Macdonald polynomials and Hilbert schemes. Mark Haiman U.C. Berkeley LECTURE

Transcription

1 Macdonald polynomials and Hilbert schemes Mark Haiman U.C. Berkeley LECTURE I Introduction to Hall-Littlewood and Macdonald polynomials, and the n! and (n + 1) (n 1) theorems 1

2 Hall-Littlewood polynomials from geometry A flag is a chain of subspaces F = (0 F 1 F 2 F n 1 C n ). Denote the usual basis of C n by {e 1,..., e n }. The standard flag E is given by E i = e 1,..., e i. Let G = GL n (C) B = {upper triangular matrices} G. Then G acts transitively on X = {flags}, and B is the stabilizer of E, hence X = G/B. 2

3 Fix a partition of n, µ = (µ 1 µ 2 µ l ), and a unipotent matrix g µ G with Jordan block sizes µ i. Example (n = 5): g µ = µ = (3, 2) Let X µ = {F X : g µ X = X} be the set of flags fixed by g µ. X µ is a Springer variety. 3

4 Examples: X (1 n ) = X, X (n) = {E } is a point, X (2,1) = is a union of two P 1 s meeting at E. Definition. R µ = H (X µ, C). Some facts: 1. H i (X µ ) = 0 for i odd; so R µ is a commutative, graded C-algebra. 2. R (1 n ) = H (X) = C[x 1,..., x n ]/(S n -invariants). 3. X µ X induces a surjection R (1 n ) R µ, so R µ = R (1 n ) /I µ. 4. The ideal I µ is S n -invariant, so S n acts on R µ. Problem. Describe the action of S n on the graded ring R µ. 4

5 I ll write the solution in terms of the characteristic map F : S n -characters symmetric functions FV = µ =n of Frobenius. Here V is an S n -module dim(v Sµ ) m µ (z) µ is a partition of n S µ = S µ1 S µl S n is a Young subgroup m µ (z) = (z µ 1 1 zµ l l + symmetric terms) is a monomial symmetric function. Theorem (Frobenius). The characteristic of the irreducible representation V λ is the Schur function FV λ = S λ (z). 5

6 For a graded S n -module V = d V d, define the Frobenius series FV (z; t) = d F(V d ) t d. Example (n = 3): R (1 3 ) R (2,1) R (3) degree 3 V (1 3 ) 2 V (2,1) 1 V (2,1) V (2,1) 0 V (3) V (3) V (3) Therefore FR (1 3 ) = S (3) + (t + t 2 )S (2,1) + t 3 S (1 3 ) FR (2,1) = S (3) + ts (2,1) FR (3) = S (3) 6

7 We work with symmetric functions in infinitely many variables Λ Q(t) (z) = Q(t)[z 1, z 2,...] S = Q(t)[p 1, p 2,...], where p k = m (k) = z1 k +zk 2 + are the Newton power-sums. Define Q(t)-algebra automorphism ε t : Λ Λ ε t (p k ) = (1 t k )p k and introduce the notation f[z(1 t)] def = ε t (f). As motivation, the inverse of ε t is f f(z, tz, t 2 z,...), which we might naturally denote by f [ Z 1 t ]. 7

8 The partial ordering on partitions of n is λ µ λ λ k µ µ k k. The transpose of a partition is, e.g., µ = (3, 2) =, µ = (2, 2, 1) =. Theorem/Definition. The algebra Λ Q(t) has a basis of Hall-Littlewood polynomials H µ (z; t) characterized by (i) H µ (z; t) Q(t){S λ (z) : λ µ}; (ii) H µ [Z(1 t); t] Q(t){S λ (z) : λ µ }; (iii) H λ, S (n) = 1. Theorem (Hotta Springer). FR µ = H µ (z; t). 8

