Hilbert schemes, polygraphs and the Macdonald positivity conjecture
We study the isospectral Hilbert scheme defined as the reduced fiber product of , with the Hilbert scheme of points in the plane over the symmetric power , By a theorem of Fogarty, . is smooth. We prove that is normal, Cohen-Macaulay and Gorenstein, and hence flat over We derive two important consequences. .
(1) We prove the strong form of the conjecture of Garsia and the author, giving a representation-theoretic interpretation of the Kostka-Macdonald coefficients This establishes the Macdonald positivity conjecture, namely that ..
(2) We show that the Hilbert scheme is isomorphic to the scheme -Hilbert of Nakamura, in such a way that is identified with the universal family over From this point of view, . describes the fiber of a character sheaf at a torus-fixed point of corresponding to .
The proofs rely on a study of certain subspace arrangements called polygraphs, whose coordinate rings , carry geometric information about The key result is that . is a free module over the polynomial ring in one set of coordinates on This is proven by an intricate inductive argument based on elementary commutative algebra. .
The Hilbert scheme of points in the plane is an algebraic variety which parametrizes finite subschemes of length in To each such subscheme . corresponds an multiset, or unordered -element with possible repetitions, -tuple of points in where the , are the points of repeated with appropriate multiplicities. There is a variety , finite over , whose fiber over the point of , corresponding to consists of all ordered -tuples whose underlying multiset is We call . the isospectral Hilbert scheme.
By a theorem of Fogarty Reference 14, the Hilbert scheme is irreducible and nonsingular. The geometry of is more complicated, but also very special. Our main geometric result, Theorem 3.1, is that is normal, Cohen-Macaulay and Gorenstein.
Earlier investigations by the author Reference 24 unearthed indications of a far-reaching correspondence between the geometry and sheaf cohomology of and on the one hand, and the theory of Macdonald polynomials on the other. The Macdonald polynomials
are a basis of the algebra of symmetric functions in variables with coefficients in the field , of rational functions in two parameters and They were introduced in 1988 by Macdonald .Reference 39 to unify the two well-known one-parameter bases of the algebra of symmetric functions, namely, the Hall-Littlewood polynomials and the Jack polynomials (for a thorough treatment see Reference 40). It promptly became clear that the discovery of Macdonald polynomials was fundamental and sure to have many ramifications. Developments in the years since have borne this out, notably, Cherednik’s proof of the Macdonald constant-term identities Reference 9 and other discoveries relating Macdonald polynomials to the representation theory of quantum groups Reference 13 and affine Hecke algebras Reference 32Reference 33Reference 41, the Calogero-Sutherland model in particle physics Reference 35, and combinatorial conjectures on diagonal harmonics Reference 3Reference 16Reference 22.
The link between Macdonald polynomials and Hilbert schemes comes from work by Garsia and the author on the Macdonald positivity conjecture. The Schur function expansions of Macdonald polynomials lead to transition coefficients known as Kostka-Macdonald coefficients. As defined, they are rational functions of , and but conjecturally they are polynomials in , and with nonnegative integer coefficients:
The positivity conjecture has remained open since Macdonald formulated it at the time of his original discovery. For it reduces to the positivity theorem for coefficients, which has important algebraic, geometric and combinatorial interpretations -KostkaReference 7Reference 10Reference 17Reference 27Reference 31Reference 34Reference 36Reference 37Reference 38Reference 45. Only recently have several authors independently shown that the Kostka-Macdonald coefficients are polynomials, but these results do not establish the positivity ,Reference 18Reference 19Reference 32Reference 33Reference 44.
In Reference 15, Garsia and the author conjectured an interpretation of the Kostka-Macdonald coefficients as graded character multiplicities for certain doubly graded -modules The module . is the space of polynomials in variables spanned by all derivatives of a certain simple determinant (see §2.2 for the precise definition). The conjectured interpretation implies the Macdonald positivity conjecture. It also implies, in consequence of known properties of the that for each partition , of the dimension of , is equal to This seemingly elementary assertion has come to be known as the . conjecture.
It develops that these conjectures are closely tied to the geometry of the isospectral Hilbert scheme. Specifically, in Reference 24 we were able to show that the Cohen-Macaulay property of is equivalent to the conjecture. We further showed that the Cohen-Macaulay property of implies the stronger conjecture interpreting as a graded character multiplicity for Thus the geometric results in the present article complete the proof of the Macdonald positivity conjecture. .
Another consequence of our results, equivalent in fact to our main theorem, is that the Hilbert scheme is equal to the scheme -Hilbert of Ito and Nakamura Reference 28, for the case , The . scheme is of interest in connection with the generalized McKay correspondence, which says that if -Hilbert is a complex vector space, is a finite subgroup of and is a so-called crepant resolution of singularities, then the sum of the Betti numbers of equals the number of conjugacy classes of In many interesting cases .Reference 6Reference 42, the scheme turns out to be a crepant resolution and an instance of the McKay correspondence. By our main theorem, this holds for -Hilbert ,.
