2.1 Macdonald polynomials

Given a symmetric function f, we will adopt the usual plethysm notation of f[X] when X is an element of some $\lambda$ -ring, so that $f[x_1+\cdots+x_N]$ is the substitution $f(x_1,\ldots ,x_N)$ . If $X=(x_1,x_2,\ldots )$ is some alphabet, we will use the same letter X to denote the sum in plethystic formulas. For details, we refer the reader to [Reference HaimanHai01b].

Let $\tilde{H}_\lambda=\tilde{H}_\lambda(X;q,t)$ denote the modified Macdonald polynomial [Reference Bergeron, Garsia, Haiman and TeslerBGHT99], defined by

\[\tilde{H}_\lambda(X;q,t)=t^{n(\lambda)} J_{\lambda}[X/(1-t^{-1});q,t^{-1}].\]

Alternatively, they are uniquely characterized by the following axioms [Reference Haglund, Haiman and LoehrHHL05b]:

1. (1) they are orthogonal with respect to the modified Macdonald inner product

   (2) \begin{equation}(p_\lambda,p_\mu)_*=(p_\lambda[X], \omega p_\mu[(1-q)(1-t)X]) = \delta_{\lambda,\mu} \mathfrak{z}_\lambda(-1)^{|\lambda|-l} \prod_{i=1}^{l} (1-q^{\lambda_i})(1-t^{\lambda_i}),\end{equation}
   where l is the length of $\lambda$ and $\mathfrak{z}_\lambda =(p_\lambda, p_\lambda)$ , $\omega$ is the involution $\omega p_\lambda = (-1)^{|\lambda|-l} p_\lambda$ ;

2. (2) they are triangular in the modified Schur basis

   (3) \begin{equation}\tilde{H}_\lambda[X;q,t]= \sum_{\mu \trianglelefteq \lambda'} a_{\lambda',\mu}(q,t) s_{\mu}[X(1-q)^{-1}]\end{equation}
   and similarly with t in place of q;

3. (3) $(\tilde{H}_\lambda,h_n)=1$ .

Now let $\nabla$ be the Garsia–Haiman–Bergeron–Tesler operator

(4) \begin{equation}\nabla \tilde{H}_\lambda(X;q,t)=q^{n(\lambda')} t^{n(\lambda)} \tilde{H}_\lambda(X;q,t),\end{equation}

where

\[n(\lambda)=\sum_{i} (i-1)\lambda_i\]

is the usual statistic from Macdonald’s book [Reference MacdonaldMac98]. In this paper, $\nabla$ will always denote an operator applied to the X variables.

2.2 Cauchy identities

The standard identities

\[\sum_{n\geq 0} h_n[X] = \exp\left(\sum_{n=1}^\infty \frac{p_n[X]}{n}\right) = \sum_{n\geq 0} \sum_{\lambda\vdash n}\frac{p_\lambda[X]}{\mathfrak{z}_\lambda}\]

imply

\[h_n[X] = \sum_{\lambda\vdash n}\frac{p_\lambda[X]}{\mathfrak{z}_\lambda}.\]

Using $(p_\lambda, p_\lambda) = \mathfrak{z}_\lambda$ and $p_k[XY]=p_k[X] p_k[Y]$ this implies

\[h_n[XY] = \sum_{\lambda\vdash n}\frac{p_\lambda[X] p_\lambda[Y]}{(p_\lambda, p_\lambda)}.\]

By a standard argument, for any orthogonal basis $(f_\lambda)_{\lambda\vdash n}$ of symmetric functions of degree n, for instance for Schur functions, we obtain what we call the Cauchy identity:

\[h_n[XY] = \sum_{\lambda\vdash n}\frac{f_\lambda[X] f_\lambda[Y]}{(f_\lambda, f_\lambda)}.\]

Similarly, for the modified scalar product we have

\[e_n\left[\frac{XY}{(1-q)(1-t)}\right] = \sum_{\lambda\vdash n} \frac{p_\lambda[X] p_\lambda[Y]}{(p_\lambda, p_\lambda)_*},\]

which implies that the same identity holds for any basis of symmetric functions of degree n which is orthogonal with respect to the modified scalar product. In particular, for the Macdonald basis we have

(5) \begin{equation}e_n\left[\frac{XY}{(1-q)(1-t)}\right] = \sum_{\lambda\vdash n} \frac{\tilde H_\lambda[X;q,t] \tilde H_\lambda[Y;q,t]}{(\tilde H_\lambda, \tilde H_\lambda)_*}.\end{equation}

Suppose L is an interesting operator on the space of symmetric functions of degree n, for instance $L=\nabla^k$ . Then we have, for any symmetric function g of degree n,

\[(L g)[X] = (-1)^n \left(L_X e_n\left[\frac{XY}{(1-q)(1-t)}\right], g[Y]\right)_*^Y,\]

where the subscripts/superscripts X, respectively Y, indicate that we apply the operator/scalar product with respect to the alphabet X, respectively Y. So computing the expression $L_Xe_n\left[{XY}/({(1-q)(1-t)})\right]$ allows one to compute the operator L applied to any function simply by taking the scalar product. Equivalently, we may write using the standard scalar product

(6) \begin{equation}(L g)[X] = (-1)^n \left(L_X e_n\left[\frac{XY}{(1-q)(1-t)}\right], g[-Y(1-q)(1-t)]\right)_Y.\end{equation}

2.3 Combinatorial definitions

Fix n and define a label to be an n-tuple of positive integers ${\mathbf{a}} =(a_1,\ldots ,a_n)$ with $a_i\geq 1$ . We will write $\mathrm{labs}(n)$ for the set of all labels of length n, and will also call the individual $a_i$ labels. For any label ${\mathbf{a}}$ , we have a multiset $A=A({\mathbf{a}})=(|A|,m_A)$ where $|A|=\{a_1,\ldots ,a_n\}$ is the total set, and $m_A:|A|\rightarrow \mathbb{Z}_{\geq 1}$ is the multiplicity. We define a (strict) composition of n

\[\alpha({\mathbf{a}})=(\alpha_1,\ldots ,\alpha_{l}),\quad|A|=\{c_1<\cdots < c_l\},\quad \alpha_i=m_A(c_i).\]

In other words, $\alpha({\mathbf{a}})$ is the result of sorting ${\mathbf{a}}$ in increasing order, and reading off the sizes of the groups, for instance

\[\alpha((1,1,1,4,4,2,1,4))=(4,1,3).\]

The multiset in this example would be $A=\{1^4,2,4^3\}$ . We may also define the corresponding partition $\mu({\mathbf{a}})=\mu(\alpha({\mathbf{a}}))$ which is the result of sorting $\alpha({\mathbf{a}})$ in decreasing order, so $\mu({\mathbf{a}})=(4,3,1)$ in the above example. Given a multiset A, let $\mathrm{labs}(A)$ denote the set of labels ${\mathbf{a}}$ with $A({\mathbf{a}})=A$ , with similar definitions for $\mathrm{labs}(\alpha)$ and $\mathrm{labs}(\mu)$ .

If $\mathcal{A},\mathcal{B},\ldots $ are totally ordered sets, we define the ordering on $\mathcal{A}\times \mathcal{B}\times \cdots $ as the corresponding lexicographic order, breaking ties from left to right. If ${\mathbf{a}} \in \mathcal{A}^n$ , ${\mathbf{b}} \in \mathcal{B}^n,\ldots$ are some elements, we define $[{\mathbf{a}},{\mathbf{b}},\ldots ]$ to be the sorted representative of the simultaneous action of $S_n$ on all components. In other words, view $({\mathbf{a}},{\mathbf{b}},\ldots ,)$ as a matrix, transpose the matrix, sort according to the order on $\mathcal{A}\times \mathcal{B}\times \cdots $ , and transpose back. For instance, in the case ${\mathbf{a}}\in \mathcal{A}^n$ , ${\mathbf{b}}\in \mathcal{B}^n$ for $\mathcal{A}=\mathcal{B}=\mathbb{Z}_{\geq 1}$ , we have

\[[(1,2,1,1,2,1),(3,2,3,1,1,3)]=((1,1,1,1,2,2),(1,3,3,3,1,2)).\]

We can then define $\alpha({\mathbf{a}},{\mathbf{b}},\ldots )$ using the same rules as above, so in the above example $\alpha({\mathbf{a}},{\mathbf{b}})=(1,3,1,1)$ . We make a similar definition for $\mu$ , which also applies when the sets are unordered.

2.4 The dinv statistic

Let ${\mathbf{a}},{\mathbf{b}}$ be labels, let ${\mathbf{m}} \in \mathbb{Z}_{\geq 0}^n$ , with the decreasing order on the $m_i$ , so that a triple $[{\mathbf{m}},{\mathbf{a}},{\mathbf{b}}]$ means one sorted as in the following way.

1. (1) $m_i \geq m_j$ ;

2. (2) if $m_i=m_j$ then $a_i\leq a_j$ ; and

3. (3) if $m_i=m_j$ and $a_i=a_j$ then $b_i\leq b_j$ .

For instance,

\[[(1,0,1,0),(2,1,1,1),(1,2,2,1)]=((1,1,0,0),(1,2,1,1),(2,1,1,2)).\]

We will often write such lists as arrays, as in Example 2.5 below.

We now define a statistic $\mathrm{dinv}_k({\mathbf{m}},{\mathbf{a}},{\mathbf{b}})$ on triples which are sorted according to Definition 2.1.

Definition 2.2. Let ${\mathbf{a}},{\mathbf{b}}\in \mathrm{labs}(n)$ , and suppose that $({\mathbf{m}},{\mathbf{a}})$ are sorted, although $({\mathbf{m}},{\mathbf{a}},{\mathbf{b}})$ may not be. We define

(7a) \begin{equation} \mathrm{dinv}_k({\mathbf{m}},{\mathbf{a}},{\mathbf{b}})=\sum_{i<j} \mathrm{dinv}^{i,j}_k({\mathbf{m}},{\mathbf{a}},{\mathbf{b}}),\end{equation}

where

(7b) \begin{equation}\mathrm{dinv}_k^{i,j}({\mathbf{m}},{\mathbf{a}},{\mathbf{b}})=\max\left(m_j-m_i-1+k+\delta(a_i>a_j)+\delta(b_i>b_j),0\right),\end{equation}

and $\delta(a_1>a_2)$ is one if $a_1>a_2$ , zero otherwise.

We similarly define $\mathrm{dinv}_k({\mathbf{m}},{\mathbf{a}})$ as the result of removing $\delta(b_i>b_j)$ , which is the same as setting ${\mathbf{b}}=(1^n)$ by default.