9 Remark: Define the t-kostka coefficients K λµ (t) by H µ (z; t) = λ K λµ (t)s λ (z) The Hotta Springer theorem implies they are non-negative polynomials, K λµ (t) N[t]. A priori we only have K λµ (t) Q(t). Macdonald polynomials Now our symmetric functions will involve two parameters, coefficient ring Q(q, t). Theorem/Definition (Macdonald). The algebra Λ Q(q,t) has a basis of Macdonald polynomials H µ (z; q, t) characterized by (i) H µ [Z(1 q); q, t] Q(t){S λ (z) : λ µ}; (ii) H µ [Z(1 t); q, t] Q(t){S λ (z) : λ µ }; (iii) H µ, S (n) = 1. 9

10 Comparing definitions, we see that H µ (z; 0, t) = H µ (z; t). New definition has more symmetry: H µ (z; q, t) = H µ (z; t, q). Define the q, t-kostka coefficients by H µ (z; q, t) = λ K λµ (q, t)s λ (z). A priori, K λµ (q, t) Q(q, t), but... Integrality Theorem (Garsia Remmel, Garsia Tesler, Knop, Kirillov Noumi, Lapointe, Sahi ca. 1995). K λµ (q, t) Z[q, t]. Positivity Theorem (H 2001). K λµ (q, t) N[q, t]. Macdonald conjectured integrality & positivity in

11 An interpretation of H µ (z; q, t) Recall from rep n theory of S n that the sign representation ε = V (1 n ) occurs in V λ V µ if and only if λ = µ. The top degree in R µ, i.e., dim C (X µ ) is n(µ) = i (i 1)µ i, and we have (R µ ) n(µ) = Vµ V λ occurs in (R µ ) d for d < n(µ) λ > µ. Then R µ R µ contains ε uniquely, in its top bi-degree (n(µ), n(µ )). Definition. R µ (x, y) = R µ R µ /J, where J is the unique largest S n -invariant ideal not containing ε. 11

12 Elementary description let the boxes in the diagram of µ be (p 1, q 1 ),..., (p n, q n ), as shown for µ = (3, 2): (1, 0) (1, 1) (0, 0) (0, 1) (0, 2) In C[x, y] = C[x 1, y 1,..., x n, y n ], define the polynomial µ (x, y) = det and consider the ideal x p 1 1 yq 1 1 x p n 1 y q n 1.. x p 1 n y q 1 n x p n n y q n n J µ = {f C[x, y] : f( x, y ) µ = 0}. Proposition. R µ (x, y) = C[x, y]/j µ. 12

13 Theorem 1. The Frobenius series of R µ (x, y) as a doubly-graded S n -module is FR µ (x, y) = H µ (z; q, t). This implies the Positivity Theorem, since K λµ (q, t) = r,s mult(v λ, R µ (x, y) r,s )t r q s. Example: R (3,1) (x, y). stands for V (2,2), so K (2,2),(3,1) (q, t) = qt + q 2, and so on. x- degree + + y-degree Left column shows Springer ring R µ = R (3,1) ; bottom row is R µ = R (2,1,1). 13

14 Proposition (Macdonald). Let V = CS n be the regular representation. Then H µ (z; 1, 1) = p 1 (z) n = FV, for any µ. Hence Theorem 1 implies R µ (x, y) = CS n as an ungraded S n -module; in particular, dim(r µ (x, y)) = n! for every partition µ of n. For µ = (1 n ), when R (1 n ) (x, y) = R (1 n ) (x) = H (X), this is classical. For general µ, I call it the n! theorem. It is equivalent to the existence of certain rank n! vector bundle on the Hilbert scheme (tomorrow s lecture). 14

15 The (n + 1) n 1 theorem Define the diagonal coinvariant ring R n = C[x, y]/(s n -invariants) = C[x, y]/(c[x, y] S n + ), a bivariate analog of the classical coinvariant ring R (1 n ) = C[x]/(S n-invariants) = H (X, C). The rings R µ in the n! theorem are quotients, R n R µ. Theorem 2. Let be the linear operator on Λ Q(q,t) given by H µ (z; q, t) = t n(µ) q n(µ ) H µ (z; q, t), and let e n (z) be the n-th elementary symmetric function. The Frobenius series of the diagonal coinvariant ring is FR n = e n (z). 15

