Affine Schubert calculus and double coinvariants
We first define an action of the double coinvariant algebra on the homology of the affine flag variety in type , and use affine Schubert calculus to prove that it preserves the image of the homology of the rational -affine Springer fiber under the pushforward of the inclusion map. In our main result, we define a filtration by -submodules of indexed by compositions, whose leading terms are the Garsia-Stanton “descent monomials” in the -variables. We find an explicit presentation of the subquotients as submodules of the single-variable coinvariant algebra , by identifying the leading torus fixed points with a subset of the torus fixed points of the regular nilpotent Hessenberg variety, and comparing them to a cell decomposition of due to Goresky, Kottwitz, and MacPherson. We also discover an explicit monomial basis of , and in particular an independent proof of the Haglund-Loehr formula.
- 1 Introduction
- 2 Geometric preliminaries
- 3 Combinatorial results
- 4 Affine Schubert calculus
- 5 Double Coinvariants
The double coinvariant algebra is the quotient space of the polynomials algebra in variables by the ideal generated by nonconstant diagonally symmetric polynomials
In , Haiman proved that this space has dimension , and in , Haglund and Loehr conjectured the combinatorial formula for the bigraded Hilbert series in terms of certain parking function statistics
Several articles due to Lusztig-Smelt, Gorsky-Mazin, Hikita, and Gorsky-Mazin-Vazirani have connected the combinatorics of the rational version of the Haglund-Loehr formula with a basis of the affine Springer fiber in type , in which the -grading corresponded with the homological grading, and the -grading was a new statistic related to the “expected” dimension of a corresponding affine paving [9, 10, 11, 19, 27]. On the other hand, Oblomkov and Yun have shown that that cohomology of this affine Springer fiber is an irreducible module over the rational Cherednik algebra . It was known from  that there is an isomorphism of graded spaces and , but this did not result in a proof of the Haglund-Loehr formula, because Hikita’s statistic could not be connected to the grading by the -variables in .
The first main result of this paper defines an action of the double coinvariant algebra on the homology of the affine flag variety in type , and shows that it preserves the homology of the -affine Springer fibers:
There is an action of on the homology of the affine flag variety that preserves , the Borel-Moore homology of the affine Springer fiber in type . Here the -variables act by dual Chern class operators which lower the homological degree, while the -variables are homogeneous operators defined using the usual action of the (extended) affine Weyl group, which is compatible with the Springer action. In the case , this action induces an isomorphism by applying to the generator , .
We then define a filtration by -submodules, which is generated by monomials for in for a certain total order, whose leading terms are the “descent monomials” , which are well known to be a basis of the single coinvariant algebra . In our second main theorem, we describe this filtration under the isomorphism of Theorem A, and find an explicit presentation for the subquotients:
Given a composition , the following three -modules are isomorphic:
The subquotients , where , for .
The corresponding subquotients of a certain filtration on , defined using a coarsening of the Bruhat order on the fixed point basis in the nil Hecke algebra, i.e. the equivariant Borel-Moore homology of the affine flag variety.
A certain submodule of the single coinvariant algebra generated by an explicit polynomial depending on .
Moreover, the third description actually has an explicit basis by monomials in the -variables, leading to an explicit formula for the Hilbert series of of all three modules, which in particular gives an independent proof the Haglund-Loehr formula.
This theorem resolves the problems of connecting the degree in the -variables to Hikita’s statistics, produces a monomial basis of the double coinvariant algebra, and characterizes the subquotients as -modules. We hope for several future directions:
The definition we present of the filtration on is defined algebraically using the equivariant homology groups. However, it seems clear that there should be a geometric description of this filtration, through topological subspaces , for . Our combinatorics turn out to be closely related to the description of the torus-fixed point set of the regular nilpotent Hessenberg varieties, but with Hessenberg functions (i.e. Dyck paths) replaced with Dyck paths [20, 31]. It is likely that related varieties will be a part of any such geometric formulation.
We expect that Theorem B will be important for generalizing the Haglund-Loehr formula to other root systems, for which there are currently no availble conjectures. We particularly hope that the desired geometric subspaces will admit a natural generalization.
We have tested numerically that this filtration is compatible with taking invariants by Young subgroups, in the sense that the subquotients of
While Theorem B applies to the case , our filtration makes sense more generally. For coprime , the replacement for should be an associated graded module of the finite-dimensional representations of the rational Cherednik algebra. One might also consider the intersection with affine Springer fibers which are not necessarily compact, such as the non-coprime case.
