω(e)Y ω(e)X −1

if e ∈ H p2,d2−ω(e)X ω(e)+1Y −1X −1 if e ∈ V Define YD =

  • ω compatible

YD(ω) for YD(ω) = wtω(1)wtω(2) · · · wtω(a1 + a2) Set a0 = (0, 1), a1 = (−1, 0), a2 = (0, −1), a3 = (1, 0) and define ak ∈ Z2

≥0 for k ∈ Z \ {0, 1, 2, 3} recursively by

ak−1 + ak+1 =

  • d2ak

if k is odd d1ak if k is even

Theorem

For k ≥ 1, Yk = YDak .

  • D. Rupel (ND)

Quiver Grass. from Non-Comm. Recursions November 20, 2017 6 / 12

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SLIDE 30

Quiver Grass. from Non-Comm. Recursions Proof - Graded Compatible Pairs Non-Commutative Weights

For ω : E → Z, define a non-commutative weight wtω(e) ∈ K associated to each edge e of D as follows: wtω(e) =

  • p1,ω(e)Y ω(e)X −1

if e ∈ H p2,d2−ω(e)X ω(e)+1Y −1X −1 if e ∈ V Define YD =

  • ω compatible

YD(ω) for YD(ω) = wtω(1)wtω(2) · · · wtω(a1 + a2) Set a0 = (0, 1), a1 = (−1, 0), a2 = (0, −1), a3 = (1, 0) and define ak ∈ Z2

≥0 for k ∈ Z \ {0, 1, 2, 3} recursively by

ak−1 + ak+1 =

  • d2ak

if k is odd d1ak if k is even

Theorem

For k ≥ 1, Yk = YDak .

  • D. Rupel (ND)

Quiver Grass. from Non-Comm. Recursions November 20, 2017 6 / 12

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SLIDE 31

Quiver Grass. from Non-Comm. Recursions Proof - Graded Compatible Pairs Non-Commutative Weights

For ω : E → Z, define a non-commutative weight wtω(e) ∈ K associated to each edge e of D as follows: wtω(e) =

  • p1,ω(e)Y ω(e)X −1

if e ∈ H p2,d2−ω(e)X ω(e)+1Y −1X −1 if e ∈ V Define YD =

  • ω compatible

YD(ω) for YD(ω) = wtω(1)wtω(2) · · · wtω(a1 + a2) Set a0 = (0, 1), a1 = (−1, 0), a2 = (0, −1), a3 = (1, 0) and define ak ∈ Z2

≥0 for k ∈ Z \ {0, 1, 2, 3} recursively by

ak−1 + ak+1 =

  • d2ak

if k is odd d1ak if k is even

Theorem

For k ≥ 1, Yk = YDak .

  • D. Rupel (ND)

Quiver Grass. from Non-Comm. Recursions November 20, 2017 6 / 12

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SLIDE 32

Quiver Grass. from Non-Comm. Recursions Proof - Graded Compatible Pairs Non-Commutative Weights

For ω : E → Z, define a non-commutative weight wtω(e) ∈ K associated to each edge e of D as follows: wtω(e) =

  • p1,ω(e)Y ω(e)X −1

if e ∈ H p2,d2−ω(e)X ω(e)+1Y −1X −1 if e ∈ V Define YD =

  • ω compatible

YD(ω) for YD(ω) = wtω(1)wtω(2) · · · wtω(a1 + a2) Set a0 = (0, 1), a1 = (−1, 0), a2 = (0, −1), a3 = (1, 0) and define ak ∈ Z2

≥0 for k ∈ Z \ {0, 1, 2, 3} recursively by

ak−1 + ak+1 =

  • d2ak

if k is odd d1ak if k is even

Theorem

For k ≥ 1, Yk = YDak .