16 Some remarkable consequences... Corollary 1. We have dim(r n ) = (n + 1) n 1, and R n is isomorphic as an S n -module to ε V, where V is the permutation representation of S n on the finite abelian group (Z/(n + 1)Z) n / (1, 1,..., 1). Corollary 2. Ignoring the y-grading and considering only x-degree, dim(r n ) d, is equal to the number of rooted forests on the vertex set {1,..., n} with d inversions [example: has three inversions: (1, 3), (2, 6), (2, 5)]

17 In Lecture 2, we ll explain these symmetric function formulas by interpreting the rings R µ and R n in terms of the Hilbert scheme of points in the plane. Now consider any Weyl group W, its root lattice Q and defining representation h = Q Z C. Theorem (I. Gordon). The diagonal coinvariant ring R W = O(h h)/(w -invariants) has a natural quotient R W such that dim( R W ) = (h + 1) r, where h is the Coxeter number and r = dim(h) is the rank. Moreover, R W is isomorphic as a W -module to ε V, where V is the permutation representation of W on Q/(h + 1)Q. Example. For W = B 4, dim(r W ) = , but dim( R W ) = 9 4. Gordon s method doesn t explain the fact that R W = R W for W = S n. 17

18 Macdonald polynomials and Hilbert schemes Mark Haiman U.C. Berkeley LECTURE II The connection between Macdonald polynomials and the Hilbert scheme of points in the plane 18

19 Hilbert scheme H n = Hilb n (C 2 ) As a set... H n = {finite subschemes S C 2 of length n} = {ideals I C[x, y] : dim C (C[x, y]/i) = n} As a scheme (in coordinates)... Set M µ = {x p y q : (p, q) µ}, e.g. (n = 5) M (3,2) = x xy 1 y y 2. H n is covered by open affines U µ = {I : M µ spans C[x, y]/i}. Given (r, s) µ, have unique coefficients s.t. x r y s (p,q) µ C rs pq x p y q (mod I). I ideal certain equations in C rs pq s hold. 19

20 As a scheme (functorially)... Have tautological family F H n C 2 π H n with fibers π 1 (I) = Spec(C[x, y]/i). Universal property: any family T Z C 2 Z flat & finite of degree n over Z, is the pullback of F by a unique morphism T F π Z φ H n. (H n represents the functor of such families.) 20

21 More info... Theorem (Fogarty). H n is non-singular, reduced and irreducible of dimension 2n. The Chow morphism H n Sym n σ (C 2 ) σ(s) = P (length O S,P ) P. is projective & birational. Torus T = (C ) 2 acts on C 2 & H n. Explicitly, (t, q) I = I x t 1 x, y q 1 y. T -fixed points of H n are ideals e.g. I µ = (x r y s : (r, s) µ), I (3,2) = (x 2, xy 2, y 3 ) x 2 xy 2 y 3. 21

22 G-Hilbert schemes Let G = finite group acting on C d. The Hilbert scheme of regular G-orbits G -Hilb(C d ) = {G-invariant subschemes S C d such that O(S) = G CG} is a closed subscheme of Hilb G (C d ). Take G = S n acting on (C 2 ) n, with O((C 2 ) n ) = C[x 1, y 1,..., x n, y n ] = C[x, y]. Let J C[x, y], J S n -Hilb(C 2n ). Now x n, y n, x r 1 ys xr n 1 ys n 1 C[x, y] S n 1, and generate (x r 1 ys xr n 1 ys n 1 ) + xr ny s n c (mod J), hence C[x n, y n ] (C[x, y]/j) S n 1 is surjective, with kernel I C[x n, y n ], I H n. 22