We expect the the description of the double coinvariants as a module will be important towards categorification, and hope for the expected applications to Khovanov-Rozansky knot homology, in which the -variables are the variables that appear in Sörgel bimodules.
The paper is divided into five sections. In section 2 we discuss the geometric results and definitions that we will need for the main construction, including the results of . In the interest of making our paper readable to combinatorialists, we have compartmentalized the necessary algebraic facts from this section into Proposition 6 of section 4, so that it may be safely skipped. In section 3 we recall combinatorial facts about affine permutations and parking functions, and we give a new description of parking functions in terms of a bijection of Haglund , which turns out to be similar to the description of the fixed points of regular nilpotent Hessenberg varieties [20, 31]. Section 4 recalls the algebraic constructions of the affine Schubert polynomials and nil Hecke algebras . Finally, in section 5, we state and prove the main results of the paper.
Acknowledgments The authors would like to thank Thomas Lam, Mark Shimozono, and S. J. Lee for interesting discussions about affine Schubert calculus. The second author was partially supported by NSF CAREER grant DMS-1352398.
2 Geometric preliminaries
We now recall some results about the affine Springer fiber and affine flag varieties that we will need for our main results in Chapter 5. The reader interested mainly in algebra can skip everything in this section, except for possibly the conventions for the root system in type , provided they are willing to take Proposition 6 of section 4 on faith.
2.1 Root systems
In this section we fix our conventions on the root system for type . Let , let be the corresponding affine Lie algebra, and let denote the Lie algebra of the maximal torus . The dual of the maximal torus is spanned by the fundamental weights together with the imaginary root
We also define define weights for all integers satisfying
and for all . The roots , form a basis of a subspace . The simple roots are given by
and the action of the affine Weyl group is given by
for . The third equation follows from the first two, and in fact holds for any integer , and defined below for extended affine permutations as well.
2.2 The affine flag variety
Let be a complex algebraic group such that its Lie algebra is simple. We define to be the ring of formal power series of , and its quotient field is . Respectively, is the group of formal loops and is the subgroup of holomorphic loops. The quotient has the structure of the ind-scheme, as an inductive limit by smooth subschemes . For more details, see the survey .
The affine flag variety is the ind scheme where is the subgroup of elements such that . In this paper we assume that and are the maximal torus and the Borel subgroup. Then the quotient has connected components, and to simplify notations we use notation for the connected component that contains .
The lattice inside of is a subspace that is preserved by and the intersection is of finite codimension inside and . The index is well-defined for a lattice. The flag variety admits the following elementary description
In this description we have tautological line bundle over has fiber at the point .
The torus acts on : the torus acts by left multiplication and acts by loop rotation for . This action has isolated fixed points which are enumerated by the bijections . Indeed, if be a basis of that is fixed by , then there is a unique flag of torus-invariant lattices satisfying
provided that satisfies:
Thus there is a natural identification between and .
There is a natural embedding such that . The Bruhat decomposition induces the decomposition of into affine cells where is the cell of dimension . The affine Schubert variety is the algebraic closure of . The variety is the union of cells and the Bruhat order is defined by condition . The varieties in the description of as an ind-variety can be taken to be the union of the cells with length at most .
We remind the reader of the construction of the equivariant Borel-Moore homology from . In this paper, all (equivariant) homology and cohomology groups will have coefficients in . Let be a scheme with a action of a linear algebraic group . Let be a representation of and let be an open subset where acts freely. Then the equivariant cohomology and Borel-Moore homology are defined by:
where , provided the complex codimension of in is greater than and .
Notice that in our definition the homological degree is bounded from above by and is not bounded from below. The main advantage of using equivariant Borel-Moore homology is we have a fundamental class , . In particular, fundamental class and cap product provide an identification and . Let us also notice that and both have a ring structure and the above mentioned identification of both spaces respect the ring structure. Thus for any with a -action, the spaces and are naturally -modules and the natural pairing between these two spaces is -linear.
The equivariant homology of the affine flag variety is defined as the direct limit
It has the structure of noncommutative ring with an explicit algebraic presentation, called the nil Hecke algebra, [22, 25]. The Schubert classes, for are defined as the fundamental classes of the closures of the Schubert cells again using Borel-Moore homology .
