Hyperplane Arrangements and Diagonal Harmonics

In 2003, Haglund's {\sf bounce} statistic gave the first combinatorial interpretation of the $q,t$-Catalan numbers and the Hilbert series of diagonal harmonics. In this paper we propose a new combinatorial interpretation in terms of the affine Weyl group of type $A$. In particular, we define two statistics on affine permutations; one in terms of the Shi hyperplane arrangement, and one in terms of a new arrangement - which we call the Ish arrangement. We prove that our statistics are equivalent to the {\sf area'} and {\sf bounce} statistics of Haglund and Loehr. In this setting, we observe that {\sf bounce} is naturally expressed as a statistic on the root lattice. We extend our statistics in two directions: to"extended"Shi arrangements and to the bounded chambers of these arrangements. This leads to a (conjectural) combinatorial interpretation for all integral powers of the Bergeron-Garsia nabla operator applied to the elementary symmetric functions.


Introduction
First we define the diagonal harmonics -which we will keep in mind throughout -then we will discuss hyperplane arrangements -which the paper is really about.
1.1.Diagonal Harmonics.The symmetric group S(n) acts on the polynomial ring S = Q[x 1 , . . ., x n ] by permuting variables.Newton showed that the subring of S(n)-invariant polynomials is generated by the algebraically independent power sum polynomials: p k = n i=1 x k i for k = 1, 2, . . ., n.It is known that the coinvariant ring R = S/(p 1 , . . ., p n ) is a graded version of the regular representation of S(n), with Hilbert series The dual ring S * = Q[∂/∂x 1 , . . ., ∂/∂x n ] acts on S via the pairing (∂/∂x i )x j = δ ij , hence the coinvariant ring is isomorphic to the quotient S * /(p * 1 , . . ., p * n ), where p * k = n i=1 (∂/∂x i ) k for k = 1, . . ., n.On the other hand, this quotient is naturally isomorphic to the submodule H ⊆ S annihilated by the p * k : H = {f ∈ S : p * k f = 0 for all k}.
This H is called the ring of harmonic polynomials since, in particular, p * 2 is the standard Laplacian operator on S. Now consider the ring DS = Q[x 1 , . . ., x n , y 1 , . . ., y n ] of polynomials in two sets of commuting variables, together with the diagonal action of S(n), which permutes the x variables and the y variables simultaneously.Weyl [29] showed that the S(n)-invariant subring of DS is generated by the polarized power sums: p k, = n i=1 x k i y i for all k + > 0. Hence the ring of diagonal coinvariants DR = DS/(p k, : k + > 0) is naturally isomorphic to the ring of diagonal harmonic polynomials: The diagonal action preserves the bigrading of DS by x-degree and y-degree, hence DH is a bigraded S(n)-module.The bigraded Hilbert series (1.1) DH(n; q, t) := n i,j=0 dim(DH) i,j q i t j has beautiful and remarkable properties.The study of DH(n; q, t) was initiated by Garsia and Haiman (see [11]) and is today an active area of research.
Since all hyperplanes in this paper contain the line e 1 + e 2 + • • • + e n = 0, we will typically think of these arrangements in the (n − 1)-dimensional quotient space R n 0 := R n /(e 1 + e 2 + • • • + e n = 0).If A is an arrangement in a space V then the connected components of the complement V − ∪ H∈A H are called chambers.We will refer to chambers of the Coxeter arrangement as cones; and refer to affine chambers as alcoves.Let C • denote the dominant cone, which satisfies the coordinate inequalities   (see [21,Chapter 7]) in his description of the Kazhdan-Lusztig cells for certain affine Weyl groups.
1.3.Symmetric Group.The symmetric group S(n) has a faithful representation as a group of isometries of R n 0 generated by the set S = {s 1 , s 2 , . . ., s n−1 }, where s i is the reflection in the hyperplane e i − e i+1 = 0.The reflection s i corresponds in S(n) to the transposition of adjacent symbols (i, i + 1).
The symmetric group acts simply-transitively on the cones of the Coxeter arrangement Cox(n).By convention, let the dominant cone C • correspond to the identity permutation; then for any permutation w ∈ S(n) the cone wC • satisfies 1.4.Affine Symmetric Group.Now let s n denote the reflection in the affine hyperplane e 1 − e n = 1.The linear reflections {s 1 , s 2 , . . ., s n−1 } together with the affine reflection a n generate the affine Weyl group of type Ãn .This group acts simply-transitively on the set of alcoves, where the fundamental alcove A • corresponds to the identity element of the group.Note that A • is a (non-regular) simplex in R n 0 whose facets are supported by the reflecting hyperplanes of the generators {s 1 , s 2 , . . ., s n }.
Lusztig [19] introduced an affine version of the symmetric group, whose combinatorial properties were developed further by Björner and Brenti [4]: Let S(n) denote the group of infinite permutations w : Z → Z satisfying: The first property says that w is periodic and the second fixes a frame of reference.The elements of S(n) are called affine permutations, and S(n) is the affine symmetric group.Following Björner and Brenti, we will usually specify an affine permutation w ∈ S(n) using the window notation: For integers i < j we will write ((i, j)) : Z → Z to denote the "affine tranposition" that swaps the elements in positions i + kn and j + kn for all k ∈ Z.We could also write ((i, j)) = k (i + kn, j + kn).Lusztig proved that the correspondence s i ↔ ((i, i + 1)) defines an isomorphism between the affine symmetric group and the affine Weyl group of type Ãn .Here the affine tranposition ((i, j)) corresponds to the reflection in the affine hyperplane where quo(x, n) and rem(x, n) are the unique quotient and remainder of x by n, with remainder taken in the set {1, 2, . . ., n}.In particular, note that the generator s i = ((i, i + 1)) corresponds to e i − e i+1 = 0 for 1 ≤ i ≤ n − 1, and s n = ((n, n + 1)) corresponds to e 1 − e n = 1.
1.5.The Ish Arrangement.To end the introduction we will introduce a new hyperplane arrangement, which we call the Ish arrangement.Like the Shi arrangement the Ish arrangement begins with the n 2 linear hyperplanes of the Coxeter arrangement and then adds another n 2 affine hyperplanes: To avoid extra notation, we will use a non-standard definition of the characteristic polynomial.This definition is due to Crapo and Rota, and was applied extensively by Athanasiadis -see Stanley [27,Lecture 5] for details.Let A be an arrangement of finitely many hyperplanes in R n .Suppose further that each of these hyperplanes has an equation with integer coefficients.Then, given a finite field F q with q elements, we may consider the reduced arrangement A q in F n q .It turns out that (for all but finitely many q), the number of points of F n q not on any hyperplane of A q is given by a polynomial in q, called the characteristic polynomial of A: The characteristic polynomial of the Shi arrangement is well known (cf.[27, Theorem 5.16]).Our new result is the following.
Theorem 1.The Shi arrangement and the Ish arrangement share the same characteristic polynomial, viz.
Proof.Let p be a large prime and consider a regular p-gon whose vertices represent the elements of the finite field F p = {1, 2, . . ., p}, in clockwise order.We will think of a vector p as a labeling of the vertices, as follows: if v i = k ∈ F p , then place the label v i on the vertex k.
To say that v ∈ F n p is in the complement of the reduced Ish arrangement Ish(n) p , means that v i − v j = 0 for all 1 ≤ i < j ≤ n (that is, labels v i and v j do not occupy the same vertex) and v i = v n + a for 1 ≤ a ≤ n − i (that is, the label v i does not occur within the n − i vertices clockwise of v n ).To count the vectors in the complement, first note that there are p ways to place the label v n .After this, we may place v 1 in (p − n) ways, since it must avoid the position of v n and the n − 1 positions just clockwise of this.Next, we may place v 2 in (p − n) ways since it must avoid the position of v n , the n − 2 positions just clockwise of this, and also the position of v 1 .Continuing in this way, we find that there are p (p − n) n−1 vectors in the complement.
The following is a standard result on real hyperplane arrangements.
Zaslavsky's Theorem (see, e.g., Theorem 2.5 of [27]).Let A be an arrangement in R d in which the intersection of all hyperplanes has dimension k.Then: Open Problem.Find a bijective proof of the corollary.
The observation that the Shi and Ish arrangements are (in some undefined sense) "dual" to each other is at the heart of this paper.

