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

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

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 F P : K → K – algebra automorphism defined by � X �→ XYX − 1 F P : Y �→ P ( Y ) X − 1 D. Rupel (ND) Quiver Grass. from Non-Comm. Recursions November 20, 2017 2 / 12

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 F P : K → K – algebra automorphism defined by � X �→ XYX − 1 F P : Y �→ P ( Y ) X − 1 D. Rupel (ND) Quiver Grass. from Non-Comm. Recursions November 20, 2017 2 / 12

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 F P : K → K – algebra automorphism defined by � X �→ XYX − 1 F P : Y �→ P ( Y ) X − 1 D. Rupel (ND) Quiver Grass. from Non-Comm. Recursions November 20, 2017 2 / 12

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 F P : K → K – algebra automorphism defined by � X �→ XYX − 1 F P : Y �→ P ( Y ) X − 1 D. Rupel (ND) Quiver Grass. from Non-Comm. Recursions November 20, 2017 2 / 12

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 F P : K → K – algebra automorphism defined by � X �→ XYX − 1 F P : Y �→ P ( Y ) X − 1 D. Rupel (ND) Quiver Grass. from Non-Comm. Recursions November 20, 2017 2 / 12

Quiver Grass. from Non-Comm. Recursions Polynomial Kontsevich Automorphisms Setup Let P 1 , P 2 ∈ k [ z ] be monic polynomials with P i (0) = 1, say P 1 ( z ) = p 1 , 0 + p 1 , 1 z + · · · + p 1 , d 1 − 1 z d 1 − 1 + p 1 , d 1 z d 1 P 2 ( z ) = p 2 , 0 + p 2 , 1 z + · · · + p 2 , d 2 − 1 z d 2 − 1 + p 2 , d 2 z d 2 with p 1 , 0 = p 1 , d 1 = p 2 , 0 = p 2 , d 2 = 1. Take p 1 , i = 0 = p 2 , j for i , j < 0, i > d 1 , j > d 2 . Set A + = Z ≥ 0 [ p 1 , i , p 2 , j : 0 < i < d 1 , 0 < j < d 2 ] and call this the pseudo-positive semiring associated to P 1 and P 2 . For k ∈ Z , define  z d 2 P 2 ( z − 1 ) if k ≡ 0 mod 4     P 1 ( z ) if k ≡ 1 mod 4  P k ( z ) = P 2 ( z ) if k ≡ 2 mod 4    z d 1 P 1 ( z − 1 )  if k ≡ 3 mod 4  D. Rupel (ND) Quiver Grass. from Non-Comm. Recursions November 20, 2017 3 / 12

Quiver Grass. from Non-Comm. Recursions Polynomial Kontsevich Automorphisms Setup Let P 1 , P 2 ∈ k [ z ] be monic polynomials with P i (0) = 1, say P 1 ( z ) = p 1 , 0 + p 1 , 1 z + · · · + p 1 , d 1 − 1 z d 1 − 1 + p 1 , d 1 z d 1 P 2 ( z ) = p 2 , 0 + p 2 , 1 z + · · · + p 2 , d 2 − 1 z d 2 − 1 + p 2 , d 2 z d 2 with p 1 , 0 = p 1 , d 1 = p 2 , 0 = p 2 , d 2 = 1. Take p 1 , i = 0 = p 2 , j for i , j < 0, i > d 1 , j > d 2 . Set A + = Z ≥ 0 [ p 1 , i , p 2 , j : 0 < i < d 1 , 0 < j < d 2 ] and call this the pseudo-positive semiring associated to P 1 and P 2 . For k ∈ Z , define  z d 2 P 2 ( z − 1 ) if k ≡ 0 mod 4     P 1 ( z ) if k ≡ 1 mod 4  P k ( z ) = P 2 ( z ) if k ≡ 2 mod 4    z d 1 P 1 ( z − 1 )  if k ≡ 3 mod 4  D. Rupel (ND) Quiver Grass. from Non-Comm. Recursions November 20, 2017 3 / 12

Quiver Grass. from Non-Comm. Recursions Polynomial Kontsevich Automorphisms Setup Let P 1 , P 2 ∈ k [ z ] be monic polynomials with P i (0) = 1, say P 1 ( z ) = p 1 , 0 + p 1 , 1 z + · · · + p 1 , d 1 − 1 z d 1 − 1 + p 1 , d 1 z d 1 P 2 ( z ) = p 2 , 0 + p 2 , 1 z + · · · + p 2 , d 2 − 1 z d 2 − 1 + p 2 , d 2 z d 2 with p 1 , 0 = p 1 , d 1 = p 2 , 0 = p 2 , d 2 = 1. Take p 1 , i = 0 = p 2 , j for i , j < 0, i > d 1 , j > d 2 . Set A + = Z ≥ 0 [ p 1 , i , p 2 , j : 0 < i < d 1 , 0 < j < d 2 ] and call this the pseudo-positive semiring associated to P 1 and P 2 . For k ∈ Z , define  z d 2 P 2 ( z − 1 ) if k ≡ 0 mod 4     P 1 ( z ) if k ≡ 1 mod 4  P k ( z ) = P 2 ( z ) if k ≡ 2 mod 4    z d 1 P 1 ( z − 1 )  if k ≡ 3 mod 4  D. Rupel (ND) Quiver Grass. from Non-Comm. Recursions November 20, 2017 3 / 12

