Stable bases for moduli of sheaves

Transcription

1 Columbia University 02 / 03 / 2015

2 Moduli of sheaves on P 2 Let M v,w be the moduli space of deg v torsion-free sheaves of rank w on P 2, which compactifies the space of instantons

3 Moduli of sheaves on P 2 Let M v,w be the moduli space of deg v torsion-free sheaves of rank w on P 2, which compactifies the space of instantons When w = 1, it is the well-known Hilbert scheme of v points

4 Moduli of sheaves on P 2 Let M v,w be the moduli space of deg v torsion-free sheaves of rank w on P 2, which compactifies the space of instantons When w = 1, it is the well-known Hilbert scheme of v points When v = 1, it is isomorphic to T P w 1 A 2. In general, M v,w is a holomorphic symplectic variety of dimension 2vw.

5 Moduli of sheaves on P 2 Let M v,w be the moduli space of deg v torsion-free sheaves of rank w on P 2, which compactifies the space of instantons When w = 1, it is the well-known Hilbert scheme of v points When v = 1, it is isomorphic to T P w 1 A 2. In general, M v,w is a holomorphic symplectic variety of dimension 2vw. By the ADHM presentation, the space M v,w is isomorphic to the space of quadruples (X, Y, A, B) of linear maps: X, Y : C v C v, A : C w C v, B : C v C w such that [X, Y ] + AB = 0, modulo the action of GL(v).

6 Fixed points and K theory The torus T = C C (C ) w acts on M v,w via: (q, t, U) (X, Y, A, B) q ( Xt, Yt 1, AU 1, UB )

7 Fixed points and K theory The torus T = C C (C ) w acts on M v,w via: (q, t, U) (X, Y, A, B) q ( Xt, Yt 1, AU 1, UB ) The fixed points are indexed by collections of partitions: whose sizes add up to v. λ = (λ 1,..., λ w )

8 Fixed points and K theory The torus T = C C (C ) w acts on M v,w via: (q, t, U) (X, Y, A, B) q ( Xt, Yt 1, AU 1, UB ) The fixed points are indexed by collections of partitions: whose sizes add up to v. λ = (λ 1,..., λ w ) These fixed points form a basis of localized K theory groups: K(w) = v N K v,w, where K v,w = K 0 T (M v,w )

9 Fixed points and K theory The torus T = C C (C ) w acts on M v,w via: (q, t, U) (X, Y, A, B) q ( Xt, Yt 1, AU 1, UB ) The fixed points are indexed by collections of partitions: whose sizes add up to v. λ = (λ 1,..., λ w ) These fixed points form a basis of localized K theory groups: K(w) = v N K v,w, where K v,w = K 0 T (M v,w ) is a module over KT 0 (pt) = Z[q±1, t ±1, u 1 ±1,..., u±1 w ].

10 Fixed points and K theory The torus T = C C (C ) w acts on M v,w via: (q, t, U) (X, Y, A, B) q ( Xt, Yt 1, AU 1, UB ) The fixed points are indexed by collections of partitions: whose sizes add up to v. λ = (λ 1,..., λ w ) These fixed points form a basis of localized K theory groups: K(w) = v N K v,w, where K v,w = K 0 T (M v,w ) is a module over KT 0 (pt) = Z[q±1, t ±1, u 1 ±1,..., u±1 w ]. As a vector space, K(w) is isomorphic to (Fock space) w

11 The U q,t ( gl 1 ) action The last remark suggests that the K(w) are meaningful representations, so we must find algebras which act on them

12 The U q,t ( gl 1 ) action The last remark suggests that the K(w) are meaningful representations, so we must find algebras which act on them For pairs of integers such that v + = v + 1, define: Z v +,v,w = {(F + F )} π + π M v +,w M v,w and the line bundle L (F + F ) = RΓ(P 2, F /F + ) on Z v +,v,w

13 The U q,t ( gl 1 ) action The last remark suggests that the K(w) are meaningful representations, so we must find algebras which act on them For pairs of integers such that v + = v + 1, define: Z v +,v,w = {(F + F )} π + π M v +,w M v,w and the line bundle L (F + F ) = RΓ(P 2, F /F + ) on Z v +,v,w Theorem (Schiffmann-Vasserot, Nakajima): The operators p ±1,m = π ± ( Lm π ) : K,w K ±1,w give rise to an action of U q,t ( gl 1 ) on K(w), for any given w.