Two Statistics on Shi Chambers
Now we define two statistics -called shi and ish -on the chambers of a Shi arrangement (more generally, on the elements of the group S(n)).The first statistic is well known and the second is new.Each statistic counts a certain kind of inversions of an affine permutation, and so we begin by discussing inversions.
2.1.Affine Inversions.Let w be an element of the (finite) symmetric group S(n).If w(i) > w(j) for indices 1 ≤ i < j ≤ n we say that the tranposition (i, j) is an inversion of w -equivalently, this means that the hyperplane e i − e j = 0 separates the cone wC • from the dominant cone C • .The number of inversions of w is called its length.
In the affine symmetric group S(n), there is again a correspondence between hyperplanes and transpositions.Recall that the affine transpositions ((i, j)) and ((i , j )) coincide if i = i + kn and j = j + kn for some k ∈ Z, in which case they represent the same hyperplane (1.2).Hence, each affine transposition has a standard representative in the set Given an affine permutation w ∈ S(n) and an affine transposition ((i, j)) ∈ T such that w(i) > w(j), we say that ((i, j)) is an affine inversion of w -equivalently, the hyperplane (1.2) separates the alcove wA • from the fundamental alcove A • .Again, the (affine) length of w is its number of affine inversions.
2.2.The shi statistic.Each chamber of the Shi arrangement contains a set of alcoves and we will see (Theorem 2) that among these is a unique alcove of minimum length -which we call the representing alcove of the chamber, or just a Shi alcove.This defines an injection from Shi chambers into the affine symmetric group.Figure 2.1 displays the representing alcoves for Shi(3), labeled by affine permutations.We have labeled the Shi hyperplanes with their corresponding affine transpositions, Definition 2.1.Given a Shi chamber with representing alcove A, let shi(A) denote the number of Shi hyperplanes separating A from the fundamental alcove A • .Equivalently, if A = wA • for affine permutation w ∈ S(n), then shi( w) is the number of affine inversions ((i, j)) of w satisfying i < j < n + i.