Since we define as inductive limit of finite-dimensional schemes , it is natural to define the cohomology as inverse limit with respect to the pullback maps:
as graded modules, as described in the last paragraph of . Then is a module over the equivariant cohomology of the point , which may be identified as a submodule
Then the affine Schubert polynomials may be defined as a dual basis to , see [23, 24, 25]. We will denote by the first Chern class . These classes, together with the pullback of the equivariant cohomology of the affine Grassmannian, generate the equivariant cohomology as an -module, with relations described in section 4.1.
2.3 The affine Springer fiber
Given an element the authors of  attach a subset of :
The lattice consisting of elements commuting with naturally acts on . It is shown in  that if is a topologically nilpotent and regular semi-simple then the quotient is a quasi-projective finite type scheme. The (ind)-scheme is a variety called the affine Springer fiber.
The element is called homogeneous if is conjugate to for all . The topologically nilpotent regular semi-simple elements are classified in  and the corresponding affine Springer fibers have a natural -action. Their homologies provide a geometric model for the representations of the graded and rational Cherednik algebra of the corresponding type [29, 32, 33]. This paper deals only with the Springer theory in type , and we now recall the relevant results.
Let us denote by an element such that
This element is homogeneous and regular semi-simple, as is the element for . If are coprime, then the affine Springer fiber is a projective variety, that was first studied in . Let be the inclusion map.
The full torus does not preserve the Springer fiber, but the one-dimensional subtorus ,
preserves . We fix our conventions by setting . Strictly speaking, this map is not quite defined because the exponents may be fractions, but this has no effect, and the normalization is preferred for Cherednik algebras. Since , the fixed point set is naturally a subset of . This set is denoted , and has explicit description given in section 3.1.
It was shown in  that , is an affine space of dimension . Respectively, we denote by the closure of the intersection . As in , there is a well-defined fundamental class . Then we have the following proposition:
For with coprime, we have
The pushforward map is injective.
The restriction map is surjective.
The localization map to the fixed point set is injective.
The equivariant Borel-Moore homology is freely generated over by the fundamental classes .
The equivariant Borel-Moore cohomology is freely generated by dual elements , such that the pairing of with is the delta funciton .
2.4 Action of the Cherednik algebra
Let us recall the definition of the graded Cherednik algebra . As a -vector space,
with grading given by
Let us fix notation . The algebra structure is defined by
and are subalgebras
The element is also central, and thus for we can define an algebra
This is the the graded Cherednik algebra with the central charge . We set the image of to be . If we specialize to we obtain the algebra which is the trigonometric algebra in the literature.
The subalgebra has a trivial representation and the induced representation
is called polynomial representation of . The subalgebra acts on the right by the left multiplication on this representation. On the other hand there is a standard action of on given by (1). The action of is a deformation of the standard action, the generator , acts by the (right) operator
The equivariant Chern classes , generate localized equivariant cohomology . Hence there is a natural isomorphism . Under this identification acquires structure of -module. Respectively, becomes an -module. The embedding induces the pullback map between the cohomology group. This map was studied in [29, 30]:
3 Combinatorial results
We review some combinatorial preliminaries about parking functions and the restricted affine permutations, i.e. the torus fixed points of the affine Springer fiber. These were given a combinatorial description by Hikita, generalizing several previous bijections [9, 10, 19]. This is described by the map called by Gorsky, Vazirani, Mazin in , who also discovered a second bijection, , that will play a major role in this paper. We then give a different description of parking functions in terms of normal permutations satisfying a condition that is similar to one that in the fixed points of Hessenberg varieties [20, 31].
3.1 Affine permutations
Let denote the affine permutations, i.e. those bijections satisfying
If the second condition is dropped, then is called an extended affine permutation. The set of extended affine permutations will be denoted . The window notation for any such permutation may be used since is determined by its values on . We will single out two particularly important extended permutations by
where is the class of modulo .
3.2 Parking function statistics
An -parking function is an -Dyck path with rows labeled by numbers, like the one shown in Figure 1. They are uniquely determined by the composition of length whose th value is the number of boxes above the path in the row containing . In the example, for instance, the composition is .
In , Gorsky, Mazin, Vazirani defined two maps from to parking functions, described in terms of their associated compositions. Their first map,
named for Pak-Stanley, is defined by
They proved that this map is in bijection with parking functions in the case , and conjectured that it is a bijection for all . For the second, they defined a parking function for each as follows: let
For each , there is a unique way to express as for , which necessarily implies . Now define . They conjectured that
is the symmetric parking function generating function that appears in the rational shuffle conjecture.
For each , let us define a composition by
where is the permutation for which is the number in row of the parking function diagram, and is the maximum number of boxes in row . Unlike the corresponding definition for , we claim that this function does not depend on , and in fact can be described by
where is the unique left-shifted permutation whose minimum value in window notation is zero.