23 We now have a morhpism φ: S n -Hilb(C 2n ) H n, which is generically the obvious one: S n (a 1, b 1,..., a n, b n ) {(a 1, b 1 ),..., (a n, b n )}. φ Theorem 1. S n -Hilb(C 2n ) = H n. To prove it, need to construct a family of regular S n orbits over H n, so universal property of S n -Hilb(C 2n ) will give φ 1 : H n S n -Hilb(C 2n ). Consider the reduced fiber product X n C 2n ρ H n σ Sym n (C 2 ) = C 2n /S n. Theorem. X n is Cohen-Macaulay (i.e., ρ is flat) and Gorenstein. 23

24 Proof sketch... let A = (C[x, y] ε ) be the ideal in C[x, y] generated by antisymmetric polynomials. A description of X n as a blowup X n = Proj(C[x, y][ta]), plus a geometric induction on n reduces us to X n 1,n X n H n 1,n H n X n 1 H n 1, Proposition. A d is a free C[x]-module for all d. We prove this by brute force, constructing a basis. 24

25 Tying in Lecture 1 Tautological families X n ρ H n C 2n F H n C 2 π H n H n = S n -Hilb(C 2n ), give tautological vector bundles B = π O F, P = ρ O Xn on H n, with fibers B(I) = C[x, y]/i, where J = φ 1 (I). Recall from Lecture 1 P (I) = C[x, y]/j, R µ (x, y) = C[x, y]/j µ, J µ = {f C[x, y] : f( x, y ) µ = 0}. Proposition. J µ = φ 1 (I µ ), i.e., R µ (x, y) = P (I µ ). Proof: both rings are Gorenstein quotients of C[x, y] with the same socle. 25

26 Now recall Macdonald polynomials (i) H µ [Z(1 q); q, t] Q(t){S λ (z) : λ µ}; (ii) H µ [Z(1 t); q, t] Q(t){S λ (z) : λ µ }; (iii) H µ, S (n) = 1. Proposition. If FV = f(z, t) is the Frobenius series of a graded S n C[x]-module V, then f[z(1 t)] = i ( 1) i F Tor C[x] i (V, C). Let f µ (z; q, t) = FR µ (x, y). Using the Proposition, read off f µ [Z(1 q); q, t] and f µ [Z(1 t); q, t] from the Koszul homology of O Xn,ρ 1 (I µ ) w.r.t. x and y. But x and y are regular sequences in O Xn, so this is easy! We verify that f µ satisfies (i)-(iii) above, hence f µ = H µ (z; q, t). 26

27 Next, the diagonal coinvariants In the diagram R n = C[x, y]/(s n -invariants). X n C 2n ρ H n ψ σ Sym n (C 2 ) = C 2n /S n, Spec(R n ) is the scheme-theoretic fiber ψ 1 (0). So X n C 2n induces a map R n H 0 (ρ 1 σ 1 (0), O) = H 0 (σ 1 (0), P ). Theorem 2. The (scheme-theoretic) zerofiber Z n = σ 1 (0) is reduced & Cohen-Macaulay, and O Zn has an explicit O Hn -locally free resolution. Theorem 3. H i (Z n, P ) = 0 for i > 0, and the above map R n H 0 (Z n, P ) is an isomorphism. 27

28 About proofs... for Theorem 2, the zero-fiber in the tautological family F turns out to be a a local complete intersection in F, and Z n is its isomorphic image. Theorem 3 follows from Theorem 2 plus a general vanishing theorem Theorem 4. H i (H n, P B k ) for i > 0. This in turn follows from Theorem 1, a theorem of Bridgeland King Reid, and the polygraph theorem (an intermediate result in the proof of Theorem 1). We can now write down FR n using Thomason s generalized Atiyah Bott Lefschetz formula. FR n = µ =n where... (1 q)(1 t)π µ (q, t)b µ (q, t) H µ (z; q, t) x µ(1 q a(x) t l(x)+1 )(1 q a(x)+1 t l(x) ), 28