Recall that a Dyck path is a path of North and East steps in the $n\times n$ grid beginning at the origin (0,0), placed in the South-West, or lower left corner, and ending at (n,n), which never goes below the diagonal. It is determined uniquely by the set

\[D(\pi)=\left\{(i,j) : \text{$1\leq i<j\leq n \, $is between the path and the diagonal}\right\}.\]

Definition 2.3. Fix $k\geq 0$ , suppose $({\mathbf{m}},{\mathbf{a}})$ is sorted, and let $i<j$ . We will say that i k-attacks j (or just attacks) if

\[m_j-m_i-1+k+\delta(a_i>a_j)\geq 0.\]

In other words, i k-attacks j if switching the order of $b_i,b_j$ has an effect on $\mathrm{dinv}_k$ . For instance, for $k=1$ we have that i attacks j if:

1. (1) $m_i=m_j+1$ and $a_i>a_j$ ; or

2. (2) $m_i=m_j$ and $a_i\leq a_j$ .

We now have that

(8) \begin{equation}\mathrm{dinv}_k({\mathbf{m}}, {\mathbf{a}}, {\mathbf{b}}) =\mathrm{dinv}_k({\mathbf{m}},{\mathbf{a}})+\mathrm{inv}_{\pi_k({\mathbf{m}},{\mathbf{a}})}({\mathbf{b}}),\end{equation}

where

(9) \begin{equation} \mathrm{inv}_\pi({\mathbf{b}}) = \#\{(i,j)\in D(\pi) : b_i>b_j\}.\end{equation}

To see this, we just check that, for any $i<j$ , we have

\[\mathrm{dinv}_k^{i,j}({\mathbf{m}},{\mathbf{a}},{\mathbf{b}})-\mathrm{dinv}^{i,j}_k({\mathbf{m}},{\mathbf{a}})=\begin{cases}1, & (i,j)\in D(\pi_k({\mathbf{m}},{\mathbf{a}})), b_i>b_j, \\0, & \mbox{otherwise.}\end{cases}\]

Example 2.5. Let ${\mathbf{m}},{\mathbf{a}}$ be given in array notation by

\[\left(\begin{array}{c|cccccc}{\mathbf{m}}&3&3&3&2&0&0\\{\mathbf{a}}&1&1&5&4&2&5\\\end{array}\right),\]

which is a sorted term for $n=6$ . Then we find that $\pi_2({\mathbf{m}},{\mathbf{a}})$ is the Dyck path given in Figure 1, as the attacking pairs are the elements of $D(\pi)$ .

Figure 1: A Dyck path of size $(6,\,6)$ with area sequence $a(\pi)=(0,\, 1,\, 2,\, 3,\, 1,\, 1)$ , and $D(\pi)\,=\{(1,2), (1,3),(1,4),(2,3),(2,4),(3,4),(4,5),(5,6)\}$ .

2.5 Examples

A sum over all ${\mathbf{a}}$ will mean the infinite sum over all labels, unless some upper bound is specified, $a_i\leq N$ . We will adopt a convenient convention that a sum over $[{\mathbf{a}},{\mathbf{b}},\ldots ]$ means a sum over orbits with respect to the diagonal $S_n$ action, with the assumption that $({\mathbf{a}},{\mathbf{b}},\ldots )$ is the sorted representative in the summand. We will also allow for some summands in which only some of the summands are grouped, which means that just those terms are sorted. For instance, the symbol

\[\sum_{[{\mathbf{a}},{\mathbf{b}}],{\mathbf{c}},[{\mathbf{d}}]}\cdots\]

indicates the sum over quadruples $({\mathbf{a}},{\mathbf{b}},{\mathbf{c}},{\mathbf{d}})$ so that for every $i<j$ we have $a_i\leq a_j$ , $b_i\leq b_j$ if $a_i=a_j$ , $d_i\leq d_j$ , and there are no constraints on ${\mathbf{c}}$ . We also define automorphism factors for the orbits

\[\mathrm{aut}({\mathbf{a}},{\mathbf{b}},\ldots )=\prod_i \mu_i!,\quad \mathrm{aut}_q({\mathbf{a}},{\mathbf{b}},\ldots )=\prod_i [\mu_i]_q!,\]

where $\mu=\mu({\mathbf{a}},{\mathbf{b}},\ldots )$ , and

\[[k]_q=1+q+\cdots q^{k-1},\quad[k]_q!=\prod_{j=1}^k [j]_q,\]

are the q-number and q-factorial.

We give some examples in symmetric functions. Let

\[X_{\mathbf{a}}=x_{a_1}\cdots x_{a_n}=\prod_{a\in A} x_a^{m(a)}\]

be the associated monomial to ${\mathbf{a}}$ , where (A,m) is the associated multiset.

Example 2.6. The complete and monomial symmetric functions are given by

\begin{eqnarray*}h_n(x_1,x_2,\ldots )&=&\sum_{{\mathbf{a}}} \frac{1}{\mathrm{aut}({\mathbf{a}})} X_{{\mathbf{a}}}=\sum_{[{\mathbf{a}}]} X_{{\mathbf{a}}}=\sum_{\mu}m_\mu(x_1,x_2,\ldots ),\\m_{\mu}(x_1,x_2,\ldots )&=&\frac{1}{\mu_1!\cdots \mu_l!}\sum_{\mu({\mathbf{a}})=\mu} X_{{\mathbf{a}}}.\end{eqnarray*}

We also have the quasi-symmetric monomials defined by

\[m_\mu=\sum_{\mu(\alpha)=\mu} M_\alpha,\quad M_\alpha(x_1,x_2,\ldots )=\frac{1}{\alpha_1!\cdots \alpha_l!} \sum_{\alpha({\mathbf{a}})=\alpha}X_{{\mathbf{a}}}.\]

Example 2.7. We have

\[e_n\left[\frac{X}{1-q}\right]=\sum_{[{\mathbf{a}}]} \frac{q^{n(\mu({\mathbf{a}})')}}{(1-q)^n \mathrm{aut}_q({\mathbf{a}})} X_{{\mathbf{a}}},\]

which can be deduced from the Cauchy product,

\[e_n[XY]=\sum_{\mu} e_\mu[Y] m_{\mu}[X]\]

at $Y=(1-q)^{-1}$ and the well-known specializations for $e_k(1,q,\ldots )$ . Replacing $e_n$ with $h_n$ simply removes the $q^{n(\mu({\mathbf{a}})')}$ factor.

Example 2.8. A more involved example has the same form as our main theorem, but without the nabla operator:

(10) \begin{equation}e_n\left[\frac{XY}{(1-q)(1-t)}\right]=\sum_{[{\mathbf{m}},{\mathbf{a}},{\mathbf{b}}]} \frac{t^{|{\mathbf{m}}|}q^{n(\mu({\mathbf{a}})')}}{(1-q)^n \mathrm{aut}_q({\mathbf{m}},{\mathbf{a}},{\mathbf{b}})}X_{{\mathbf{a}}}Y_{{\mathbf{b}}}.\end{equation}

This can be deduced by successively applying the methods of the previous example. For instance, we would have

\[e_n\left[\frac{XY}{(1-q)(1-t)}\right]=\sum_{\mu} e_\mu\left[\frac{X}{(1-q)(1-t)}\right]m_{\mu}[Y].\]

We then apply the same expansion to each of the $e_{\mu_i}[\ldots]$ factors, extracting the expressions of X, followed by $(1-t)^{-1}$ . Applying

\[m_{\mu}[(1-t)^{-1}]=m_\mu(1,t,\ldots ) =\sum_{[{\mathbf{m}}]=\mu} t^{|{\mathbf{m}}|}\]

and the above expression for $e_n[(1-q)^{-1}]$ , we arrive at (10).

3.1 Main theorem

Recall the conventions for summations of sorted representatives described in § 2.5. We have our main theorem.

Theorem 3.1. For any $k\geq 1$ , we have

(11) \begin{equation}\nabla^k e_n\left[\frac{XY}{(1-q)(1-t)}\right]= \sum_{[{\mathbf{m}}, {\mathbf{a}}, {\mathbf{b}}]} \frac{t^{|{\mathbf{m}}|}q^{\mathrm{dinv}_k({\mathbf{m}}, {\mathbf{a}},{\mathbf{b}})}}{(1-q)^n \mathrm{aut}_q({\mathbf{m}},{\mathbf{a}},{\mathbf{b}})}X_{{\mathbf{a}}} Y_{{\mathbf{b}}}.\end{equation}

Notice that we may recover $\nabla^k$ as an operator from (11) using

(12) \begin{equation} \nabla^k f = (-1)^{n} (\Omega_k[X,Y],f[-(1-q)(1-t)Y])_Y,\end{equation}

where $\Omega_k[X,Y]$ denotes the expression on the right-hand side of (11), and n is the homogeneous degree of f.

Before proving Theorem 3.1, we state a few consequences. Let

(13) \begin{equation}\xi_{\pi}[Y;q]=\sum_{{\mathbf{b}}} q^{\mathrm{inv}_{\pi}({\mathbf{b}})}Y_{{\mathbf{b}}},\end{equation}

where $\pi$ is a Dyck path, and $\mathrm{inv}_{\pi}({\mathbf{b}})$ is defined in § 2.4.

Proposition 3.2. The right-hand side of (11) is given by

\[\sum_{[{\mathbf{m}},{\mathbf{a}}]}\frac{t^{|{\mathbf{m}}|} q^{\mathrm{dinv}_k({\mathbf{m}}, {\mathbf{a}})}}{(1-q)^n \mathrm{aut}_q({\mathbf{m}},{\mathbf{a}})}X_{{\mathbf{a}}} \xi_{\pi_k({\mathbf{m}},{\mathbf{a}})}[Y;q].\]

We have the following interpretation of $\xi_{\pi}[Y;q]$ . Let

(14) \begin{equation}\mathcal{X}_{\pi}[Y;q]=\sum_{{\mathbf{b}}:(i,j)\in D(\pi)\Rightarrow b_i\neq b_j}q^{\mathrm{inv}_\pi({\mathbf{b}})}Y_{{\mathbf{b}}}\end{equation}

be Stanley’s chromatic symmetric function.

Proposition 3.3. We have that

(15) \begin{equation} \xi_{\pi}[Y;q]=(1-q)^n\omega\mathcal{X}_{\pi}[Y(1-q)^{-1};q].\end{equation}

In particular, it is a symmetric function.

In particular, since we have already shown that $\Omega_k[X,Y]$ is a symmetric function in the Y-variables in Proposition 3.3, it is symmetric in the X variables as well.

Proof. It suffices to check that the bijection $[{\mathbf{m}},{\mathbf{a}},{\mathbf{b}}]\leftrightarrow[{\mathbf{m}},{\mathbf{b}},{\mathbf{a}}]$ , which is well defined on the collection of diagonal orbits, preserves the $\mathrm{dinv}_k$ statistic. For this, we may assume that ${\mathbf{m}}=(0,\ldots ,0)$ , since the $\mathrm{dinv}_k$ pairs between different blocks of ${\mathbf{m}}$ is not changed by this operation. The symmetry follows from

\[\mathrm{dinv}_k((0^n),{\mathbf{a}},{\mathbf{b}})=\#\left\{i<j: \mbox{$a_i>a_j$ and $b_i<b_j$}\right.\left.\mbox{or $a_i<a_j$ and $b_i>b_j$} \right\},\]

for sorted pairs $[{\mathbf{a}},{\mathbf{b}}]$ , which is unchanged by simultaneous reordering. We may also see the symmetry using Proposition 5.4 below.