Quiver Grass. from Non-Comm. Recursions Polynomial Kontsevich Automorphisms Setup Let P 1 , P 2 ∈ k [ z ] be monic polynomials with P i (0) = 1, say P 1 ( z ) = p 1 , 0 + p 1 , 1 z + · · · + p 1 , d 1 − 1 z d 1 − 1 + p 1 , d 1 z d 1 P 2 ( z ) = p 2 , 0 + p 2 , 1 z + · · · + p 2 , d 2 − 1 z d 2 − 1 + p 2 , d 2 z d 2 with p 1 , 0 = p 1 , d 1 = p 2 , 0 = p 2 , d 2 = 1. Take p 1 , i = 0 = p 2 , j for i , j < 0, i > d 1 , j > d 2 . Set A + = Z ≥ 0 [ p 1 , i , p 2 , j : 0 < i < d 1 , 0 < j < d 2 ] and call this the pseudo-positive semiring associated to P 1 and P 2 . For k ∈ Z , define  z d 2 P 2 ( z − 1 ) if k ≡ 0 mod 4     P 1 ( z ) if k ≡ 1 mod 4  P k ( z ) = P 2 ( z ) if k ≡ 2 mod 4    z d 1 P 1 ( z − 1 )  if k ≡ 3 mod 4  D. Rupel (ND) Quiver Grass. from Non-Comm. Recursions November 20, 2017 3 / 12

Quiver Grass. from Non-Comm. Recursions Polynomial Kontsevich Automorphisms Main Theorem Theorem (R. 2017) For k ≥ 1 , the elements X k := F P 1 F P 2 · · · F P k ( X ) Y k := F P 1 F P 2 · · · F P k ( Y ) and 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 P k ( z ) = 1 + z 2 Di Francesco-Kedem 2009: Laurentness and positivity when P 1 ( z ) = 1 + z d 1 and P 2 ( z ) = 1 + z d 2 with d 1 d 2 = 4 Usnich 2010: Laurentness when P k ( z ) is independent of k Berenstein-Retakh 2010: Laurentness when P 1 ( z ) = 1 + z d 1 and P 2 ( z ) = 1 + z d 2 Lee-Schiffler 2011: Laurentness and positivity when P k ( z ) = 1 + z d R. 2012: Laurentness and positivity when P 1 ( z ) = 1 + z d 1 and P 2 ( z ) = 1 + z d 2 D. Rupel (ND) Quiver Grass. from Non-Comm. Recursions November 20, 2017 4 / 12

Quiver Grass. from Non-Comm. Recursions Polynomial Kontsevich Automorphisms Main Theorem Theorem (R. 2017) For k ≥ 1 , the elements X k := F P 1 F P 2 · · · F P k ( X ) Y k := F P 1 F P 2 · · · F P k ( Y ) and 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 P k ( z ) = 1 + z 2 Di Francesco-Kedem 2009: Laurentness and positivity when P 1 ( z ) = 1 + z d 1 and P 2 ( z ) = 1 + z d 2 with d 1 d 2 = 4 Usnich 2010: Laurentness when P k ( z ) is independent of k Berenstein-Retakh 2010: Laurentness when P 1 ( z ) = 1 + z d 1 and P 2 ( z ) = 1 + z d 2 Lee-Schiffler 2011: Laurentness and positivity when P k ( z ) = 1 + z d R. 2012: Laurentness and positivity when P 1 ( z ) = 1 + z d 1 and P 2 ( z ) = 1 + z d 2 D. Rupel (ND) Quiver Grass. from Non-Comm. Recursions November 20, 2017 4 / 12

Quiver Grass. from Non-Comm. Recursions Polynomial Kontsevich Automorphisms Main Theorem Theorem (R. 2017) For k ≥ 1 , the elements X k := F P 1 F P 2 · · · F P k ( X ) Y k := F P 1 F P 2 · · · F P k ( Y ) and 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 P k ( z ) = 1 + z 2 Di Francesco-Kedem 2009: Laurentness and positivity when P 1 ( z ) = 1 + z d 1 and P 2 ( z ) = 1 + z d 2 with d 1 d 2 = 4 Usnich 2010: Laurentness when P k ( z ) is independent of k Berenstein-Retakh 2010: Laurentness when P 1 ( z ) = 1 + z d 1 and P 2 ( z ) = 1 + z d 2 Lee-Schiffler 2011: Laurentness and positivity when P k ( z ) = 1 + z d R. 2012: Laurentness and positivity when P 1 ( z ) = 1 + z d 1 and P 2 ( z ) = 1 + z d 2 D. Rupel (ND) Quiver Grass. from Non-Comm. Recursions November 20, 2017 4 / 12