29 the sum is over partitions µ of n, B µ (q, t) = Π µ (q, t) = (r,s) µ t r q s, (r,s) µ (r,s) (0,0) (1 t r q s ), t qt 1 q q 2 and arm a(x) and leg l(x) of a box x µ are l l x a a a a(x) = 3, l(x) = 2. Numerator factors (1 q)(1 t)π µ (q, t)b µ (q, t) come from the free resolution of O Zn ; H µ (z; q, t) comes from the fiber P (I µ ). Denominator factors x µ (1 q a(x) t l(x)+1 )(1 q a(x)+1 t l(x) ) come from torus action on T I µ H n. 29

30 Proposition (Garsia H ). The expansion of the n-th elementary symmetric function e n (z) in terms of Macdonald polynomials is e n (z) = µ =n Hence t n(µ) q n(µ ) (1 q)(1 t)π µ B µ H µ (z; q, t) x µ(1 q a(x) t l(x)+1 )(1 q a(x)+1 t l(x) ) FR n = e n (z), where H µ = t n(µ) q n(µ ) H µ. Set O(1) = n B. The miraculous identity in the Proposition reduces to an instance of Atiyah Bott for FH 0 (Z n, O( 1) P ) = FV (1 n ) = e n(z), assuming the truth of Conjecture. H i (Z n, O( 1) P ) = 0 for i > 0. More generally (since O( 1) is a summand of P ), for i > 0 H i (H n, P P B k ) = 0. 30

31 A bigger picture Fix Γ SL 2 (C) finite. G = S n Γ acts on C 2n. Γ ( 1) corresponds to a Dynkin diagram of type A, D, or E. Conjecture. Quiver varieties M(Λ 0, ν) associated to affine Dynkin diagrams Â, D, Ê and the basic weight Λ 0 are moduli spaces of stable G-constellations. Our Theorem 1 on the Hilbert scheme is the case Γ = 1. Nakajima & Grojnowski constructed a level- (0,1) representation V Λ0 of the quantum double loop algebra U q (ĝ) on ν K0 C (M(Λ 0, ν)). The Conjecture would supply a basis consisting of distinguished vector bundles. One expects this to be a canonical basis of V Λ0 in some suitable sense. 31

32 In type Âr 1, Γ = Z/rZ is Abelian and commutes with T = (C ) 2, which acts on M(Λ 0, ν) with isolated fixed points. The conjecture gives a tautological bundle P of G-constellations on M(Λ 0, ν). Its fibers P (I) at fixed points I M(Λ 0, ν) T are doubly graded G-modules. Their characters should be wreath Macdonald polynomials H I N[q, t] X(G), determined (conjecturally) by an analog of the definition we gave in Lecture 1 for usual Macdonald polynomials. Plenty of computational evidence suggests that wreath Macdonald polynomials do indeed exist and have coefficients in N[q, t]. 32

33 Macdonald polynomials and Hilbert schemes Mark Haiman U.C. Berkeley LECTURE III New combinatorial developments in Macdonald theory 33

34 Combinatorial formula for H µ (z; q, t) Motivation H µ (z; 1, 1) = p 1 (z) n = (z 1 + z 2 + ) n for any µ, where n = µ. Assign each filling σ : µ Z + the weight e.g. z σ = x µ z σ(x), σ = , z σ = z 1 z 3 2 z2 3 z 4z 5. Then p 1 (z) n = σ : µ Z + z σ, 34

35 and we may expect H µ (z; q, t) = σ : µ Z + q? t? z σ. Definitions. Descents and major index of σ: b a a < b maj(σ) = x Des(σ) (recall arm a(x) and leg l(x) l(x) + 1 l l x a a a a(x) = 3, l(x) = 2). Example maj(σ) = = 3 35

36 Inversions of σ b a or a b a < b inv(σ) = Inv(σ) x Des(σ) a(x). We subtracted forced inversions b a c a < b c < b or a < c. Example inv(σ) = 5 2 = 3 Theorem (Haglund Loehr H 2004, conj. by Haglund). H µ (z; q, t) = σ : µ Z + q inv(σ) t maj(σ) z σ. 36