As a corollary, we have the expression for $(\nabla^k e_1^n,e_n)$ from [Reference Elias and HogancampEH16]. It was later proved in [Reference Gorsky and HogancampGH22], where it was shown that both sides equal the Poincaré polynomial for the Khovanov–Rozansky knot homology of kth power of the full twist, and also the Hilbert series for the kth power of Haiman’s alternant ideal $J_n^k$ . Here, $J_n \subset\mathbb{C}[{\mathbf{x}},{\mathbf{y}}]$ is defined as the ideal generated by the alternating elements under the diagonal action.

Corollary 3.6. We have that

(16) \begin{equation} \frac{1}{M^n}(\nabla^k e_1^n,e_n)=\frac{1}{M^n} (\nabla^{k+1} e_1^n,h_n)=\frac{1}{(1-q)^n}\sum_{{\mathbf{m}}}t^{|{\mathbf{m}}|}q^{d_k({\mathbf{m}})},\end{equation}

where $M=(1-q)(1-t)$ , ${\mathbf{m}} \in \mathbb{Z}_{\geq 0}$ ranges over all compositions (not just the sorted ones), and

\[d_k({\mathbf{m}})=\sum_{i<j} \max(k-m_i+m_j,k+1-m_j+m_i).\]

Proof. Taking $f=e_1^n$ and applying (12), we obtain that

\[[X_{{\mathbf{a}}}] \Omega_k[X,Y]=\frac{1}{M^n}\nabla^k e_1^n,\]

where $[X_{\mathbf{a}}]$ is the coefficient for ${\mathbf{a}}=(1,\ldots ,n)$ . We then obtain that

\[[X_{{\mathbf{a}}} Y_{1^n}] \Omega_k[X,Y]=\frac{1}{M^n} (\nabla^k e_1^n,h_n).\]

Notice now that the compositions ${\mathbf{m}}\in \mathbb{Z}^n$ are in bijection with sorted pairs $[{\mathbf{m}}',{\mathbf{a}}]$ when ${\mathbf{a}}$ has distinct elements. The second equality of (16) then follows from checking that $d_k({\mathbf{m}})=\mathrm{dinv}_{k+1}({\mathbf{m}}',{\mathbf{a}})$ when $({\mathbf{m}}',{\mathbf{a}})$ is sorted, and ${\mathbf{m}}={\mathbf{m}}'|_{\sigma^{-1}}\in \mathbb{Z}_{\geq 0}^n$ for $\sigma=\mathrm{Std}({\mathbf{a}})$ .

The first equality is simple identity in $\nabla$ which holds for all symmetric functions in place of $e_1^n$ . It follows from the formulas

\[(\tilde{H}_{\lambda},h_n)=1,\quad(\tilde{H}_{\lambda},e_n)= q^{n(\lambda')}t^{n(\lambda)}.\]

Then it is also the same as taking the coefficient of $X_{{\mathbf{a}}}$ and $Y_{{\mathbf{b}}}$ for ${\mathbf{a}}=(1,\ldots ,n)$ and ${\mathbf{b}}=(1,\ldots ,1)$ on the right-hand side. Since the entries of ${\mathbf{a}}$ are distinct, there is no automorphism factor. Now notice that the compositions ${\mathbf{m}}$ are in bijection with sorted pairs $[{\mathbf{m}}',{\mathbf{a}}]$ where ${\mathbf{a}}$ has distinct elements, meaning it is a permutation, and that $d_k({\mathbf{m}})=\mathrm{dinv}_k({\mathbf{m}}',{\mathbf{a}})$ .

3.2 Proof of Theorem A

As above, let $\Omega_k[X,Y]$ denote the expression on the right-hand side of equation (11). Since we know that $\Omega_k[X,Y]$ is a symmetric function in each set of variables, we may define an operator $\nabla'_{k}$ on symmetric functions by

(17) \begin{equation}(\nabla'_{k} f)[X]=(-1)^{n} (f[-Y(1-q)(1-t)],\Omega_k[X,Y])_Y.\end{equation}

Then, in light of the expression (12), Theorem 3.1 is equivalent to the statement that $\nabla'_k=\nabla^k$ . We will give our first proof of Theorem 3.1 by verifying that $\nabla'_k$ satisfies the defining properties of $\nabla^k$ , similar to the approach in [Reference Haglund, Haiman and LoehrHHL05b].

We will prove the following proposition.

1. (1) It is symmetric in the two sets of variables, $\Omega_k[X,Y]=\Omega_k[Y,X]$ .

2. (2) If $\lambda,\mu$ are partitions, then the coefficient of $X^{\lambda} Y^{\mu}$ in $\Omega_k[X(t-1),Y(q-1)]$ is zero unless $\lambda \trianglelefteq \mu'$ in the dominance order.

3. (3) The leading coefficient in front of $X^{\lambda}Y^{\lambda'}$ is $q^{kn(\lambda')}t^{kn(\lambda)}$ .

Then we have the following.

1. (1) It is self-adjoint with respect to the Macdonald inner product,

   \[(\nabla^k f,g)_*=(f,\nabla^k g)_*.\]

2. (2) It is triangular in the modified Schur basis,

   \[\nabla s_\lambda[X(q-1)^{-1}]=\sum_{\mu \trianglelefteq \lambda} a_{\lambda,\mu}(q,t) s_{\mu}[X(q-1)^{-1}], \]
   and similarly with t in place of q.

3. (3) The leading coefficient is given by $a_{\lambda,\lambda}(q,t)=q^{kn(\lambda')}t^{kn(\lambda)}$ .

That $\nabla^k$ satisfies these properties is a consequence of the defining properties of the modified Macdonald polynomials from § 2.1. To see the uniqueness, the triangularity statement implies the triangularity of the matrix of $\nabla^k$ in the modified Macdonald basis. On the other hand, the self-adjointness implies that the matrix is symmetric, so it is also diagonal.

We therefore need to check that $\nabla_k'$ satisfies these properties as well, using the corresponding parts from Proposition 3.7. First, item (1) implies that the operator given by $T(f)=(f,\Omega_k[X,Y])_Y$ is self-adjoint in the usual Hall inner product. It follows that the operator $f\mapsto T(f[-X(1-q)$ $(1-t)])$ is self-adjoint with respect to the modified Macdonald inner product (2), and therefore so is $\nabla_k'$ .

To see the triangularity statement, we check that

\[\nabla_k' e_{\lambda}[X(t-1)^{-1}]=(-1)^n (e_{\lambda}[Y(1-q)],\Omega_k[X,Y])_{Y}=(h_{\lambda}[Y],\Omega_k[X,Y(q-1)])_{Y}.\]

Then part (2) of the proposition says that the expansion of the resulting symmetric function in the modified monomial basis $m_{\mu}[X(t-1)^{-1}]$ contains only nonzero terms for $\mu\trianglelefteq \lambda'$ . Since $e_{\lambda'},s_\lambda$ , and $m_\lambda$ are all triangular with respect to each other, their modified versions are as well, and the triangularity statement follows. The leading term statement is immediate from item (3) of the proposition.

To prove Proposition 3.7, we first note that the symmetry statement is just Proposition 3.4. We now turn to the hard part which is showing triangularity. We first evaluate the plethystic substitution $Y\mapsto Y(q-1)$ . Recall the definition of k-attacks from Definition 2.4, which depends only on the sorted pair $[{\mathbf{m}},{\mathbf{a}}]$ .

Lemma 3.9. We have

(18) \begin{equation} \Omega_k[X,Y(q-1)]=(-1)^n\sum_{\substack{[{\mathbf{m}},{\mathbf{a}},{\mathbf{b}}] \\i \,k\text{-attacks } j\Rightarrow b_i\neq b_j}} t^{|{\mathbf{m}}|}q^{\mathrm{dinv}_k({\mathbf{m}},{\mathbf{a}},{\mathbf{b}})}X_{{\mathbf{a}}}Y_{{\mathbf{b}}}.\end{equation}

We next give a combinatorial formula for the additional substitution $X\mapsto X(t-1)$ , so as to arrive at $\Omega_k[X(t-1),Y(q-1)]$ . We first define the combinatorial objects and their corresponding statistics.

Definition 3.10. For any pair $({\mathbf{m}},{\mathbf{b}})$ , we define

(19) \begin{equation} d_k^{i,j}({\mathbf{m}},{\mathbf{b}})=\begin{cases}k+m_j-m_i+\delta(b_i>b_j), & m_i>m_j, \\k-1+m_i-m_j+\delta(b_i<b_j), & m_i\leq m_j.\end{cases}\end{equation}

and $\delta$ is the delta function, 1 for true, 0 for false.

1. 1. the terms $(a_i,m_i,b_i)$ are sorted for $l+1 \leq i \leq n$ , and in reverse order for $1\leq i \leq l$ ;

2. 2. $m_i>0$ for $1 \leq i\leq l$ ;

3. 3. if $m_i \in \{m_j-k+1,\ldots ,m_j+k\}$ for $i<j$ , then $b_i\neq b_j$ .

Example 3.12. For instance, we would have

\[A=\left(\begin{array}{c|cc|cccc}{\mathbf{a}}&3&2&1&1&2&4\\{\mathbf{m}}&3&1&0&0&0&0\\{\mathbf{b}}&1&3&2&4&1&5\end{array}\right)\in \mathcal{A}(6,2),\]

where we are drawing a dividing line to indicate that $l=2$ . Below is the table of the contributions (before taking the max with zero) to $d_2({\mathbf{m}},{\mathbf{b}})$ :

\[(d_2^{i,j}({\mathbf{m}},{\mathbf{b}})) =\left(\begin {array}{rrrrrr} 1&\phantom{-}0&-1&-1&-1&-1\\ -1&1&2&1&2&1\\ -2&1&1&2&1&2\\ -2&0&1&1&1&2\\ -2&1&2&2&1&2\\ -2&0&1&1&1&1\end {array} \right)\!.\]

We see that $d_2({\mathbf{m}},{\mathbf{b}})=16$ , by adding up the positive entries above the diagonal.