  • D. Rupel (ND)

Quiver Grass. from Non-Comm. Recursions November 20, 2017 6 / 12

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SLIDE 33

Quiver Grass. from Non-Comm. Recursions Proof - Graded Compatible Pairs Non-Commutative Weights

For ω : E → Z, define a non-commutative weight wtω(e) ∈ K associated to each edge e of D as follows: wtω(e) =

  • p1,ω(e)Y ω(e)X −1

if e ∈ H p2,d2−ω(e)X ω(e)+1Y −1X −1 if e ∈ V Define YD =

  • ω compatible

YD(ω) for YD(ω) = wtω(1)wtω(2) · · · wtω(a1 + a2) Set a0 = (0, 1), a1 = (−1, 0), a2 = (0, −1), a3 = (1, 0) and define ak ∈ Z2

≥0 for k ∈ Z \ {0, 1, 2, 3} recursively by

ak−1 + ak+1 =

  • d2ak

if k is odd d1ak if k is even

Theorem

For k ≥ 1, Yk = YDak .

  • D. Rupel (ND)

Quiver Grass. from Non-Comm. Recursions November 20, 2017 6 / 12

slide-34
SLIDE 34

Quiver Grass. from Non-Comm. Recursions Proof - Graded Compatible Pairs Non-Commutative Weights

For ω : E → Z, define a non-commutative weight wtω(e) ∈ K associated to each edge e of D as follows: wtω(e) =

  • p1,ω(e)Y ω(e)X −1

if e ∈ H p2,d2−ω(e)X ω(e)+1Y −1X −1 if e ∈ V Define YD =

  • ω compatible

YD(ω) for YD(ω) = wtω(1)wtω(2) · · · wtω(a1 + a2) Set a0 = (0, 1), a1 = (−1, 0), a2 = (0, −1), a3 = (1, 0) and define ak ∈ Z2

≥0 for k ∈ Z \ {0, 1, 2, 3} recursively by

ak−1 + ak+1 =

  • d2ak

if k is odd d1ak if k is even

Theorem

For k ≥ 1, Yk = YDak .

  • D. Rupel (ND)

Quiver Grass. from Non-Comm. Recursions November 20, 2017 6 / 12

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SLIDE 35

Quiver Grass. from Non-Comm. Recursions Important Consequence Rank Two Valued Quiver Grassmannians

Theorem (R. 2017)

The number of points in the quiver Grassmannian Gre1,e2(Pk) is given by |Gre1,e2(Pk)| =

  • ω:E→Z

compatible

qγω, where ω(H) ⊂ {0, d1} and ω(V ) ⊂ {0, d2} |supp(ω|V )| = e2 and |supp(ω|H)| = ak,1 − e1 with γω =

e<e′ γω(e, e′) for

γω(e, e′) =            −d1d2 if e ∈ supp(ω|H) and e′ ∈ supp(ω|V ) d1 if e ∈ supp(ω|H) and e′ ∈ H \ supp(ω|H) d2 if e ∈ V \ supp(ω|V ) and e′ ∈ supp(ω|V )

  • therwise
  • D. Rupel (ND)

Quiver Grass. from Non-Comm. Recursions November 20, 2017 7 / 12

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SLIDE 36

Quiver Grass. from Non-Comm. Recursions Important Consequence Rank Two Valued Quiver Grassmannians

Theorem (R. 2017)

The number of points in the quiver Grassmannian Gre1,e2(Pk) is given by |Gre1,e2(Pk)| =

  • ω:E→Z

compatible

qγω, where ω(H) ⊂ {0, d1} and ω(V ) ⊂ {0, d2} |supp(ω|V )| = e2 and |supp(ω|H)| = ak,1 − e1 with γω =

e<e′ γω(e, e′) for

γω(e, e′) =            −d1d2 if e ∈ supp(ω|H) and e′ ∈ supp(ω|V ) d1 if e ∈ supp(ω|H) and e′ ∈ H \ supp(ω|H) d2 if e ∈ V \ supp(ω|V ) and e′ ∈ supp(ω|V )

  • therwise
  • D. Rupel (ND)

Quiver Grass. from Non-Comm. Recursions November 20, 2017 7 / 12

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SLIDE 37

Quiver Grass. from Non-Comm. Recursions Important Consequence Rank Two Valued Quiver Grassmannians