2.3.
The ish statistic.To give a natural definition for our second statistic, we must discuss the quotient group S(n)/S(n).By abuse of notation, let S(n) denote the subgroup of S(n) generated by the subset We define the ish statistic in terms of minimal coset representatives.Two notes: In order to facilitate later generalization, we have defined ish in terms of all hyperplanes of the form e i − e n = a.In our current context, however, only the Ish hyperplanes (i.e. a ∈ {1, . . ., n − i}) will contribute.We also emphasize the fact that ish is a statistic on the (representing alcoves of ) Shi chambers, not on the Ish chambers.It seems that the chambers of the Ish arrangement are not so natural.
Theorems and a Conjecture.We will make four assertions and then describe our state of knowledge about them (i.e.whether each is a Theorem or a Conjecture).We will use the following notation.
Recall from (1.1) that DH(n; q, t) denotes the bigraded Hilbert series of the ring of diagonal harmonic polynomials.Define where the sum is taken over representing alcoves A for the chambers of the arrangement Shi(n).We say that an alcove is positive if it is contained in the dominant cone C • (i.e. if A is on the "positive" side of each generating hyperplane).Let Shi + (n; q, t) denote the corresponding sum over positive Shi alcoves.Finally, consider the standard q-integer, qfactorial, and q-binomial coefficient: Assertions.
In particular, note that q ( n 2 ) Shi + (n; q, 1/q) is equal to the sum of q shi(A)+ish(A) over the positive Shi alcoves A. For n = 3 we may compute this sum using the data in Figure 2.2 to obtain 1 + q 2 + q 3 + q 4 + q which is a q-Catalan number.One may check that the other three assertions are also true in the case n = 3.
In the following section we will establish a bijection (Theorem 6) from Shi chambers to labeled lattice paths, which sends our statistics (ish, shi− n 2 ) to the statistics (bounce, area ) of Haglund and Loehr [14].This allows us to clarify the Assertions.

Status.
The following results all depend on our main theorem (Theorem 6).
(1) Conjecture.This is equivalent to a conjecture of Haglund and Loehr [14] (known in a different form to Haiman).No combinatorial explanation of the q-t symmetry is known.(2) Theorem.This is equivalent to a theorem of Loehr [15].
(3) Theorem.This follows from theorems of Garsia and Haglund [7,8].No combinatorial explanation of the q-t symmetry is known.(4) Theorem.This is equivalent to a theorem of Haglund [12], which was later proved bijectively by Loehr [16].

Shi Chambers and Lattice Paths
In this section we will prove the above stated results regarding the shi and ish statistics.To do this we will interpret Shi chambers as certain labeled lattice paths.
The root poset is a partial order on Φ + defined as follows.Given two positive roots α, β ∈ Φ + we say that α ≤ β whenever β − α can be written in the basis ∆ using nonnegative coefficients -equivalently, we have α ≤ β when β − α is in the positive cone generated by ∆.In type A this means that e j − e k ≤ e i − e if and only if i ≤ j < k ≤ .
In this paper we will visualize the root poset in a particular way.Consider an array of integer points (i, j), 1 ≤ i < j ≤ n, and place the label "ij" in the unit square with top right corner (i, j).(See Figure 3.1.)This square will represent the root e i − e j .Thus for α, β ∈ Φ + we have α ≤ β when the square labeled β occurs weakly to the left and weakly above the square labeled α.
A set of roots I ⊆ Φ + is called an ideal if α ∈ I and α ≤ β together imply β ∈ I.We may picture this as a collection of unit squares aligned up and to the left.The lower boundary of these squares defines a lattice path from (0, 0) to (n, n) which • uses only steps of the form (0, 1) and (1, 0), and • stays weakly above the diagonal.This defines a bijection between ideals in Φ + and so-called Dyck paths.For example, Figure 3.1 displays an ideal in the root poset of S( 9) and its corresponding Dyck path.Shi's Theorem.A function k : Φ + → Z is the address of an alcove if and only if, for all triples α, β, α + β of positive roots, we have We say that the alcove A is positive if it lies in the dominant cone C • .Equivalently, A is positive if and only if its address k A takes non-negative values.We observe that the address of a positive alcove is an increasing function on the root poset.Indeed, if α ≤ β then β − α is a non-negative integer combination of simple roots.Morever, there exists a way to get from α to β by successively adding these simple roots, always staying within Φ + .Since we assumed that k A (γ) ≥ 0 for all simple γ ∈ ∆ ⊆ Φ + , the result follows from Shi's Theorem.