Lemma 3.13. We have

(20) \begin{equation}\Omega_k[X(t-1),Y(q-1)]=\sum_{(l,{\mathbf{a}},{\mathbf{m}},{\mathbf{b}})\in \mathcal{A}(n,k)}(-1)^{l}t^{|{\mathbf{m}}|}q^{d_k({\mathbf{m}},{\mathbf{b}})}X_{{\mathbf{a}}} Y_{{\mathbf{b}}}.\end{equation}

Proof. We must write equation (18) as a quasi-symmetric function in the X-variables. We first sort the triples in a different order, so that $[{\mathbf{a}},{\mathbf{m}},{\mathbf{b}}]'$ is a triple in which the $a_i$ are in descending order, the $m_i$ are in increasing order to break ties, and the $b_i$ are in descending order to breaking ties. This is the reverse of the usual order, modified so that ${\mathbf{a}}$ has priority over ${\mathbf{m}}$ . Define two conditions W (wrong) and NW (not wrong) on pairs ${\mathbf{m}},{\mathbf{b}}$ :

\begin{array}{rl}\mathrm{W}_i: & \text{if}\;m_{i}>m_{i+1} \;\text{or}\; m_{i}=m_{i+1},\,b_{i}<b_{i+1},\\\mathrm{NW}_i: & \text{if}\; m_{i}<m_{i+1} \;\text{or}\; m_{i}=m_{i+1},\,b_{i}>b_{i+1}.\end{array}

We can reconstruct the condition of when a nonzero term in (18) must have the inequality $a_i>a_{i+1}$ based on the ordering ${\mathbf{m}},{\mathbf{b}}$ , to produce a quasi-symmetric expansion.

For $({\mathbf{a}},{\mathbf{m}},{\mathbf{b}})=[{\mathbf{a}},{\mathbf{m}},{\mathbf{b}}]'$ reverse sorted, we have that

\[\mathrm{dinv}_k([{\mathbf{m}},{\mathbf{a}},{\mathbf{b}}])=d_k({\mathbf{m}},{\mathbf{b}})=\sum_{i<j}\max(d^{i,j}_k({\mathbf{m}},{\mathbf{b}}),0)\]

for every nonzero term in (18).

Now let

(21) \begin{equation} \mathcal{B}(n,k)= \left\{({\mathbf{m}},{\mathbf{b}}) : m_i\in \{m_j-k+1,\ldots ,m_j+k\}\mbox{ for $i<j$} \Rightarrow b_i\neq b_j\right\}\end{equation}

for all pairs $({\mathbf{m}},{\mathbf{b}})$ , not necessarily sorted. We can now write

(22) \begin{equation} \Omega_k[X,(q-1)Y]= (-1)^n\sum_{({\mathbf{m}}, {\mathbf{b}})\in \mathcal{B}(n,k)} t^{|{\mathbf{m}}|} q^{d_k({\mathbf{m}}, {\mathbf{b}})} Y_{{\mathbf{b}}} \sum_{\substack{a_1\geq \cdots \geq a_n:\\ \mathrm{W}_i\;\Rightarrow\; a_i>a_{i+1}}} X_{{\mathbf{a}}},\end{equation}

which is a quasi-symmetric expansion. We may apply the operator of the substitution $F[X]\mapsto F[(t-1)X]$ using the standardization approach from [Reference Haglund, Haiman and LoehrHHL05b] to obtain

(23) \begin{align}\Omega_k[(t-1)X,(q-1)Y]\ &= \sum_{({\mathbf{m}}, {\mathbf{b}}) \in \mathcal{B}(n,k)} t^{|{\mathbf{m}}|} q^{d_k({\mathbf{m}}, {\mathbf{b}})} Y_{{\mathbf{b}}} \sum_{l=0}^n(-t)^l\sum_{\substack{a_1\leq \cdots \leq a_{n-l}:\\ \mathrm{NW}_i\;\Rightarrow\; a_i<a_{i+1}}}\sum_{\substack{a_{n-l+1}\geq \cdots \geq a_{n}:\\\mathrm{W}_i\;\Rightarrow\; a_i>a_{i+1}}} X_{{\mathbf{a}}}\nonumber\\\ &= \sum_{({\mathbf{m}}, {\mathbf{b}}) \in \mathcal{B}(n,k)} t^{|{\mathbf{m}}|} q^{\mathrm{dinv}_k({\mathbf{m}}, {\mathbf{b}})} Y_{{\mathbf{b}}} \sum_{l=0}^n (-t)^{l} \sum_{\substack{a_1\leq \cdots \leq a_{n-l}:\\ a_i=a_{i+1}\;\Rightarrow\;\mathrm{W}_i}} \sum_{\substack{a_{n-l+1}\geq \cdots \geq a_{n}:\\ a_i=a_{i+1}\;\Rightarrow\;\mathrm{NW}_i}} X_{{\mathbf{a}}}.\end{align}

Before proving the vanishing, it will be helpful to write equation (23) in a more convenient form by collecting powers of t. Define the rotation operator $\rho$ on pairs $({\mathbf{m}},{\mathbf{b}})$ by $\rho({\mathbf{m}}, {\mathbf{b}}) = ({\mathbf{m}}', {\mathbf{b}}')$ , where

(24) \begin{equation} m'_i=m_{i-1},\quad b'_i=b_{i-1},\quad m'_1=m_n+1, \quad b'_1=b_n,\end{equation}

which satisfies

\[d_k(\rho({\mathbf{m}}, {\mathbf{b}}))=d_k({\mathbf{m}}, {\mathbf{b}}),\quad \mathrm{area}(\rho({\mathbf{m}},{\mathbf{b}}))=\mathrm{area}({\mathbf{m}}, {\mathbf{b}})+1,\]

where $\mathrm{area}({\mathbf{m}},{\mathbf{b}})=|{\mathbf{m}}|$ . Moreover, for $1\leq i<n-1$ , $\mathrm{W}_i$ for $({\mathbf{m}}, {\mathbf{b}})$ is equivalent to $\mathrm{W}_{i+1}$ for $\rho({\mathbf{m}}, {\mathbf{b}})$ . The triples in (23) are then bijectively mapped via $\rho^l$ to triples satisfying $m_1,\ldots,m_l\geq 1$ , so the right-hand side of (23) becomes equation (20).

We now demonstrate the triangularity of equation (20) by finding an involution $\iota_k:\mathcal{A}(n,k)\rightarrow \mathcal{A}(n,k)$ which sends a quadruple $(l,{\mathbf{a}},{\mathbf{m}},{\mathbf{b}})$ to itself, or sends it to one which cancels it in (20). We then show that the set of fixed points are empty unless the dominance order property is satisfied.

Definition 3.14. For any i and any quadruple $(l,{\mathbf{a}},{\mathbf{m}},{\mathbf{b}})$ , we define

\[\mathrm{move}_i(l,{\mathbf{a}},{\mathbf{m}},{\mathbf{b}})=(l',{\mathbf{a}}',{\mathbf{m}}',{\mathbf{b}}'),\]

where $l'=l-1$ if $i\leq l$ or $l+1$ if $i>l$ , and $({\mathbf{a}}',{\mathbf{m}}',{\mathbf{b}}')$ is the result of inserting $(a_i,m_i,b_i)$ in the unique position on the opposite side of the dividing line l so that $(l',{\mathbf{a}}',{\mathbf{m}}',{\mathbf{b}}')$ is sorted as in condition (1) of Definition 3.11.

Notice that for any element of $\mathcal{A}(n,k)$ , we never have $(a_i,m_i,b_i)=(a_j,m_j,b_j)$ unless $i=j$ , because of condition (3). We therefore have a unique permutation $\sigma$ so that $({\mathbf{a}}_{\sigma},{\mathbf{m}}_{\sigma},{\mathbf{b}}_{\sigma})$ is overall sorted, not in reverse order for $i\leq l$ .

Definition 3.15. Given $A=(l,{\mathbf{a}},{\mathbf{m}},{\mathbf{b}})\in \mathcal{A}(n,k)$ , we will say that i is k-movable if $\mathrm{move}_i(A)\in \mathcal{A}(n,k)$ , and for any j with $\sigma_j<\sigma_i$ , we have $d_k^{i,j}({\mathbf{m}},{\mathbf{b}}),d_k^{j,i}({\mathbf{m}},{\mathbf{b}})\leq 0$ . Let $\iota_k : \mathcal{A}(n,k)\rightarrow \mathcal{A}(n,k)$ be the involution defined by setting

$\iota_k(A)=\begin{cases}A, & \mbox {no element}\, i \, \mbox {is} \, k-\mbox movable,\\\mathrm{move}_i(A), & i,\mbox{the movable element with smallestvalue of}\, \sigma_i.\end{cases}$

Lemma 3.16. The map $\iota_k$ is an involution. For any non-fixed element given by $(l',{\mathbf{a}}',{\mathbf{m}}',{\mathbf{b}}')=\iota_k(l,{\mathbf{a}},{\mathbf{m}},{\mathbf{b}})$ , we have

(25) \begin{equation} d_k({\mathbf{m}}',{\mathbf{b}}')=d_k({\mathbf{m}},{\mathbf{b}}),\quad|{\mathbf{m}}'|=|{\mathbf{m}}|,\quad(-1)^l=-(-1)^{l'}.\end{equation}

To see this, suppose that j’ is movable in A’ and that $\sigma_{j}<\sigma_{i'}$ . Let j be the index with $\sigma_j=\sigma'_{j'}$ , which by assumption is not movable in A. Then we must have that $\mathrm{move}_j(A) \notin \mathcal{A}(n,k)$ , which can only happen by condition (3) of Definition 3.11 being satisfied. The second index for which the condition is violated must be i, otherwise j’ would not have been movable in A’. But since j,j’ are both on the same side of i, this cannot happen either.

Equation (25) is clear.

Example 3.17. Let us compute the involution on the term $A\in \mathcal{A}(6,2)$ from Example 3.12. We have that $\sigma=(5,3,1,2,4,6)$ . The smallest element is therefore $i=3$ , which is not moveable because $m_3=0$ , so we cannot move it to the left of the dividing line $l=2$ without violating condition (2) of the definition of $\mathcal{A}(n,k)$ . The next smallest values of $i=4,2,5$ cannot be moved because we have $d_2^{i,j}({\mathbf{m}},{\mathbf{b}})>0$ or $d_2^{j,i}({\mathbf{m}},{\mathbf{b}})>0$ for some j earlier in the list. However, $i=1$ is moveable, and we end up with

\[\iota_k(A)=\left(\begin{array}{c|c|cccccc}{\mathbf{a}}& 2& 1&1&2&3&4\\{\mathbf{m}}&1&0&0&0&3&0\\{\mathbf{b}}& 3&2&4&1&1&5\end{array}\right).\]

Let $\mathcal{A}'(n,k)$ denote the fixed points of $\iota_k$ .