37 No combinatorial formula for K λµ (q, t), as we wrote H µ (z; q, t) in terms of monomials, not Schur functions. Open Problem 1: explain q t symmetry H µ (z; q, t) = H µ (z; t, q), generalizing Foata Schützenberger bijection for µ = (1 n ), (n). inv = 0 maj is classical maj = 0 inv is classical. Open Problem 2: connect combinatorics to R µ and Hilbert scheme. A puzzle: why is our formula a symmetric function in z? 37

38 LLT polynomials Recall that a semistandard Young tableau is a filling, increasing weakly on rows & strictly on columns. Schur functions are given by S λ (z) = T SSYT(λ) z T. Fix a tuple ν of (skew) diagrams ν (1) ν (2) ν (3) A semistandard tableau on ν is a tuple T SSYT(ν (1) ) SSYT(ν (k) ), e.g

39 Mark the content c(x) = (row column) of each box, e.g (we may fix a separate origin for each ν (i) ). Definition. Inversions of SSYT T on ν b x ν (i) a y ν (j) a y ν (j) or b x ν (i) a < b, c(y) = c(x), i < j, a < b, c(y) = c(x) 1, i > j. Example. T = inv(t ) = 10 39

40 Definition. LLT polynomial G ν (z; q) = T SSYT(ν) q Inv(T ) z T. Note G ν (z; 1) is a product of skew Schur functions S ν (1)(z) S ν (k)(z). Theorem (Lascoux Leclerc Thibon). G ν (z; q) is a symmetric function. Proposition. Fix µ and D µ. Then F µ,d (z; q, t) def = σ : µ Z + Des(σ)=D q Inv(σ) z σ = G ν (z; q) for a suitable tuple of ribbon skew diagrams ν. Picture proof. D = { } µ ν, c(x) 40

41 Corollary. The combinatorial expression for H µ (z; q, t) is a symmetric function, equal to D t x D (l(x)+1) q x D a(x) F µ,d (z; q, t). About the proof of our formula... let C µ (z; q, t) = σ : µ Z + q inv(σ) t maj(σ) z σ be the combinatorial expression. Given that C µ is symmetric, we can make sense of C µ [Z(1 q); q, t], C µ [Z(1 t); q, t]. We construct sign-reversing involutions to verify C µ satisfies (i) (ii) in the def n of Macdonald polynomials: (i) C µ [Z(1 q); q, t] Q(t){S λ (z) : λ µ}; (ii) C µ [Z(1 t); q, t] Q(t){S λ (z) : λ µ }; (iii) C µ, S (n) = 1. For (iii), C µ, S (n) is the coefficient of z n 1 in C µ. The all-1 s filling σ has maj(σ) = inv(σ) = 0. 41

42 Diagonal coinvariants Recall R n = C[x, y]/(s n -invariants) = H 0 (Z n, P ) FR n = e n (z). Let δ n = (n 1, n 2,..., 1). Consider partitions λ δ n and tableaux T SSY T (λ+(1 n )/λ), e.g. (n = 6) λ = (3, 2, 2) T Theorem (Garsia H ). ( e n )(z; 1, t) = λ δ n t δ n/λ S λ+(1 n )/λ (z). 42

43 Definition. Inversions of T SSYT(λ+(1 n )/λ) Then b a or a T SSYT(λ+(1 n )/λ) b q Inv(T ) z T a < b is an LLT polynomial (hence symmetric), e.g. λ + (1 n )/λ ν, c(x) 4 Conjecture (Haglund Loehr Remmel Ulyanov H ). e n (z) = q Inv(T ) z t. λ δ n t δ n/λ T SSYT(λ+(1 n )/λ) 43

44 Open Problem 3. Prove the conjecture. Open Problem 4. Relate it to geometry. Open Problem 5. Exhibit the q t symmetry of e n (z) in the combinatorial formula. Open Problem 6. Find a combinatorial expression for the doubly-graded character of Gordon s module R W for other Weyl groups W. 44