Theorem (R. 2017)

The number of points in the quiver Grassmannian Gre1,e2(Pk) is given by |Gre1,e2(Pk)| =

  • ω:E→Z

compatible

qγω, where ω(H) ⊂ {0, d1} and ω(V ) ⊂ {0, d2} |supp(ω|V )| = e2 and |supp(ω|H)| = ak,1 − e1 with γω =

e<e′ γω(e, e′) for

γω(e, e′) =            −d1d2 if e ∈ supp(ω|H) and e′ ∈ supp(ω|V ) d1 if e ∈ supp(ω|H) and e′ ∈ H \ supp(ω|H) d2 if e ∈ V \ supp(ω|V ) and e′ ∈ supp(ω|V )

  • therwise
  • D. Rupel (ND)

Quiver Grass. from Non-Comm. Recursions November 20, 2017 7 / 12

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SLIDE 38

Quiver Grass. from Non-Comm. Recursions Important Consequence Rank Two Valued Quiver Grassmannians

Theorem (R. 2017)

The number of points in the quiver Grassmannian Gre1,e2(Pk) is given by |Gre1,e2(Pk)| =

  • ω:E→Z

compatible

qγω, where ω(H) ⊂ {0, d1} and ω(V ) ⊂ {0, d2} |supp(ω|V )| = e2 and |supp(ω|H)| = ak,1 − e1 with γω =

e<e′ γω(e, e′) for

γω(e, e′) =            −d1d2 if e ∈ supp(ω|H) and e′ ∈ supp(ω|V ) d1 if e ∈ supp(ω|H) and e′ ∈ H \ supp(ω|H) d2 if e ∈ V \ supp(ω|V ) and e′ ∈ supp(ω|V )

  • therwise
  • D. Rupel (ND)

Quiver Grass. from Non-Comm. Recursions November 20, 2017 7 / 12

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SLIDE 39

Quiver Grass. from Non-Comm. Recursions Important Consequence Rank Two Valued Quiver Grassmannians

Theorem (R. 2017)

The number of points in the quiver Grassmannian Gre1,e2(Pk) is given by |Gre1,e2(Pk)| =

  • ω:E→Z

compatible

qγω, where ω(H) ⊂ {0, d1} and ω(V ) ⊂ {0, d2} |supp(ω|V )| = e2 and |supp(ω|H)| = ak,1 − e1 with γω =

e<e′ γω(e, e′) for

γω(e, e′) =            −d1d2 if e ∈ supp(ω|H) and e′ ∈ supp(ω|V ) d1 if e ∈ supp(ω|H) and e′ ∈ H \ supp(ω|H) d2 if e ∈ V \ supp(ω|V ) and e′ ∈ supp(ω|V )

  • therwise
  • D. Rupel (ND)

Quiver Grass. from Non-Comm. Recursions November 20, 2017 7 / 12

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SLIDE 40

Quiver Grass. from Non-Comm. Recursions Important Consequence Rank Two Valued Quiver Grassmannians

Theorem (R. 2017)

The number of points in the quiver Grassmannian Gre1,e2(Pk) is given by |Gre1,e2(Pk)| =

  • ω:E→Z

compatible

qγω, where ω(H) ⊂ {0, d1} and ω(V ) ⊂ {0, d2} |supp(ω|V )| = e2 and |supp(ω|H)| = ak,1 − e1 with γω =

e<e′ γω(e, e′) for

γω(e, e′) =            −d1d2 if e ∈ supp(ω|H) and e′ ∈ supp(ω|V ) d1 if e ∈ supp(ω|H) and e′ ∈ H \ supp(ω|H) d2 if e ∈ V \ supp(ω|V ) and e′ ∈ supp(ω|V )

  • therwise
  • D. Rupel (ND)

Quiver Grass. from Non-Comm. Recursions November 20, 2017 7 / 12

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SLIDE 41

Quiver Grass. from Non-Comm. Recursions Rank Two Valued Quiver Grassmannians Big Question

Question: Is there a decomposition of Gre1,e2(Pk) into affine cells which explains the existence of these counting polynomials? If so, the cells should be labeled by compatible gradings ω with the dimension of the cell corresponding to ω given by γω.