Positive Shi
Alcoves.The Shi arrangement consists of the hyperplanes H α,k for all α ∈ Φ + and k ∈ {0, 1}.Given an alcove A, we would like to understand in which chamber of the Shi arrangement it occurs.This problem is easiest to solve for positive alcoves; in this case we need only specify for which roots k A is zero and for which roots it is positive.To this end, we define Since the address of a positive alcove A is increasing, we observe in this case that I A ⊆ Φ + is an ideal in the root poset.It turns out that this defines a bijection between positive Shi chambers and ideals.For this result we refer to Sommers [25, Lemmas 5.1 and 5.2].
Theorem 2 (Representing Alcoves).Given an ideal I ⊆ Φ + of positive roots, there exists a unique alcove of minimum length such that I = I A .The address of this alcove is given by k I : Φ + → Z where k I (α) is the maximum number r such that α can be expressed as a sum of r roots in the ideal I.
We call the unique minimum alcove in a positive Shi chamber its representing alcove, or just a positive Shi alcove.Recall that a given positive Shi chamber C corresponds to an ideal I ⊆ Φ + of positive roots: given a positive root α = e i − e j , the chamber C lies on the positive side of H α,1 when α ∈ I and C lies between H α,0 and H α,1 when α ∈ I.In addition, the minimal roots α ∈ I (such that I − α is also an ideal) correspond exactly to the hyperplanes H α,1 that support a facet of the chamber and also separate it from the fundamental alcove A • .We call these the floors of the chamber.In the language of Dyck paths, these are the squares contained in the "valleys" of the path.For instance, the valleys in Figure 3.1 contain roots e 1 − e 4 , e 2 − e 6 , e 6 − e 7 and e 7 − e 9 .Now consider a non-positive Coxeter cone wC • , with w ∈ S(n).The Shi hyperplanes that intersect the dominant cone C • have the form e i − e j = 1 for 1 ≤ i < j ≤ n.The images of these hyperplanes in wC • have the form e w(i) − e w(j) = 1 for 1 ≤ i < j ≤ n, and such a plane is actually a member of the Shi arrangement exactly when w(i) < w(j).That is, the Shi planes that intersect wC • correspond to the non-inversions (i, j) of w ∈ S(n).If we then map a positive Shi alcove into wC • , it will remain a Shi alcove if and only if its floors continue to exist.In summary, we have the following.
Theorem 3 (Pak and Stanley [26]).The chambers of the Shi arrangement are in bijection with pairs (w, I) where w ∈ S(n) is a permutation and I ⊆ Φ + is an ideal of positive roots (a Dyck path) such that the minimal elements of I (labels in the valleys of the path) are non-inversions of w.Let us interpret the statistics shi and ish in terms of labeled Dyck paths.