14 Subalgebras of U q,t ( gl 1 ) U q,t ( gl 1 ) is generated by symbols p n,m for all m, n Z

15 Subalgebras of U q,t ( gl 1 ) U q,t ( gl 1 ) is generated by symbols p n,m for all m, n Z For any rational number m n, the subalgebra: B m n = p kn,km k Z U q,t ( gl 1 ) is isomorphic to the q Heisenberg algebra U q ( gl 1 ).

16 Subalgebras of U q,t ( gl 1 ) U q,t ( gl 1 ) is generated by symbols p n,m for all m, n Z For any rational number m n, the subalgebra: B m n = p kn,km k Z U q,t ( gl 1 ) is isomorphic to the q Heisenberg algebra U q ( gl 1 ). U q,t ( gl 1 ) is quasitriangular Hopf w.r.t. the coproduct: (p k,0 ) = p k,0 1 + c k p k,0 (p k,1 ) = p k,1 1 + n 0 h n,0 p k n,1

17 Subalgebras of U q,t ( gl 1 ) U q,t ( gl 1 ) is generated by symbols p n,m for all m, n Z For any rational number m n, the subalgebra: B m n = p kn,km k Z U q,t ( gl 1 ) is isomorphic to the q Heisenberg algebra U q ( gl 1 ). U q,t ( gl 1 ) is quasitriangular Hopf w.r.t. the coproduct: (p k,0 ) = p k,0 1 + c k p k,0 (p k,1 ) = p k,1 1 + n 0 h n,0 p k n,1 and with it one associates a universal R matrix: R U q,t ( gl 1 ) U q,t ( gl 1 )

18 R matrices according to Maulik-Okounkov R factors in terms of the R matrices of the subalgebras: ( R = R(B m ) = ) p kn,km p kn, km exp n k m n [0; 0) m n [0; 0) k=1

19 R matrices according to Maulik-Okounkov R factors in terms of the R matrices of the subalgebras: ( R = R(B m ) = ) p kn,km p kn, km exp n k m n [0; 0) m n [0; 0) k=1 In geometry, the above R matrix gives rise to operators: K(w) K(w ) R K(w) K(w )

20 R matrices according to Maulik-Okounkov R factors in terms of the R matrices of the subalgebras: ( R = R(B m ) = ) p kn,km p kn, km exp n k m n [0; 0) m n [0; 0) k=1 In geometry, the above R matrix gives rise to operators: K(w) K(w ) R K(w) K(w ) Maulik-Okounkov have another way to construct these operators, via a construction known as the stable basis

21 R matrices according to Maulik-Okounkov R factors in terms of the R matrices of the subalgebras: ( R = R(B m ) = ) p kn,km p kn, km exp n k m n [0; 0) m n [0; 0) k=1 In geometry, the above R matrix gives rise to operators: K(w) K(w ) R K(w) K(w ) Maulik-Okounkov have another way to construct these operators, via a construction known as the stable basis It is defined for any symplectic resolution T X, w.r.t. σ Lie(T symp ), L Pic(X ) Q