Proposition 3.18. If $\lambda,\mu$ are partitions, then the set

\[\mathcal{A}'(n,k)_{\lambda,\mu}=\left\{(l,{\mathbf{a}},{\mathbf{m}},{\mathbf{b}})\in \mathcal{A}'(n,k): \alpha({\mathbf{a}})=\lambda,\ \alpha({\mathbf{b}})=\mu\right\}\]

is empty unless $\lambda\trianglelefteq \mu'$ in the dominance order. If $\lambda=\mu'$ , then it contains just one element, namely

(26) \begin{equation} l=0,\quad {\mathbf{a}}=(1^{\lambda_1},\ldots ,l^{\lambda_r}),\quad{\mathbf{m}}=(0^{\lambda_1},\ldots ,(rk)^{\lambda_r}),\quad{\mathbf{b}}=(1,\ldots ,\lambda_1,\ldots ,1,\ldots ,\lambda_r).\end{equation}

Example 3.19. If $\lambda=\mu'=(2,1,1)$ , then $\mathcal{A}'_{\lambda,\mu}(4,2)$ contains only the element

\[\left(\begin{array}{c||cccc}{\mathbf{a}} & 1 & 1 & 2 & 3 \\{\mathbf{m}} & 0 & 0 & 2 & 4 \\{\mathbf{b}} & 1 & 2 & 1 & 1\end{array}\right).\]

We will prove Proposition 3.18 through some lemmas. Let $T(A)=T({\mathbf{a}},{\mathbf{m}},{\mathbf{b}})$ be the result of filling the rth row of the composition $\alpha=\alpha({\mathbf{a}})$ with the pairs $(m_i,b_i)$ for which $a_i=\alpha_r$ , in sorted order left to right. For instance, we have the following.

(27)

Notice that the diagram does not depend on the ordering or on l.

Let us refer to the numbers on the left and right in each box of the diagram as the ‘m’ and ‘b’-numbers, respectively.

Proof. Suppose the number b appears $r+1$ times in rows 1 through r. Let $x_1,\ldots ,x_r$ denote the set of the corresponding values of $m_i$ in the order they appear in ${\mathbf{m}}$ , for instance (1,0,1,0) for the b-value of 1 in (27). Then we must have that

(28) \begin{equation}x_s \leq x_t-k\quad \mbox{or}\quad x_s \geq x_t+k+1\end{equation}

for $1\leq s<t \leq r+1$ by condition (3) of Definition 3.11. Now let $0\leq y_1\leq \cdots \leq y_{r+1}$ denote the same set of numbers as the $x_s$ but in sorted order. By (28) we have that $y_{s+1}\geq y_s+k$ , so that $y_{r+1}\geq rk$ , which contradicts Lemma 3.20.

To prove the second statement, define $0\leq y_1\leq \cdots \leq y_r$ as above. Then, by the same reasoning, we have $y_r\geq (r-1)k$ and also $y_r\leq (r-1)k$ by the same lemma, so we must have $y_s=sk$ . Then only the first case is possible in (28), and so all the $x_s$ are same order $x_s=y_s$ . Since $x_1=0$ , it must be to the right of the dividing line because of condition (2), and so the rest are as well. Then if two b-values appear in the same row, there will be increasing m-values for the same a-value, so in the wrong order for the entries to the right of the dividing line.

We now prove Proposition 3.18.

For the second statement, if $\lambda=\mu'$ , then the ith lowest b-number appears $\lambda'_i$ times. By the second statement of Lemma 3.21, it appears once in every row up to $\lambda'_i$ , to the right of the dividing line, and (by the proof) with corresponding m-values $0,k,2k,\ldots$ . The unique term with these properties is precisely the one from (26).

Finally, we can prove Proposition 3.7, and therefore Theorem 3.1.

3.3 A new proof of the shuffle conjecture

We now show how to recover the shuffle theorem from Theorem 3.1.

Notice that if we have $b_i=b_j$ for any $i\neq j$ in (18), then we cannot have $m_i=m_j$ , or $m_i=m_j+1$ and $a_i>a_j$ . By the first condition, we can uniquely sort the orbits so that the $b_i$ are sorted in reverse order, $b_1\geq \cdots \geq b_n$ , and if $b_i=b_{i+1}$ then $m_{i}<m_{i+1}$ . Then the second condition says

\[b_{i}=b_{i+1}\Rightarrow m_{i+1}>m_i+1 \text{ or } m_{i+1}=m_i+1\text{ and } a_{i+1}\leq a_i.\]

For a pair of sequences $({\mathbf{m}},{\mathbf{a}})$ and a position i we will define two conditions, ‘parking function at i,’ and ‘not parking function at i’, noting that one is the negation of the other:

\[\begin{array}{rl}\mathrm{PF}_i: & \text{if}\;m_{i+1}\leq m_{i} \;\text{or}\; m_{i+1}=m_i+1,\,a_{i+1}>a_i,\\\mathrm{NPF}_i: & \text{if}\; m_{i+1}>m_i+1 \;\text{or}\; m_{i+1}=m_i+1,\,a_{i+1}\leq a_i.\end{array}\]

We can reformulate dinv as we did in § 3.2:

\[d({\mathbf{m}}, {\mathbf{a}}) = \#\{i<j:\;m_i=m_j,\,a_i<a_j\;\text{or}\;m_i=m_j+1,a_i>a_j\},\]

noting that $d({\mathbf{m}},{\mathbf{a}})=\mathrm{dinv}([{\mathbf{m}},{\mathbf{a}},{\mathbf{b}}])$ .

We can now write

(29) \begin{equation} \nabla h_n\left[\frac{XY}{1-t}\right]= \sum_{{\mathbf{m}}, {\mathbf{a}}} X_{{\mathbf{a}}} t^{|{\mathbf{m}}|} q^{d({\mathbf{m}}, {\mathbf{a}})} \sum_{\substack{b_1\geq \cdots \geq b_n:\\ \mathrm{PF}_i\;\Rightarrow\; b_i>b_{i+1}}} Y_{{\mathbf{b}}}.\end{equation}

We would like to evaluate the substitution $Y\rightarrow (1-t)Y$ . To do this, notice that (29) is a sum of quasi-symmetric functions in Y. We can therefore compute the substitution using the standardization approach from [Reference Haglund, Haiman and LoehrHHL05b]. The result is

(30) \begin{equation}\nabla h_n[XY] = \sum_{{\mathbf{m}},{\mathbf{a}}} t^{|{\mathbf{m}}|} q^{d({\mathbf{m}}, {\mathbf{a}})} \sum_{l=0}^n (-t)^l X_{{\mathbf{a}}} \sum_{\substack{b_1\geq \cdots \geq b_{n-l}:\\ \mathrm{PF}_i\;\Rightarrow\; b_i>b_{i+1}}} Y_{{\mathbf{b}}} \sum_{\substack{b_{n-l+1}\leq \cdots \leq b_{n}:\\ \mathrm{NPF}_i\;\Rightarrow\; b_i<b_{i+1}}} Y_{{\mathbf{b}}}.\end{equation}

Finally, let us make the evaluation $Y=-1$ in (30) adding an extra sign $(-1)^n$ , which amounts to counting only the terms in which the quasi-symmetric functions have strict inequalities. We obtain

(31) \begin{equation}\nabla e_n[X] = \sum_{l=0}^n (-t)^l \sum_{\substack{{\mathbf{m}}, {\mathbf{a}}:\\\mathrm{PF}_i \;\text{for}\; 1\leq i\leq n-l-1,\\\mathrm{NPF}_i \;\text{for}\; n-l+1\leq i\leq n-1}} X_{{\mathbf{a}}} t^{|{\mathbf{m}}|} q^{d({\mathbf{m}}, {\mathbf{a}})}.\end{equation}

We would like to cancel certain terms in the right-hand side of (31), this time using the rotation operator $\rho$ defined in (24). For $1\leq i<n-1$ , notice that $\mathrm{PF}_i$ for $({\mathbf{m}}, {\mathbf{a}})$ is equivalent to $\mathrm{PF}_{i+1}$ for $\rho({\mathbf{m}}, {\mathbf{a}})$ . Consider those triples $(l,{\mathbf{m}},{\mathbf{a}})$ satisfying:

1. (1A) $l>0$ ;

2. (2A) $PF_1$ for $\rho({\mathbf{m}}, {\mathbf{a}})$ if $l<n$ .

The image of $\rho$ on these triples is the set of triples $(l,{\mathbf{m}},{\mathbf{a}})$ satisfying:

1. (1B) $l<n$ ;

2. (2B) $NPF_{n-1}$ for $\rho^{-1}({\mathbf{m}},{\mathbf{a}})$ if $l>0$ ;

3. (3B) $m_1>0$ .

We can now check the following proposition, which implies that the two sets have no elements in common, and so the terms coming from the two sets cancel each other out in (31).

We can now give a new proof of the shuffle theorem [Reference Haglund, Haiman and LoehrHHL05b, Reference Carlsson and MellitCM18], noting that the conditions of the summation in (32) mean that ${\mathbf{m}}$ is the area sequence of a Dyck path, and ${\mathbf{a}}$ is a word parking function, see [Reference HaglundHag08].

Theorem 3.23. We have

(32) \begin{equation}\nabla e_n[X] = \sum_{\substack{{\mathbf{m}}, {\mathbf{a}}: PF_i \;\text{for all} \, i,\; m_1=0}} X_{{\mathbf{a}}} t^{|{\mathbf{m}}|} q^{d({\mathbf{m}}, {\mathbf{a}})}.\end{equation}

4.1 Counting formula

On the first side, we will need a result from [Reference MellitMel20, §5] for counting bundles on $\mathbb{P}^1$ over a finite field. Let q be a prime power, and let $\mathbf{k}$ be the finite field with $|\mathbf{k}|=q$ elements. Let $S=\{s_1,\ldots,s_k\}\subset \mathbb{P}^1(\mathbf{k})$ be a collection of distinct rational points. Let N be a big integer (this will correspond to the number of variables in each alphabet). We need k alphabets $X_1,\ldots,X_k$ . The variables in alphabet $X_i$ are denoted $x_{i,j}$ ( $1\leq i\leq k$ , $1\leq j\leq N$ ).

Definition 4.1. A parabolic bundle is a pair $(\mathcal{E}, \mathbf{F})$ , where $\mathcal{E}$ is a vector bundle on $\mathbb{P}^1$ over $\mathbf{k}$ , and $\mathbf{F}=(F_{i,j})_{1\leq i\leq k,0\leq j\leq N}$ is a collection of vector spaces so that for each i we have

$0=F_{i,0}\subseteq F_{i,1} \subseteq \cdots \subseteq F_{i,N-1} \subseteq F_{i,N}=\mathcal{E}(s_i).$

An endomorphism of $(\mathcal{E}, \mathbf{F})$ is an endomorphism of $\mathcal{E}$ preserving each $F_{i,j}$ . An endomorphism $\theta$ is nilpotent if $\theta^n=0$ for some n.

Here, $\mathcal{E}(s_i)$ is the fiber of $\mathcal{E}$ over $s_i$ . If $\mathcal{E}$ had rank n, then $\mathcal{E}(s_i)$ is an n-dimensional vector space.

Parabolic bundles have the following discrete invariants:

• – $\mathrm{rank}(\mathcal{E})$ = rank of $\mathcal{E}$ ;

• – $\deg(\mathcal{E})$ = degree of $\mathcal{E}$ ;

• – $r_{i,j}=\dim(F_{i,j}/F_{i,j-1})$ .