n
2 − shi = area .In [14] Haglund and Loehr defined two statistics on labeled Dyck paths -called area and bounce -and they conjectured that the generating function q area t bounce equals the bigraded Hilbert series DH(n; q, t) of diagonal harmonic polynomials.
We first deal with area , which Haglund and Loehr defined as the number of noninversions of w below the labeled Dyck path (w, I).When w is the identity permutation, this is just the number of unit squares fully between the path and the diagonal, i.e. the "area" of the path.Proof.Recall that shi(A) is the number of Shi hyperplanes separating A from the fundamental alcove A • .These come in two classes.First, it is well known that the hyperplanes separating A • from wA • are exactly e i − e j = 0 such that 1 ≤ i < j ≤ n and w(i) > w(j).These are the ×'s in the diagram.Second, the hyperplanes of the form e i − e j = 1 separating A from wA • correspond to unit squares above the path.Such a hyperplane is a Shi hyperplane whenever w(i) < w(j), so these correspond to 's above the path.Since the total number of symbols is n 2 we conclude that n 2 − shi(A) is the number of 's below the path.
3.2.5.ish = bounce.The bounce statistic was discovered by Haglund in 2003 [12].It provided the first combinatorial interpretation of the q, t-Catalan numbers of Garsia and Haiman.Haglund and Loehr [14] later extended the statistic to labeled Dyck paths (w, I) by defining bounce(w, I) = bounce(I).Definition 3.1 (Haglund).Given a Dyck path I, we construct its bounce path as follows.Begin at (n, n) and travel left until we hit the path, then travel down until we hit the diagonal.Repeat these two steps until we hit (0, 0).Then bounce(I) is the sum of i between 1 and n − 1 such that the bounce path contains the diagonal point (i, i).
For example, the bounce path in Figure 3.4 is defined by the white vertices.The numbers along the bottom show that bounce for this path is 2 + 6 + 7 = 15.
Theorem 5. Given a Shi alcove A (positive or non-positive) and its corresponding labeled Dyck path (w, I) we have ish(A) = bounce(w, I).
Proof.Suppose that A = wA • where w is an affine permutation w ∈ S(n).Suppose further that w = w I wI where w I ∈ S(n) ⊆ S(n) is a finite permutation and wI is a minimal coset representative.The alcove A thus corresponds to a labeled Dyck path (w I , I) and the positive alcove wI A • corresponds to the "unlabeled" Dyck path (1, I).
Recall that ish(A) is the number of hyperplanes between wI A • and A • of the form e i −e n = a.Given α = e i −e n the number of these hyperplanes is exactly k A (α), where k A : Φ + → Z is the address of the positive Shi alcove wI A • .By Theorem 2, k A (α) = k I (α) is the maximum r such that α can be written as a sum of r roots in the ideal I (i.e.above the path I).For example, ish(A) is the sum of the entries in the top row of Figure 3.2.Now consider the bounce path of I and extend it to the left from each point that it hits I.This decomposes the collection of squares above the path into "blocks".We have done this in Figure 3.4; note here that there are 3 blocks.Given α ∈ Φ + , suppose that we have α = γ 1 + • • • + γ r where each γ i is above the path.In this case we can reorder the summands such that γ 1 + • • • + γ q is above the path for all 1 ≤ q ≤ r (see [25,Lemma 3.2]).In other words, if γ i = e i 1 − e i 2 , we must have i 2 = (i + 1) 1 for all 1 ≤ i ≤ r − 1.This means there can be at most one γ i from each block that intersects the column containing α.
If α = e i − e n , we claim that in fact k I (α) is equal to the number of blocks that intersect the ith column.Indeed, set γ 1 = e i − e j with j minimal such that e i − e j ∈ I. Thus γ 1 is in the lowest block below α.Then we travel right from γ 1 , bounce off the diagonal, and travel up until we reach γ 2 such that: γ 2 is in the block above the block containing γ 1 , and γ 2 is in the top row of this block.Continuing in this way, we will obtain α = γ 1 + • • • + γ r such that there is one summand from each block intersecting the ith column.Since this was an upper bound, the claim is proved.For example, in Figure 3 Finally, ish(A) is the sum of the values k I (e i − e n ) for 1 ≤ i ≤ n − 1.In other words, we sum over the number of blocks that intersect each column.This is the same as summing the number of squares in the top row of each block.Note also that there exists a block whose top row contains j squares if and only if the bounce path touches the diagonal at (j, j).We conclude that ish(A) = bounce(I).
In conclusion, here is the main result of the paper.Theorem 6.The bijection A → (w, I) from Shi alcoves to labeled Dyck paths sends the pair of statistics ( n 2 − shi, ish) to the pair (area , bounce).

The Inverse Statistics
We chose the definitions of shi and ish to emphasize their connection with the Ish hyperplane arrangement.However, we will obtain a more natural interpretation of ish when we compose it with inversion in the Weyl group.That is, let us define the following inverse statistics.First let us say why we care about the inverse statistics.
4.1.Inverse Shi Alcoves.Let E denote the set of representing alcoves for the chambers of the Shi arrangement Shi(n) (see Theorem 2).Thinking of these alcoves as elements of the affine symmetric group S(n) we may invert them.J.-Y.Shi showed that the set E −1 of inverted alcoves has a remarkable shape (see [23]).
Theorem 7. The inverted Shi alcoves E −1 are precisely the alcoves inside the simplex which is congruent to the dilation (n + 1)A • of the fundamental alcove A • .
Since the dimension of the space R n 0 is n − 1, the simplex D n+1 (n) contains (n + 1) n−1 alcoves.Shi concluded that his arrangement has (n + 1) n−1 chambers.Following Theorem 6, we assert that the joint-distribution of shi −1 and ish −1 on the simplex D n+1 (n) is the bigraded Hilbert series of diagonal harmonic polynomials.In fact, since the shape D n+1 (n) (Figure 4.1) is much nicer than the distribution of Shi alcoves (Figure 2.1), it seems that the inverse statistics shi −1 and ish −1 are more natural than the originals.Thus we would like to understand them directly, without reference to inversion in S(n).