We wish to say a little at this point about how the discoveries presented here came about. It has long been known Reference 27Reference 45 that the coefficients -Kostka are graded character multiplicities for the cohomology rings of Springer fibers. Garsia and Procesi Reference 17 found a new proof of this result, deriving it directly from an elementary description of the rings in question. In doing so, they hoped to reformulate the result for in a way that might generalize to the two-parameter case. Shortly after that, Garsia and the author began their collaboration and soon found the desired generalization, in the form of the conjecture. Based on Garsia and Procesi’s experience, we initially expected that the conjecture itself would be easy to prove and that the difficulties would lie in the identification of as the graded character multiplicity. To our surprise, however, the conjecture stubbornly resisted elementary attack.
In the spring of 1992, we discussed our efforts on the conjecture with Procesi, along with another related conjecture we had stumbled upon in the course of our work. The modules involved in the conjecture are quotients of the ring of coinvariants for the action of on the polynomial ring in variables. This ring is isomorphic to the space of diagonal harmonics. Computations suggested that its dimension should be and that its graded character should be related to certain well-known combinatorial enumerations (this conjecture is discussed briefly in §5.3 and at length in Reference 16Reference 22). Procesi suggested that the Hilbert scheme and what we now call the isospectral Hilbert scheme should be relevant to the determination of the dimension and character of Specifically, he observed that there is a natural map from . to the ring of global functions on the scheme-theoretic fiber in over the origin in With luck, this map might be an isomorphism, and—as we are now able to .confirm— might be flat over so that its structure sheaf would push down to a vector bundle on , Then . would coincide with the space of global sections of this vector bundle over the zero-fiber in and it might be possible to compute its character using the Atiyah-Bott Lefschetz formula. ,
The connection between and the conjecture became clear when the author sought to carry out the computation Procesi had suggested, assuming the validity of some needed but unproven geometric hypotheses. More precisely, it became clear that the spaces in the conjecture should be the fibers of Procesi’s vector bundle at distinguished torus-fixed points in a fact which we prove in § ,3.7. These considerations ultimately led to a conjectured formula for the character of in terms of Macdonald polynomials. This formula turned out to be correct up to the limit of practical computation ( Furthermore, Garsia and the author were able to show in ).Reference 16 that the series of combinatorial conjectures in Reference 22 would all follow from the conjectured master formula. Thus we had strong indications that Procesi’s proposed picture was indeed valid, and that a geometric study of should ultimately lead to a proof of the and Macdonald positivity conjectures, as is borne out here. By now the reader should expect the geometric study of also to yield a proof of the character formula for diagonal harmonics and the conjecture. This subject will be taken up in a separate article.
The remainder of the paper is organized as follows. In §2 we give the relevant definitions concerning Macdonald polynomials and state the positivity, and graded character conjectures. Hilbert scheme definitions and the statement and proof of the main theorem are in §3, along with the equivalence of the main theorem to the conjecture. In §3.9 we review the proof from Reference 24 that the main theorem implies the conjecture of Garsia and the author on the character of the space and hence implies the Macdonald positivity conjecture. ,
The proof of the main theorem uses a technical result, Theorem 4.1, that the coordinate ring of a certain type of subspace arrangement we call a polygraph is a free module over the polynomial ring generated by some of the coordinates. Section 4 contains the definition and study of polygraphs, culminating in the proof of Theorem 4.1. At the end, in §5, we discuss other implications of our results, including the connection with schemes, along with related conjectures and open problems. -Hilbert
2. The and Macdonald positivity conjectures
2.1. Macdonald polynomials
We work with the transformed integral forms of the Macdonald polynomials, indexed by integer partitions and homogeneous of degree , These are defined as in .Reference 24, eq. (2.18), to be
where denotes Macdonald’s integral form as in Reference 40, VI, eq. (8.3), and is the partition statistic
(not to be confused with ).
The square brackets in Equation 3 stand for plethystic substitution. We pause briefly to review the definition of this operation (see Reference 24 for a fuller discussion). Let be the algebra of formal series over the coefficient field in variables , For any . we denote by , the result of replacing each indeterminate in by its power. This includes the indeterminates -th and as well as the variables The algebra of symmetric functions . is freely generated as an by the power-sums -algebra
Hence there is a unique homomorphism -algebra
In general we write for for any , With this notation goes the convention that . stands for the sum of the variables, so we have and hence for all Note that a plethystic substitution like . such as we have on the right-hand side in ,Equation 3, yields again a symmetric function.
There is a simple direct characterization of the transformed Macdonald polynomials .
We set where , is the Kostka-Macdonald coefficient defined in Reference 40, VI, eq. (8.11). This is then related to the transformed Macdonald polynomials by
It is known that has degree at most in so the positivity conjecture ,Equation 2 from the introduction can be equivalently formulated in terms of .