Theorem (R.-Weist, coming soon)

Each quiver Grassmannian Gre1,e2(Pk) has such a decomposition into affine cells.

  • D. Rupel (ND)

Quiver Grass. from Non-Comm. Recursions November 20, 2017 8 / 12

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SLIDE 42

Quiver Grass. from Non-Comm. Recursions Rank Two Valued Quiver Grassmannians Big Question

Question: Is there a decomposition of Gre1,e2(Pk) into affine cells which explains the existence of these counting polynomials? If so, the cells should be labeled by compatible gradings ω with the dimension of the cell corresponding to ω given by γω.

Theorem (R.-Weist, coming soon)

Each quiver Grassmannian Gre1,e2(Pk) has such a decomposition into affine cells.

  • D. Rupel (ND)

Quiver Grass. from Non-Comm. Recursions November 20, 2017 8 / 12

slide-43
SLIDE 43

Quiver Grass. from Non-Comm. Recursions Rank Two Valued Quiver Grassmannians Big Question

Question: Is there a decomposition of Gre1,e2(Pk) into affine cells which explains the existence of these counting polynomials? If so, the cells should be labeled by compatible gradings ω with the dimension of the cell corresponding to ω given by γω.

Theorem (R.-Weist, coming soon)

Each quiver Grassmannian Gre1,e2(Pk) has such a decomposition into affine cells.

  • D. Rupel (ND)

Quiver Grass. from Non-Comm. Recursions November 20, 2017 8 / 12

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SLIDE 44

Quiver Grass. from Non-Comm. Recursions Rank Two Valued Quiver Grassmannians Quiver Schubert calculus?

Question: Is there a decomposition of Gre1,e2(Pk) into affine cells which explains the existence of these counting polynomials? If so, the cells should be labeled by compatible gradings ω with the dimension of the cell corresponding to ω given by γω.

Theorem (R.-Weist, coming soon)

Each quiver Grassmannian Gre1,e2(Pk) has such a decomposition into affine cells. Schubert-like conditions cutting out these cells?

  • D. Rupel (ND)

Quiver Grass. from Non-Comm. Recursions November 20, 2017 9 / 12

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SLIDE 45

Quiver Grass. from Non-Comm. Recursions Quiver Schubert calculus? Quiver Schubert calculus?

Question: Is there a decomposition of Gre1,e2(Pk) into affine cells which explains the existence of these counting polynomials? If so, the cells should be labeled by compatible gradings ω with the dimension of the cell corresponding to ω given by γω.

Theorem (R.-Weist, coming soon)

Each quiver Grassmannian Gre1,e2(Pk) has such a decomposition into affine cells. Schubert-like conditions cutting out these cells? Combinatorial description of closures of cells?

  • D. Rupel (ND)

Quiver Grass. from Non-Comm. Recursions November 20, 2017 10 / 12

slide-46
SLIDE 46

Quiver Schubert calculus? Quiver Schubert calculus? Quiver Schubert calculus?

Question: Is there a decomposition of Gre1,e2(Pk) into affine cells which explains the existence of these counting polynomials? If so, the cells should be labeled by compatible gradings ω with the dimension of the cell corresponding to ω given by γω.

Theorem (R.-Weist, coming soon)

Each quiver Grassmannian Gre1,e2(Pk) has such a decomposition into affine cells. Schubert-like conditions cutting out these cells? Combinatorial description of closures of cells? Intersection theory?

  • D. Rupel (ND)

Quiver Schubert calculus? November 20, 2017 11 / 12

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SLIDE 47

Quiver Grass. from Non-Comm. Recursions End Thank You

Thank you!

  • D. Rupel (ND)

Quiver Grass. from Non-Comm. Recursions November 20, 2017 12 / 12