The Inverse shi Statistic.
To do this we need to discuss the realization of the affine symmetric group S(n) as a semi-direct product of the finite symmetric group S(n) and the root lattice . By abuse of notation, we think of Q as an abelian group by associating the root r ∈ Q with the translation t r : R n Note in particular that inversion satisfies (wt r ) −1 = w −1 t −w −1 (r) .
We can now describe the inverse shi statistic explicitly.Theorem 8.For any affine permutation w ∈ S(n) we have Proof.First recall that the Shi arrangement Shi(n) consists of all the affine hyperplanes H α,k that touch the fundamental alcove A • .There are two of these perpendicular to each root α ∈ Φ + ; namely H α,0 and H α,1 .
The inversions of w ∈ S(n) are the affine hyperplanes H separating the alcoves wA • and A • .These biject under the map w−1 to the hyperplanes w−1 H separating the alcoves A • and w−1 A • .If w = wt r , note that wH α,k = H w(α),k+(r,α) .This implies that the inversions of w parallel to the root α biject to the inversions of w−1 parallel to the root w −1 (α).Finally, since every such set contains a unique Shi hyperplane (if it contains anything at all), and since the finite permutation w −1 is a bijection on the roots Φ = Φ + ∪ −Φ + , we conclude that w −1 defines a bijection from the Shi hyperplane inversions of w to the Shi hyperplane inversions of w−1 .Hence these two sets have the same cardinality.
For example, the inverse of the affine permutation [−2, 2, 6] is [4,2,0].Each of these affine permutations has 4 inversions, among which there are 3 Shi hyperplanes.In Section 5 we will need a more general version of this result, whose proof is the same.Theorem 9. Given an affine permutation w ∈ S(n), define its inversion partition (I 0 ≥ I 1 ≥ . ..) by letting I k denote the number of affine transpositions ((i, j)) such that w(j)− w(i) n = k.Then w and w−1 have the same inversion partition.

4.3.
The Inverse ish Statistic.Next we will compute a formula for the ish −1 statistic.We will find that ish −1 depends only on the root lattice Q.
To do this we need a lemma about the original ish statistic, which follows directly from Björner and Brenti [4, Lemma 4.2].The proof is instructive, so we reproduce it here.
So let ũ = u + nr, where u ∈ S(n) is a finite permutation and r ∈ Q is an element of the root lattice.Next fix an index 1 ≤ i ≤ n − 1 and consider the integer this number is always non-negative and it counts the pairs n < i + kn such that ũ(n Somehow, everything balances to create a simple formula.From this formula we get an expression for ish −1 . Theorem 10.Given an affine permutation w = w + nr, where w ∈ S(n) is a finite permutation and r ∈ Q is an element of the root lattice, choose the largest index j ∈ {1, . . ., n} such that the value of r(j) is a minimum.Then ish −1 ( w) = j + n(−r(j) − 1).
For example, Figure 2.2 displays the shi and ish −1 statistics on the simplex D 4 (3).(The darker shaded alcoves have positive inverses.)This is the inverse of Figure 4.2.
Finally, we wish to emphasize the following.The value of ish −1 ( w) depends only on the element r ∈ Q of the root lattice.(This is the analogue of the fact that ish( w) depends only on the minimal coset representative wI .)Combining this observation with Theorem 6, we conclude that Haglund's bounce statistic is really a statistic on the root lattice of type A.