22 R matrices according to Maulik-Okounkov R factors in terms of the R matrices of the subalgebras: ( R = R(B m ) = ) p kn,km p kn, km exp n k m n [0; 0) m n [0; 0) k=1 In geometry, the above R matrix gives rise to operators: K(w) K(w ) R K(w) K(w ) Maulik-Okounkov have another way to construct these operators, via a construction known as the stable basis It is defined for any symplectic resolution T X, w.r.t. σ Lie(T symp ), L Pic(X ) Q as a uniquely determined class supported on the stable leaf: s σ,l λ K T (attracting set of λ), λ X Tsymp

23 The geometric R matrix When X = M v,w, Maulik-Okounkov define their R matrices as the change of stable basis R 0 0 σ σ, where we define: R l l σ σ : K(w) K(w ) Stabl σ K(w +w ) Stabl σ K(w) K(w ) Stab l σ : s σ,l λ sσ,l µ s σ,l λ µ

24 The geometric R matrix When X = M v,w, Maulik-Okounkov define their R matrices as the change of stable basis R 0 0 σ σ, where we define: R l l σ σ : K(w) K(w ) Stabl σ K(w +w ) Stabl σ K(w) K(w ) Stab l σ : s σ,l λ sσ,l µ s σ,l λ µ On general grounds, the above R matrix can be factored as: R 0 0 σ σ = 0 l= R l ε l+ε σ σ l=0 R l ε l+ε σ σ

25 The geometric R matrix When X = M v,w, Maulik-Okounkov define their R matrices as the change of stable basis R 0 0 σ σ, where we define: R l l σ σ : K(w) K(w ) Stabl σ K(w +w ) Stabl σ K(w) K(w ) Stab l σ : s σ,l λ sσ,l µ s σ,l λ µ On general grounds, the above R matrix can be factored as: R 0 0 σ σ = 0 l= R l ε l+ε σ σ l=0 R l ε l+ε σ σ These match the R matrices from the U q,t ( gl 1 ) factorization: R = R 0 0 σ σ, R(B l) = R l ε l+ε σ σ

26 Explicit formulas in the stable basis Fix σ and write l = m n. The above predicts that the map: Stab m n : s m n λ s m n µ s m n λ µ intertwines the U q,t ( gl 1 ) actions.

27 Explicit formulas in the stable basis Fix σ and write l = m n. The above predicts that the map: Stab m n : s m n λ s m n µ s m n λ µ intertwines the U q,t ( gl 1 ) actions. With respect to the above coproduct, the generators p kn,km are primitive. Hence we expect them to act component-wise: p kn,km s m n λ = µ obtained from λ by changing only one constituent partition s m n µ coefficient

28 Explicit formulas in the stable basis Fix σ and write l = m n. The above predicts that the map: Stab m n : s m n λ s m n µ s m n λ µ intertwines the U q,t ( gl 1 ) actions. With respect to the above coproduct, the generators p kn,km are primitive. Hence we expect them to act component-wise: p kn,km s m n λ = µ obtained from λ by changing only one constituent partition s m n µ coefficient In fact, when gcd(m, n) = 1 we have (N): p n,m s m n λ = s m n µ ( 1) ht R χ m (R) µ=λ+added n ribbon R

29 The m n Pieri rule For general k N, let e kn,km be to p kn,km as elementary symmetric functions are to power sum functions.

30 The m n Pieri rule For general k N, let e kn,km be to p kn,km as elementary symmetric functions are to power sum functions. Theorem (N) the m n Pieri rule: for any rational number m n, any k N and any fixed point λ, we have: µ=λ+k added e kn,km s m n ribbons n λ = s m n µ no two next to each other k ( 1) ht R i χ m (R i ) i=1

31 The m n Pieri rule For general k N, let e kn,km be to p kn,km as elementary symmetric functions are to power sum functions. Theorem (N) the m n Pieri rule: for any rational number m n, any k N and any fixed point λ, we have: µ=λ+k added e kn,km s m n ribbons n λ = s m n µ no two next to each other k ( 1) ht R i χ m (R i ) i=1 R 1 R 2 w = 1, n = 5, k = 2 λ = (1) µ = (4, 4, 3)