Note that $r_{i,\bullet}$ is a composition of n for each $i=1,\ldots,k$ (of length N with zeros allowed). These invariants are packaged in the following weight:

\[\mathrm{weight}(\mathcal{E},\mathbf{F}) = t^{\deg} \prod_{i=1}^k \prod_{j=1}^N x_{i,j}^{r_{i,j}}.\]

It is well known that over $\mathbb{P}^1$ every vector bundle is a sum of line bundles, so we can write $\mathcal{E}=O(m_1)\oplus\cdots\oplus O(m_n)$ . We write $\mathcal{E}\geq 0$ if all $m_i\geq 0$ and $\mathcal{E}\leq 0$ if all $m_i\leq 0$ .

Fix $n\geq 0$ and $\lambda\vdash n$ . In [Reference MellitMel20, §5.2] the second author introduced the counting functionFootnote ¹

(33) \begin{equation}\Omega_{\lambda, S}[X_1,\ldots,X_k] =\sum_{\substack{(\mathcal{E},\theta):\mathcal{E}\leq0,\;\theta\in\mathrm{Nilp}(\mathcal{E})\\ \mathrm{type}\,\theta =\lambda}} \frac{t^{-\deg \mathcal{E}}}{|\mathrm{Aut}(\mathcal{E},\theta)|} \prod_{i=1}^k \tilde H_{\mathrm{type}\,\theta(s_i)}[X_i;q,0].\end{equation}

The summation runs over the isomorphism classes of pairs $(\mathcal{E},\theta)$ of a vector bundle over $\mathbb{P}^1$ of rank n and an endomorphism, $\mathrm{Nilp}(\mathcal{E})$ denotes the set of all nilpotent endomorphisms of $\mathcal{E}$ , and $\mathrm{Aut}(\mathcal{E},\theta)$ is the set of automorphisms of $\mathcal{E}$ which commute with $\theta$ . The notation $\mathrm{type}\ \theta(s_i)$ respectively $\mathrm{type}\ \theta$ stands for the partition whose conjugate specifies the sizes of the Jordan blocks of $\theta$ restricted to the fiber $\mathcal{E}(s_i)$ respectively over the generic point.

The specialized Macdonald polynomials $\tilde H_{\mu}[X;q,0]$ , also called the Hall–Littlewood polynomials have an interpretation as counting partial flags preserved by a nilpotent matrix [Reference MellitMel20, Corollary 2.13]. Let M be a nilpotent matrix of type $\mu\vdash n$ over $\mathbb{F}_q$ . Then

\[\widetilde H_\mu[X;q,0] = \sum_{\substack{0=F_0\subset F_1\subset\cdots\subset F_N=\mathbb{F}_q^n\\ M F_j\subset F_j}} \prod_{j=1}^N x_j^{\dim F_j/F_{j-1}}.\]

Thus we can rewrite the counting function as followsFootnote ² :

(34) \begin{equation}\Omega_{\lambda, S}[X_1,\ldots,X_k] = \sum_{\substack{(\mathcal{E},\mathbf{F},\theta):\mathcal{E}\leq 0,\;\theta\in\mathrm{Nilp}(\mathcal{E},\mathbf{F})\\ \mathrm{type}\,\theta =\lambda}} \frac{t^{-\deg \mathcal{E}} \prod_{i=1}^{k}\prod_{j=1}^N x_{i,j}^{r_{i,j}}}{|\mathrm{Aut}(\mathcal{E},\mathbf{F},\theta)|}.\end{equation}

This is essentially [Reference MellitMel20, (5.2)]. Now the summation runs over the isomorphism classes of parabolic bundles with an endomorphism, $\mathrm{Nilp}(\mathcal{E},\mathbf{F})$ denotes the set of all nilpotent endomorphisms of $\mathcal{E}$ which preserve $\mathbf{F}$ , and $\mathrm{Aut}(\mathcal{E},\mathbf{F},\theta)$ is the set of automorphisms of $(\mathcal{E},\mathbf{F})$ commuting with $\theta$ .

Between (33) and (34) one can also stop midway. Assume $s_1=0$ and $s_2=\infty$ and expand the Hall–Littlewood polynomials $\tilde H_{\mathrm{type}\,\theta(s_i)}[X_i;q,0]$ only for $i=1,2$ . We obtain

(35) \begin{equation}\Omega_{\lambda, S}[X_1,\ldots,X_k] = \sum_{\substack{(\mathcal{E},\mathbf{F}^0,\mathbf{F}^\infty,\theta):\mathcal{E}\leq 0,\\\theta\in\mathrm{Nilp}(\mathcal{E},\mathbf{F}^0,\mathbf{F}^\infty)\\ \mathrm{type}\,\theta =\lambda}} \frac{t^{-\deg \mathcal{E}} \prod_{i=1}^2 \prod_{j=1}^N x_{i,j}^{r_{i,j}}}{|\mathrm{Aut}(\mathcal{E},\mathbf{F}^0,\mathbf{F}^1,\theta)|} \prod_{i=3}^k \tilde H_{\mathrm{type}\,\theta(s_i)}[X_i;q,0].\end{equation}

Here, we are summing over parabolic bundles with flags at 0 and $\infty$ , denoted $\mathbf{F}^0$ and $\mathbf{F}^\infty$ .

The following explicit formula has been proved in [Reference MellitMel20, Corollary 5.9]Footnote ³ :

(36) \begin{equation}\Omega_{\lambda, S}[X_1,\ldots,X_k] = (-1)^n \frac{\prod_{i=1}^k \tilde H_{\lambda}[X_i;q,t]}{(\widetilde H_\lambda, \widetilde H_\lambda)_*}.\end{equation}

Corollary 4.2. We have

\begin{eqnarray*}&&\sum_{(\mathcal{E},\mathbf{F}^0,\mathbf{F}^\infty):\mathcal{E}\geq 0}\frac{t^{\deg \mathcal{E}} \prod_{j=1}^N x_{1,j}^{r_{1,j}}x_{2,j}^{r_{2,j}}}{|\mathrm{Aut}(\mathcal{E},\mathbf{F}^0,\mathbf{F}^\infty)|}\sum_{\theta\in\mathrm{Nilp}(\mathcal{E},\mathbf{F}^0,\mathbf{F}^\infty)}\prod_{i=3}^k \tilde H_{\mathrm{type}\, \theta(s_i)}[X_i;q,0]\\&&\qquad= \sum_{n=0}^\infty (-1)^n \sum_{\lambda\vdash n}\frac{\prod_{i=1}^k \tilde H_{\lambda}[X_i;q,t]}{(\widetilde H_\lambda, \widetilde H_\lambda)_*},\end{eqnarray*}

where the first summation on the right-hand side runs over the isomorphism classes of parabolic bundles with marked points $0,\infty$ .

Below we are interested in expressions of the form

\[\nabla^k e_n\left[\frac{XY}{(q-1)(t-1)}\right] = \sum_{\lambda\vdash n} \left(q^{n(\lambda')} t^{n(\lambda)}\right)^k \frac{\tilde H_{\lambda}[X;q,t] \tilde H_{\lambda}[Y;q,t]}{(\tilde H_\lambda, \tilde H_\lambda)_*}.\]

Recall that

\[(\tilde H_{\lambda}[X;q,t], s_{1^n}) = q^{n(\lambda')} t^{n(\lambda)},\]

and by setting $t=0$

\[(\tilde H_{\lambda}[X;q,0], s_{1^n}) = \begin{cases} q^{\binom{n}{2}}, & \lambda=(n),\\0, & \text{otherwise}.\end{cases}\]

Applying $(-,s_{1^n})$ in the alphabets $X_3,\ldots,X_k$ to both sides of Corollary 4.2, and then replacing k by $k+2$ and relabeling $s_i$ we obtain the following.

Corollary 4.3. Let $k\geq 0$ , and let $\{s_1,\ldots,s_k\}$ be an arbitrary collection of distinct points on $\mathbb{P}(\mathbf{k})\setminus\{0,\infty\}$ . We have

\[q^{k \binom{n}{2}} \sum_{\substack{(\mathcal{E},\mathbf{F}^0,\mathbf{F}^\infty)\\ \mathrm{rank}(\mathcal{E})=n}} \frac{t^{\deg} \prod_{j=1}^N x_j^{r_{1,j}} y_{j}^{r_{2,j}}}{|\mathrm{Aut}(\mathcal{E},\mathbf{F}^0,\mathbf{F}^\infty)|}|\mathrm{Nilp}_k(\mathcal{E},\mathbf{F}^0,\mathbf{F}^\infty)| = (-1)^n\nabla^k e_n\left[\frac{XY}{(q-1)(t-1)}\right],\]

where $\mathrm{Nilp}_k(\mathcal{E},\mathbf{F}^0,\mathbf{F}^\infty)$ denotes the set of nilpotent endomorphisms $\theta$ satisfying $\theta(s_i)=0$ for $i=1,\ldots,k$ .

4.2 Parabolic bundles with two marked points

Next we will use an explicit classification of triples $(\mathcal{E},\mathbf{F}^0,\mathbf{F}^\infty)$ to give an alternative formula for the generating function in Corollary 4.3. The building blocks of the classification will be parabolic bundles of rank 1, i.e. parabolic line bundles.

Proposition 4.5. Let $(\mathcal{E},\mathbf{F}^0,\mathbf{F}^\infty)$ be a parabolic vector bundle of rank n on $\mathbb{P}^1$ with two marked points. There exists a unique multiset of triples $(m_1,a_1,b_1)$ , …, $(m_n,a_n,b_n)$ such that

\[ (\mathcal{E},\mathbf{F}^0,\mathbf{F}^\infty) = \bigoplus_{i=1}^n O(m_n;a_n,b_n). \]

This can be thought of as a generalization of the classical Bruhat decomposition for $GL_n$ . There is a tedious direct proof based on several applications of the standard Bruhat decomposition, but we will use homological algebra instead. The proof will occupy the rest of the section.

Of course, the category of parabolic bundles is not an abelian category, but it can be embedded as a full subcategory into the abelian category of parabolic coherent sheaves (see [Reference HeinlothHei04] or [Reference MellitMel20, §6.1]), which has global dimension 1, so all $\mathrm{Ext}^i$ vanish for $i>1$ . The Euler form is given by

(37) \begin{eqnarray}\dim \mathrm{Hom}(\overline{\mathcal{E}},\overline{\mathcal{E}}') -\dim \mathrm{Ext}(\overline{\mathcal{E}},\overline{\mathcal{E}}') &=&\mathrm{rank}\ \mathcal{E}\ \mathrm{rank}\ \mathcal{E}'+\mathrm{rank}\ \mathcal{E} \deg \mathcal{E}' - \mathrm{rank}\\mathcal{E}' \deg \mathcal{E}\\&& - \sum_{i=1}^k \sum_{1\leq j<j'\leq N} r_{i,j}(\overline{\mathcal{E}}) r_{i,j'}(\overline{\mathcal{E}}').\nonumber\end{eqnarray}

We denote by $\overline{\mathcal{E}}$ the pair $(\mathcal{E},\mathbf{F})$ . In the case $k=2$ we write $(\mathcal{E}, \mathbf{F}^0, \mathbf{F}^\infty)$ .