Powers of Nabla
In this final section we will describe several ideas for future research, roughly in order of increasing generality.Most of this depends on the nabla operator ∇ of F. Bergeron and Garsia [3], which we will define first.5.1.The Nabla Operator.call a formal power series in Q[[x 1 , x 2 , . . ., ]] a symmetric function if it is invariant under permuting variables.Let Λ = ⊕ n≥0 Λ n denote the ring of symmetric functions, graded by degree.Then Λ n is isomorphic to the space of (virtual) representations of the symmetric group S(n) over Q.Under this isomorphism, the role of the irreducible representations is played by the basis of Schur functions s λ ∈ Λ n , one for each partition λ = (λ 1 ≥ λ 2 ≥ • • • ) of the integer n = i λ i .
If we extend the field of coefficients from Q to Q(q, t), another remarkable basis of Λ n is the set of modified Macdonald polynomials Hµ , where again µ = (µ 1 ≥ µ 2 ≥ • • • ) is an integer partition of n.Let ν(µ) := i≥1 (i − 1)µ i , and let µ be the conjugate partition defined by µ i = #{j ≥ 1 : µ j ≥ i}.Then the Bergeron-Garsia nabla operator is the unique Q(q, t)-linear map on Λ n defined by Hµ .
That is, the modified Macdonald polynomials are a basis of eigenfunctions for ∇.It turns out that many results on diagonal harmonics can be expressed elegantly in terms of ∇.
In particular, if is the Frobenius character of the diagonal harmonics.That is, if we replace each Schur function in ∇(e n ) by its dimension, we obtain DH(n; q, t).For details, see Haglund [13].
Now we suggest some ways to generalize our earlier results, which amount to new conjectural interpretations of the ∇ operator.5.2.Extended Shi Arrangements.Recall that the set of reflections in the affine Weyl group S(n) is T = {((i, j)) : 1 ≤ i ≤ n, i < j}.
The affine transposition ((i, j)) corresponds to the hyperplane H α,a where α = e rem(j,n) − e rem(i,n) and a = quo(j, n), and where remainder is taken in the set {1, . . ., n}.We will call j−i n = k the height of the hyperplane; this is some measure of how far the hyperplane is from the fundamental alcove.Recall that the Shi arrangement consists of the hyperplanes of height 0. We may now define the m-extended Shi arrangement.Athanasiadis proved [2, Proposition 3.5] that every chamber of Shi m (n) contains a unique alcove of minimum length.It seems true, thought we cannot find a reference, that the inverses of these representing alcoves are precisely the alcoves contained in the simplex D mn+1 (n) ⊆ R n 0 bounded by the hyperplanes Sommers showed in [25, proof of Theorem 5.7] that the simplex D mn+1 (n) is congruent to the dilation (mn + 1)A • of the fundamental alcove A • .Since this occurs in the (n − 1)dimensional quotient space R n 0 , the simplex D mn+1 (n) consists of (mn + 1) n−1 alcoves.We would like to extend the statistics shi and ish −1 to these alcoves.This turns out to be very easy to do.The shi statistic generalizes naturally, and the ish statistic needs no generalization at all.Definition 5.2.Given an affine permutation w ∈ S(n), let shi m ( w) denote the number of hyperplanes of Shi m (n) separating wA • from the fundamental alcove A • .
We wish to study the joint-distribution of shi m and ish on the minimal alcoves in the arrangement Shi m (n).By Theorem 9, this is the same as the joint-distribution of shi m and ish −1 on the alcoves of the simplex D mn+1 (n).
Conjecture 1.Consider the following generating function for shi m and ish −1 over alcoves in the dilated simplex D mn+1 (n), Shi m (n; q, t) := A⊆D mn+1 (n) and let Shi m + (n; q, t) denote the same sum over alcoves whose inverses are in the dominant cone C • .We conjecture the following: (1) Shi m (n; q, t) is the Hilbert series of ∇ m (e n ).
For example, let m = 2 and n = 3. Figure 5.1 displays the statistics shi 2 and ish −1 on the alcoves of D 7 (3), and Table 1 displays the corresponding generating functions.One may observe that all four assertions hold in this case.Table 1.The generating functions Shi 2 (3 : q, t) and Shi 2 + (3; q, t) Positive powers of ∇ have been well-studied.We believe that a suitable extension of our main Theorem 6 is possible, which would make our conjectures equivalent to earlier conjectures of Haiman, Loehr and Remmel (see [17]), which are based on lattice paths from (0, 0) to (mn, n) that stay weakly above the diagonal y = x/m.5.3.Bounded Chambers.While positive powers of ∇ have been investigated by several authors, to our knowledge there has been no combinatorial conjectures for negative powers of ∇.In this section we will provide one.
As mentioned earlier, Athanasiadis showed that each chamber of Shi m (n) contains a unique alcove of minimum length.In the case m = 1, Sommers showed [25, Lemmas 5.1 and 5.2] that, moreover, every bounded chamber of Shi m (n) contains a unique alcove of maximum length.We believe that this is true for general m ≥ 1, and furthermore we believe that the inverses of these alcoves are precisely the alcoves contained in the simplex As with D mn+1 (n) above, Sommers has shown that D mn−1 (n) is congruent to the dilation (mn − 1)A • of the fundamental alcove, which implies that D mn−1 (n) contains (mn − 1) n−1 alcoves.We wish to study the statistics shi m and ish −1 on these alcoves.
Conjecture 2. Consider the following generating function for shi m and ish −1 over alcoves in the dilated simplex D mn−1 (n), and let Shi −m + (n; q, t) denote the same sum over alcoves whose inverses are in the dominant cone C • .We conjecture the following.
For example, Figure 5.2 displays the statistics shi 2 and ish −1 on the simplex D 5 (3), and Table 2 displays the corresponding generating functions.One may observe that all four assertions hold for this data.Combining Conjectures 1 and 2, we obtain a conjectural combinatorial interpretation for all integral powers of the nabla operator acting on e n .We wonder whether Athanasiadis' result [1] on Ehrhart reciprocity for Shi arrangements may reflect some sort of reciprocity theorem for the nabla operator.5.4.Interpolation.Since the forms of Conjectures 1 and 2 are so similar, one may ask whether there is a more general form encompassing them both.In this case we do not have a concrete conjecture, but we we will suggest some ideas.
The simplices D mn+1 (n) and D mn−1 (n) are both special cases of the following construction of Sommers.
Recall that the root system of type A n−1 is defined by Φ = {e i − e j : 1 ≤ i, j ≤ n}, and the basis of simple roots is ∆ = {e i − e i+1 : 1 ≤ i ≤ n − 1}.Given a root α ∈ Φ, let b denote the sum of its coefficients in the simple root basis; we say that b is the height of the root α.As with D mn+1 (n) and D mn−1 (n), Sommers showed that for any p coprime to n, D p (n) is congruent to the dilation pA • of the fundamental alcove; that it contains p n−1 alcoves; and that it contains 1 p+n p+n n alcoves whose inverses are in the dominant cone.We suggest the following: Open Problem.Define a statistic stat on the alcoves of D p (n).Consider the generating function F (p, n; q, t) := A⊆D p (n) q ish −1 (A) t (p−1)(n−1)/2−stat(A) , and let F + (p, n; q, t) denote the same sum over alcoves whose inverses lie in the dominant cone C • .These generating functions should satisfy (1) F (p, n; q, t) = F (p, n; t, q).
Note that ish −1 does not need to be modified.It is the shi statistic that is difficult to define in general.We note that the smallest mystery case is p = 2 and n = 5, which corresponds to the 4-dimensional simplex D 2 (5) in  [0, 2, 3, 4, 6] and [2,0,3,6,4].The distribution of ish −1 over the former 16 is q ish −1 (A) = 10 + 5q + q 2 and the distribution of ish −1 over the latter 3 is q ish −1 (A) = 1 + q + q 2 .We do not know what the analogue of shi is in this case, but we have checked that it cannot simply be the number of inversions coming from some special set of hyperplanes.5.5.Other Types.In this paper we have focused on the affine Weyl group of type Ãn , which is the group S(n) of affine permutations.However, we have tried to use language throughout that is general to all affine Weyl groups.Certainly, the combinatorics of Shi arrangements is completely general.Also, Sommers' simplexD p (h) is defined in general for any integer p coprime to the Coxeter number h.
Haiman observed that the most obvious generalization of the ring of harmonic polynomials to other types is "too large" (see [11,Section 7]), and he conjectured that some suitable quotient should be considered instead.Using rational Cherednik algebras, Gordon [9] was able to construct such a quotient.Gordon and Griffeth [10] have now observed that this module does have a suitable bigrading and it satisfies many of the desired combinatorial properties.Gordon-Griffeth [10] and Stump [28] have both defined q, t-Catalan numbers in general type (even in complex types), however their numbers disagree in the non-well-generated complex types.This is an active area, and we wish to emphasize: as of this writing, there is no known combinatorial interpretation for these objects beyond type A.
We suggest that the shi statistic on the simplex D h+1 (h) is a good place to start.The next step is to define an analogue of the ish −1 statistic.Unfortunately, we have checked that in type B 2 it cannot simply be a statistic on the root lattice.