2.2. The and graded character conjectures
be the polynomial ring in variables. To each subset -element we associate a polynomial , as follows. Let be the elements of listed in some fixed order. Then we define
If is a partition of its diagram is the set ,
(Note that in our definition the rows and columns of the diagram are indexed starting with zero.) In the case where is the diagram of a partition, we abbreviate
The polynomial is a kind of bivariate analog of the Vandermonde determinant which occurs as the special case ,.
Given a partition of we denote by ,
the space spanned by all the iterated partial derivatives of In .Reference 15, Garsia and the author proposed the following conjecture, which we will prove as a consequence of Proposition 3.7.3 and Theorem 3.1.
The conjecture arose as part of a stronger conjecture relating the Kostka-Macdonald coefficients to the character of as a doubly graded The symmetric group -module. acts by automorphisms of -algebra permuting the variables:
The ring is doubly graded, by degree in the and variables respectively, and the action respects the grading. Clearly is i.e., we have -alternating, for all where , is the sign character. Note that is also doubly homogeneous, of -degree and -degree It follows that the space . is and has a double grading -invariant
by subspaces -invariant.
We write for the character of an -module and denote the irreducible , characters by with the usual indexing by partitions , of The following conjecture implies the Macdonald positivity conjecture. .
Macdonald had shown that is equal to the degree of the irreducible , character or the number of standard Young tableaux of shape , Conjecture .2.2.2 therefore implies that affords the regular representation of In particular, it implies the . conjecture.
In Reference 24 the author showed that Conjecture 2.2.2 would follow from the Cohen-Macaulay property of We summarize the argument proving Conjecture .2.2.2 in §3.9, after the relevant geometric results have been established.
3. The isospectral Hilbert scheme
In this section we define the isospectral Hilbert scheme and deduce our main theorem, Theorem ,3.1 (§3.8). We also define the Hilbert scheme and the nested Hilbert scheme and develop some basic properties of these various schemes in preparation for the proof of the main theorem. ,
The main technical device used in the proof of Theorem 3.1 is a theorem on certain subspace arrangements called polygraphs, Theorem 4.1. The proof of the latter theorem is lengthy and logically distinct from the geometric reasoning leading from there to Theorem 3.1. For these reasons we have deferred Theorem 4.1 and its proof to the separate §4.
Throughout this section we work in the category of schemes of finite type over the field of complex numbers, All the specific schemes we consider are quasiprojective over . We use classical geometric language, describing open and closed subsets of schemes, and morphisms between reduced schemes, in terms of closed points. A variety is a reduced and irreducible scheme. .
Every locally free coherent sheaf of rank on a scheme of finite type over is isomorphic to the sheaf of sections of an algebraic vector bundle of rank over For notational purposes, we identify the vector bundle with the sheaf . and write for the fiber of at a closed point In sheaf-theoretic terms, the fiber is given by ..
A scheme is Cohen-Macaulay or Gorenstein if its local ring at every point is a Cohen-Macaulay or Gorenstein local ring, respectively. For either condition it suffices that it holds at closed points At the end of the section, in § .3.10, we provide a brief summary of the facts we need from duality theory and the theory of Cohen-Macaulay and Gorenstein schemes.
3.2. The schemes and
Let be the coordinate ring of the affine plane By definition, closed subschemes . are in one-to-one correspondence with ideals The subscheme . is finite if and only if has Krull dimension zero, or finite dimension as a vector space over In this case, the length of . is defined to be .
The Hilbert scheme parametrizes finite closed subschemes of length The scheme structure of . and the precise sense in which it parametrizes the subschemes are defined by a universal property, which characterizes up to unique isomorphism. The universal property is actually a property of together with a closed subscheme called the universal family. ,
To see how parametrizes finite closed subschemes of length note that the latter are exactly the families , in Proposition 3.2.1 for By the universal property they correspond one-to-one with the closed points of . in such a way that the fiber of the universal family , over the point corresponding to is itself. For notational purposes we will identify the closed points of with ideals satisfying rather than with the corresponding subschemes ,.
We have the following fundamental theorem of Fogarty Reference 14.
The generic examples of finite closed subschemes of length are the reduced subschemes consisting of distinct points. They form an open subset of and the irreducibility aspect of Fogarty’s theorem means that this open set is dense. ,
The most special closed subschemes in a certain sense are those defined by monomial ideals. If is a monomial ideal, then the standard monomials form a basis of If . the exponents , of the standard monomials form the diagram of a partition of and conversely. We use the partition , to index the corresponding monomial ideal, denoting it by Note that . for all so the subscheme , is concentrated at the origin and owes its length entirely to its nonreduced scheme structure. ,
The algebraic torus
acts on as the group of invertible diagonal matrices. The monomial ideals are the torus invariant ideals, and thus they are the fixed points of the induced action of on the Hilbert scheme. Every ideal has a monomial ideal in the closure of its ( -orbitReference 23, Lemma 2.3).
We write for the coordinates on the factor in the Cartesian product -th so we have , where ,