3.3 A Hessenberg description of parking functions
We now give another description of the restricted permutations in the case of , using a bijection in Haglund’s book. In this subsection, we will assume .
Define the runs of a permutation as the maximal consecutive increasing subsequences of , so that has descents. If there are runs, we decide that there is a st run consisting only of the number . Then for each parking function, there is a permutation such that the numbers (cars) in each row of length zero are the elements of the th run of , the numbers in rows of length one are the elements of the st run, etc. The set of parking functions associated to this permutation is denoted . Then it is easy to check that
where is the composition such that if is in the th run, so that . We let denote those restricted permutations satisfying either side of equation (6).
In Figure 4, Chapter 5 of Haglund’s book it is shown that
where , and where we are writing parking functions as compositions. The corresponding restricted permutations are
whose inverses map to the desired parking functions after applying . We can then check that these are precisely the restricted permutations satisfyig .
For each , consider the function given by
where we identify affine permutations with -tuples of integers by the window notation, and take the unique shift of the extended permutation in parentheses which is in . Then define
Let , . Then we would have
We now give a parametrization of this set, which is essentially the same as a bijection in Chapter 5 of Haglund’s book as follows: if is in the th run, we define as the set of the indices of all those elements in the th run that are greater than , together with the set of elements in the st run that are less than . For instance, writing for the list of all the , we have
We define a map as follows: first start by setting to be an arrangement starting with the number , which we will think of as . Then for from to , insert the number to the right of the th element of , where the order is the opposite of the order in which they appear in , i.e. right to left. Finally, remove the leading and let . We define as the set of all for .
For , and , the sequence would be
so would be .
We have that .
Recall from page 81 of Haglund’s book  that there is a bijection that we will call . The proof of the lemma is essentially the the same as Haglund’s proof, but rewritten in terms of restricted permutations using . To be precise, we have
It is shown in Figure 4 of page 80 of Haglund’s book that the parking functions in Example 2 with are the elements for in
On the other hand, the restricted permutations are also the images for in
We also have
where the first equality comes from , and the second is built into Haglund’s construction. In particular, we have
where is the -number, using Corollary 5.3.1, page 82 of . Then summing over all in (8) gives the character of the double coinvariant algebra , assuming the Haglund-Loehr formula. For instance, the -character of would be
We now prove a third description of this set:
We have that is the set of all such that
for all , where as above, means .
Let denote the set described by the condition in the lemma. It is clear that (10) is satisfied at every step in the construction of Defintion 1, because each number is added to the right of a number in , and adding a smaller number to the left of any digit preserves the condition. This shows that .
To see the reverse, suppose that satisfies the desired condition, and let denote the result of adding immediately to the right of at every step in Definition 1, where is the largest index satisfying , and , or if none exists. It is clear that , and it remains to show that we necessarily have , so that . To see this, we simply confirm the equation
establishing that . ∎
4 Affine Schubert calculus
We review some background on affine Schubert calculus, for which we refer to Goresky, Kottwitz, and MacPherson , as well as Lam , Kostant and Kumar , and the book of Lam, Lapointe, Morse, Schilling, Shimozono, and Zacbrocki . We follow the descriptions of the latter.
4.1 The nil Hecke and GKM rings
and consider the noncommutative algebra
with product given by
where , and the action of on is determined by equation (1).
We define a subalgebra as follows: for any , let
These operators satisfy the braid relations in type , and so we may define
whenever is a reduced word. The subring generated by these elements is called the affine nil Coxeter algebra. It is noncommutative and satisfies
The subring generated by the and is called the affine nil Hecke algebra, and is denoted by . It is graded by assigning the elements of and degree , whence has degree . Now define
Then is a free -module with a basis
As explained in section 2, we have
(Kostant, Kumar ) We have isomorphisms of graded -modules
in which the Schubert cycles map to , the dual classes in cohomology map to , and the pairing between homology and cohomology agrees with the pairing between and .
4.2 Affine Schubert polynomials
The left and right actions of by the embedding preserve , and also induce dual actions on . If we view either as an -module by left multiplication, then the action by right multiplication by is linear over , whereas left multiplication has a nontrivial internal action on . There is also a compatible action of the extended affine permutations on by conjugation, in which
We also have the dual actions on , and we will denote by the same letter , which acts by on scalars.
The cohomology ring is generated by two important sets of classes: first, we have the Chern classes of the natural line bu