Figure 1 .
Figure 1.1 displays the arrangements Cox(3), Shi(3), and Aff(3) in R 3 0 , with the dominant cone and fundamental alcove shaded.The Shi arrangement was introduced by Jian-Yi Shi

Figure 1 .
Figure 1.2 displays the arrangements Shi(3) and Ish(3).Note that each has 16 chambers and 4 bounded chambers.There is an important reason for this: the arrangements Shi(n) and Ish(n) share the same characteristic polynomial, as we now show.

Definition 2 . 2 .
Consider a Shi chamber with representing alcove A and suppose that A = wA • .Its minimal coset representative wI A • is an alcove in the dominant cone C • .Let ish(A) denote the number of hyperplanes of the form e i − e n = a (with 1 ≤ i ≤ n − 1 and a ∈ Z) separating wI A • from the fundamental alcove A • .Equivaently, let ish( w) denote the number of affine inversions of wI of the form ((n, j)).

3. 2 .
Shi Alcoves.3.2.1.The Address of an Alcove.For each root α ∈ Φ + and each real number k ∈ R let H α,k denote the hyperplane {x ∈ R n 0 : (x, α) = k}.When α = e i − e j this is the hyperplane e i − e j = k.Now let A be an alcove of the affine arrangement.For each root α ∈ Φ + there exists a unique integer k A (α) such that A lies between the hyperplanes H α,k A (α) and H α,k A (α)+1 .The function k A : Φ + → Z uniquely specifies the position of A, so we call it the address of A. An important result of J.-Y.Shi characterizes which functions can be addresses (see Sommers [25, Proposition 4.1], which is a restatement of J.-Y.Shi [22, Theorem 5.2]).

Figure 3 .
2 displays the address of the representing alcove corresponding to the ideal in Figure 3.1.

Theorem 4 . 2 −
Given a Shi alcove A (positive or non-positive) and its corresponding labeled Dyck path (w, I) we have n shi(A) = area (w, I).
Figure 4.1 displays the simplex D 4 (3) and the Shi arrangement in R 3 0 .We have labeled each alcove by the inverse of its corresponding affine permutation.Compare to Figure 2.1.

Figure 5 . 1 .
Figure 5.1.The shi 2 and ish −1 statistics on D 7 (3) It was shown by Edelman and Reiner [5, Section 3] and by Postnikov and Stanley [20, Proposition 9.8] that the characteristic polynomial of the m-extended Shi arrangement is