Lemma 4.6. The dimension of Hom between two parabolic line bundles is given by

\[\dim \mathrm{Hom}(O(m;j_1,\ldots,j_k), O(m';j_1',\ldots,j_k')) =\max(1+m'-m-\#\{i:j_i<j_i'\}, 0).\]

Combining with the formula for the Euler form we obtain

\[\dim \mathrm{Ext}(O(m;j_1,\ldots,j_k), O(m';j_1',\ldots,j_k'))=\max(m-m'-1+\#\{i:j_i< j_i'\}, 0).\]

Introduce a total order on parabolic line bundles in such a way that $O(m;j_1,\ldots,j_k)<O(m';j_1',\ldots,j_k')$ precisely when

\[(m,-j_1,\ldots,-j_k)<(m',-j_1',\ldots,-j_k') \;\text{lexicographically}.\]

This order clearly satisfies the following proposition.

Proof of Proposition 4.5 . Let $k=2$ . We prove the existence first. The proof goes by induction on the rank n. The case $n=1$ is clear. Assume $n>1$ and suppose $\overline{\mathcal{E}}$ is a parabolic bundle of rank n. Consider the set of all parabolic line bundles L such that $\mathrm{Hom}(L, \overline{\mathcal{E}})\neq 0$ . This set is non-empty because any vector bundle has a line subbundle, and we can simply induce the parabolic structure from $\overline{\mathcal{E}}$ to make it into a parabolic line subbundle. Among these choose one that is maximal in our order. The maximal one exists because the degrees of line subbundles in a given vector bundle are bounded from above.

Now consider the short exact sequence

$0 \to L \to \overline{\mathcal{E}}\to \overline{\mathcal{E}}\to 0.$

Let us show that $\overline{\mathcal{E}}'$ is a parabolic bundle. If it is not, it has some nonzero torsion part $\overline{\mathcal{E}}'_\mathrm{tor} \subset \overline{\mathcal{E}}'$ . Let K be the kernel of the map $\overline{\mathcal{E}} \to \overline{\mathcal{E}}'/\overline{\mathcal{E}}'_\mathrm{tor}$ . It is a parabolic line bundle, there is a nonzero map $L\to K$ , so by Proposition 4.7 we have $L\leq K$ . Thus $K=L$ by the maximality of L and therefore $\overline{\mathcal{E}}'_\mathrm{tor}$ has to be zero.

By the induction assumption, $\overline{\mathcal{E}}'\cong \bigoplus_{l=1}^{n-1} L_l$ . If the short exact sequence above splits, we are done. Suppose it does not split. This implies that $\mathrm{Ext}(\overline{\mathcal{E}}', L)\neq 0$ , so there exists l such that for $L'=L_l$ we have

$\mathrm{Ext}(L', L)\neq 0 \;\Leftrightarrow\; m'-m-1+\#\{i:j^{'}_i < j_i\} \geq 1,$

where we let $L=O(m;j_1,\ldots,j_k)$ and $L'=O(m';j_1',\ldots,j_k')$ . Note that, since $k=2$ , this implies $m'\geq m$ . Our plan is to construct a parabolic line bundle L” such that:

1. (1) $\mathrm{Hom}(L'', L')\neq 0$ ;

2. (2) $\mathrm{Ext}(L'', L)=0$ ;

3. (3) $L'' > L$ .

By the exact sequence

$\mathrm{Hom}(L'', \overline{\mathcal{E}}) \to \mathrm{Hom}(L'', \overline{\mathcal{E}}') \to \mathrm{Ext}(L'', L)$

these conditions would guarantee that any nonzero homomorphism $h\in\mathrm{Hom}(L'', L')\subset \mathrm{Hom}(L'', \mathcal{E}')$ can be lifted to a nonzero homomorphism $L''\to \overline{\mathcal{E}}$ , and we would obtain a contradiction with the maximality of L.

If $m'\geq m+1$ , we pick $L''=O(m+1;N,N)$ . This guarantees that $\dim \mathrm{Hom}(L'', L') = m'-m>0$ , $\dim \mathrm{Ext}(L'', L) = 0$ and $L''>L$ , so the required conditions are satisfied.

Otherwise, we must have $m'=m$ , $j_1'<j_1$ and $j_2'<j_2$ . Picking $L'' = O(m;j_1',j_2)$ (or $O(m;j_1,j_2')$ ) satisfies $\dim \mathrm{Hom}(L'', L') = 1$ , $\dim \mathrm{Ext}(L'', L) = 0$ and $L''> L$ . So the existence have been proven.

Note that we have in particular demonstrated that the maximal line subbundle is a direct summand. By Proposition 4.7 it must be present in any direct sum decomposition, and by successively splitting away the maximal subbundle we deduce the uniqueness.

4.3 Computations

We are ready to identify all the ingredients on the left-hand side of Corollary 4.3. By Proposition 4.5, the summation runs over the set of sorted triples $[{\mathbf{m}}, {\mathbf{a}}, {\mathbf{b}}]$ (see § 2.3 for combinatorial notation). Each sorted triple corresponds to a direct sum of line bundles $L_i=O(m_i;a_i,b_i)$ , which satisfy $L_1\geq \cdots \geq L_n$ . Denote

\[O({\mathbf{m}};{\mathbf{a}},{\mathbf{b}}) = \bigoplus_{i=1}^n O(m_i;a_i,b_i).\]

Proposition 4.9. Suppose $({\mathbf{m}},{\mathbf{a}},{\mathbf{b}})$ is sorted. The number of automorphisms of $O({\mathbf{m}};{\mathbf{a}},{\mathbf{b}})$ is given by

\[ |\mathrm{Aut}(O({\mathbf{m}};{\mathbf{a}},{\mathbf{b}})| = (q-1)^n \mathrm{aut}_q({\mathbf{m}},{\mathbf{a}},{\mathbf{b}}) q^{\sum_{i<j} \max(1+m_i-m_j - \delta_{a_j<a_i} - \delta_{b_j<b_i}, 0)}. \]

Proof. By Proposition 4.7, automorphisms are given by block-upper-triangular matrices with block sizes equal to the multiplicities of triples $(m_i,a_i,b_i)$ . A block-upper-triangular matrix is invertible precisely if the blocks are. So we obtain that the number of automorphisms is given by

\[ |\mathrm{Aut}(O({\mathbf{m}};{\mathbf{a}},{\mathbf{b}})| = \prod_{i<j:\; L_i\neq L_j} q^{\dim \mathrm{Hom}(L_j, L_i)} \times \prod_{i} |GL_{\mu_i}(\mathbf{k})|, \]

where $\mu_1, \mu_2, \ldots$ denote the multiplicities, $\sum_i \mu_i = n$ . The number of elements of $GL_r(\mathbf{k})$ is given by

\[ |GL_r(\mathbf{k})| = q^{\binom{r}{2}} (q-1)^r [r]_q!. \]

Since $\dim \mathrm{Hom}(L_i, L_i)=1$ , we have

\[ |\mathrm{Aut}(O({\mathbf{m}};{\mathbf{a}},{\mathbf{b}})| = \prod_{i<j} q^{\dim \mathrm{Hom}(L_j, L_i)} (q-1)^n \mathrm{aut}_q({\mathbf{m}},{\mathbf{a}},{\mathbf{b}}), \]

and applying Lemma 4.6 we obtain the formula.

Below we include the case $k=0$ for completeness. We have the following.

Proposition 4.10. Suppose $({\mathbf{m}},{\mathbf{a}},{\mathbf{b}})$ is sorted and $k\geq 0$ . We have

\[ |\mathrm{Nilp}_k(O({\mathbf{m}};{\mathbf{a}},{\mathbf{b}})| = q^{\sum_{i<j} \max(1-k+m_i-m_j - \delta_{a_j<a_i} - \delta_{b_j<b_i}, 0)} \times \begin{cases} 1 & (k>0),\\ q^{\sum_{i} \binom{\mu_i}{2}} & (k=0), \end{cases} \]

where $\mu=(\mu_1,\ldots,\mu_l)$ are the multiplicities of the triples $(m_i,a_i,b_i)$ .

The remaining pieces of the left-hand side of Corollary 4.3 are identified as follows:

\[t^{\deg} = t^{|{\mathbf{m}}|},\quad \prod_{j=1}^N x_j^{r_{1,j}} = X_{\mathbf{a}}, \quad \prod_{j=1}^N y_j^{r_{2,j}} = Y_{\mathbf{b}}.\]

Example 4.11. Let $k=0$ . The left-hand side of Corollary 4.3 becomes

\[\sum_{[{\mathbf{m}},{\mathbf{a}},{\mathbf{b}}]} \frac{t^{|{\mathbf{m}}|} X_{\mathbf{a}} Y_{\mathbf{b}} q^{\sum_{i} \binom{\mu_i}{2}}}{(q-1)^n \mathrm{aut}_q({\mathbf{m}},{\mathbf{a}},{\mathbf{b}})}.\]

So in each summand each triple (m,a,b) with multiplicity $\mu$ contributes a factor of

\[\frac{t^{\mu m} x_a^\mu y_b^\mu q^{\binom{\mu}{2}}}{(q-1)^\mu [\mu]_q!}.\]

Summing over all n we obtain every possible triple with every multiplicity, so the result can be written as an infinite product

\[\prod_{m=0}^\infty \prod_{a,b=1}^N \sum_{\mu=0}^\infty \frac{t^{\mu m} x_a^\mu y_b^\mu q^{\binom{\mu}{2}}}{(q-1)^\mu [\mu]_q!} = \prod_{m=0}^\infty \prod_{a,b=1}^N \prod_{r=0}^\infty (1-x_a y_b t^m q^r) = \mathrm{Exp}\left[-\frac{X Y}{(1-t)(1-q)}\right],\]

which matches the right-hand side of Corollary 4.3.

Our main conclusion is as follows.

Theorem 4.12. For $k\geq 1$ we have

\[\nabla^k e_n\left[\frac{XY}{(q-1)(t-1)}\right] = \sum_{[{\mathbf{m}},{\mathbf{a}},{\mathbf{b}}]} \frac{t^{|{\mathbf{m}}|} q^{\mathrm{dinv}_k({\mathbf{m}},{\mathbf{a}},{\mathbf{b}})} X_{\mathbf{a}} Y_{\mathbf{b}}}{(q-1)^n \mathrm{aut}_q({\mathbf{m}},{\mathbf{a}},{\mathbf{b}})}.\]

Proof. Applying Corollary 4.3, we write the left-hand side as a summation over isomorphism classes of parabolic bundles with two marked points. These are identified with multisets of triples $({\mathbf{m}}, {\mathbf{a}}, {\mathbf{b}})$ by Proposition 4.5. We claim that for each such multiset the corresponding terms match, i.e. we have

\[q^{k \binom{n}{2}} \frac{t^{\deg} \prod_{j=1}^N x_j^{r_{1,j}} y_{j}^{r_{2,j}}}{|\mathrm{Aut}(\mathcal{E},\mathbf{F}^0,\mathbf{F}^\infty)|}|\mathrm{Nilp}_k(\mathcal{E},\mathbf{F}^0,\mathbf{F}^\infty)| = \frac{t^{|{\mathbf{m}}|} q^{\mathrm{dinv}_k({\mathbf{m}},{\mathbf{a}},{\mathbf{b}})} X_{\mathbf{a}} Y_{\mathbf{b}}}{(q-1)^n \mathrm{aut}_q({\mathbf{m}},{\mathbf{a}},{\mathbf{b}})}. \]

The monomials $X_{\mathbf{a}}$ and $Y_{{\mathbf{b}}}$ and the t-degree match naturally. Using Propositions 4.9 and 4.10 to express the right-hand side and throwing away the resulting common factor ${1}/({(q-1)^n \mathrm{aut}_q({\mathbf{m}},{\mathbf{a}},{\mathbf{b}})})$ , we are reduced to show that the q-degree $\mathrm{dinv}_k({\mathbf{m}}, {\mathbf{a}}, {\mathbf{b}})$ equals

\begin{eqnarray*} && k \binom{n}{2} + \sum_{i<j} \max(1-k+m_i-m_j - \delta_{a_j<a_i} - \delta_{b_j<b_i}, 0)\\&&\qquad - \sum_{i<j} \max(1+m_i-m_j - \delta_{a_j<a_i} - \delta_{b_j<b_i}, 0). \end{eqnarray*}

For each pair $i<j$ let $c_{i,j} = m_i-m_j - \delta_{a_j<a_i} - \delta_{b_j<b_i}$ and note that $c_{i,j}\geq -1$ (case-by-case analysis using the assumption that the sequence of triples is ordered). Then the above sum can be written as

\[ \sum_{i<j} \left(k+\max(1-k+c_{i,j},0) - (1+c_{i,j})\right) = \sum_{i<j} \max(k-1-c_{i,j}, 0). \]

Each summand matches the corresponding summand in Definition 2.2,

\[ \max(k-1-c_{i,j}, 0) = \mathrm{dinv}_k^{i,j}({\mathbf{m}}, {\mathbf{a}}, {\mathbf{b}}), \]

and therefore the q-degrees also match.

5.1 Affine permutations

We describe the connection between affine permutations and the combinatorics of the dinv statistic and rational slope parking functions, following [Reference Gorsky, Mazin and VaziraniGMV14].

The set of affine permutations is defined as

(38) \begin{equation} W=\left\{\mbox{bijections } w:\mathbb{Z}\rightarrow \mathbb{Z} : w(i+n)=w(i)+n\right\}.\end{equation}

Each one is determined by its values in window notation, $w=(w_1,\ldots ,w_n)$ , where as usual we denote $w_i=w(i)$ . Note that we are using the unconstrained ‘ $GL_n$ ’ version of affine permutations as opposed to the ‘ $SL_n$ ’ types, which requires that $w_1+\cdots +w_n=n(n+1)/2$ . We define the set of positive affine permutations as

\[W^+_n=\left\{ w\in W:w(i+n)=w(i)+n, \mbox{$w_i\geq 1$ for $i\geq 1$}\right\}.\]

Let

\[W^+_{n,d}=\left\{w\in W^+_{n} :w_1+\cdots+w_n=dn+n(n+1)/2\right\}.\]

We may multiply $w\in W_{n,d}$ and $w'\in W_{n,d'}$ to obtain a permutation in $W_{n,d+d'}$ . For each d we have the Bruhat order $\leq_{bru}$ on $W^+_{n,d}$ , as defined in [Reference Bjorner and BrendiBB05, Reference Lam, Lapointe, Morse, Schilling, Shimozono and ZabrockiLLM⁺14].

Following [Reference Gorsky, Mazin and VaziraniGMV14], we have the following.

For integers $a,b \in \mathbb{Z}$ which are not congruent modulo n, there is a unique affine transposition $t_{a,b}$ which switches the two. Note that $t_{a,b}=t_{b,a}$ , and $t_{a,b}=t_{a+kn,b+kn}$ , so the map taking pairs of incongruent integers to W is many-to-one. Given an m-restricted permutation, let

(39) \begin{equation} E_m(w)= \left\{t_{a,b} : t_{a,b} w\leq_{bru}w,\ |a-b|<m\right\}.\end{equation}

The statistic $|a-b|$ does not depend on the representatives a,b or their order, and is called the height of the transposition. The set $E_m(w)$ represent directed edges $w\rightarrow v$ with $v=t_{a,b}w$ in the Goresky–Kottwitz–MacPherson (GKM) graph of $\mathcal{B}_{n,m}$ , corresponding to the one-dimensional orbits under $\widetilde{T}$ .

Recall that an (n,m)-rational slope Dyck path is one that begins at (0,0) and ends at (m,n), never crossing the line of slope $n/m$ . Again, we have the area and coarea sequences $\mathrm{area}(\pi),\mathrm{coarea}(\pi)$ , and also $D(\pi)$ . For any m-restricted permutation w, there is a rational (n,m)-Dyck path with coarea sequence

\[b(\tilde{\pi}_{m}(w))=\mathrm{sort}({\mathbf w}_m(w),<),\]

where

\[{\mathbf w}_m(w)_j =\#\left\{t_{a,b} \in E_m(w):w^{-1}t_{a,b}w=t_{i,j} \text{ for some $i<j$}\right\}.\]

This is the underlying Dyck path of sequence ${\mathbf w}_m(w)=\mathcal{PS}_{w^{-1}}$ of [Reference Gorsky, Mazin and VaziraniGMV14], which was shown to define a bijection from the set of m-stable affine permutations in $W_n$ to rational parking functions for (n,m) coprime in [Reference Thomas and WilliamsTW15].

We define the following.

Definition 5.2. Let

$\mathrm{dinv}_m(w)=\mathrm{area}(\pi_{n,m})-\#E_m(w)=\mathrm{area}(\tilde{\pi}_m(w)),$

where $\pi_{n,m}=(1^n0^m)$ is the (n,m)-Dyck path of maximal area.

We now explain the connection with the statistic $\mathrm{dinv}_k({\mathbf{m}},{\mathbf{a}},{\mathbf{b}})$ of § 2.3. Recall the definition of Standardization from [Reference Haglund, Haiman and LoehrHHL05b].

We will also define $\mathrm{Std}_<({\mathbf{a}})$ and $\mathrm{Std}_>({\mathbf{a}})$ with respect to the usual, and reverse order on $\mathbb{Z}_{\geq 1}$ , so that $\mathrm{Std}=\mathrm{Std}_<$ . For instance, if ${\mathbf{a}}=(3,3,3,1,2,3,1)$ , then

$\mathrm{Std}_<({\mathbf{a}})=(4, 5, 6, 1, 3, 7, 2),\quad\mathrm{Std}_>({\mathbf{a}})=(1, 2, 3, 6, 5, 4, 7).$

In particular, $\mathrm{dinv}_k$ respects standardization, i.e. $\mathrm{dinv}_k({\mathbf{m}},\mathrm{Std}({\mathbf{a}}),\mathrm{Std}({\mathbf{b}}))=\mathrm{dinv}_k({\mathbf{m}},{\mathbf{a}},{\mathbf{b}})$ .

Now given a tuple $({\mathbf{m}},{\mathbf{a}},{\mathbf{b}})$ which is sorted, we define an affine permutation

(40) \begin{equation} \mathrm{aff}({\mathbf{m}},{\mathbf{a}},{\mathbf{b}})= \mathrm{Std}_{>}(\mathrm{rev}({\mathbf{b}})) t({\mathbf{m}}) \mathrm{Std}_{<}({\mathbf{a}})^{-1},\end{equation}

where $t({\mathbf{m}})=(n+m_1 n,\ldots ,1+m_nn)$ is the maximal representative of its coset in $S_n\backslash W^+_n /S_n$ , and $\mathrm{rev}({\mathbf{b}})$ is the result of writing ${\mathbf{b}}$ in the reverse order. We similarly define $\mathrm{aff}({\mathbf{m}},{\mathbf{a}})$ as the left coset $S_n \mathrm{aff}({\mathbf{m}},{\mathbf{a}},{\mathbf{b}})$ , which is independent of ${\mathbf{b}}$ . The proof of the following proposition is tedious, and will be omitted.

Proposition 5.4. Fix multisets A,B of size n with $|A|,|B|\subset \mathbb{Z}_{\geq 1}$ . Then $\mathrm{aff}({\mathbf{m}},{\mathbf{a}},{\mathbf{b}})$ defines a bijection from the set of sorted triples

\[({\mathbf{m}},{\mathbf{a}},{\mathbf{b}})\in \mathbb{Z}_{\geq 0}^n \times \mathrm{labs}(A)\times \mathrm{labs}(B)\]

to the double coset $S_{\mathrm{rev}(\alpha(B))}W^+_n S_{\alpha(A)}$ , where $S_\alpha$ is the Young subgroup, and $\mathrm{aff}({\mathbf{m}},{\mathbf{a}},{\mathbf{b}})$ is the unique representative of its double coset of maximal length. Moreover, we have

\[\mathrm{dinv}_k({\mathbf{m}},{\mathbf{a}},{\mathbf{b}})=\mathrm{dinv}_{kn}\left(\mathrm{aff}({\mathbf{m}},{\mathbf{a}},{\mathbf{b}})\right),\]

and the Dyck path is determined by

\[a(\pi_k({\mathbf{m}},{\mathbf{a}}))=a(\tilde{\pi}_{kn}(w_{\min}))-a(\tilde{\pi}_{kn}(w_{\max})),\]

where $w_{\min},w_{\max}$ are the unique representatives of the coset $\mathrm{aff}({\mathbf{m}},{\mathbf{a}})$ which are minimal and maximal in the Bruhat order.

Example 5.5. Take $({\mathbf{m}},{\mathbf{a}},{\mathbf{b}})=((2,1,0,0),(2,3,1,1),(1,2,1,1))$ , which is sorted. Then we have

\[t({\mathbf{m}})=(12,7,2,1),\quad \mathrm{Std}_<({\mathbf{a}})=(3, 4, 1, 2),\quad \mathrm{Std}_>(\mathrm{rev}({\mathbf{b}}))=(2,3,1,4),\]

which gives $w=\mathrm{aff}({\mathbf{m}},{\mathbf{a}},{\mathbf{b}})=(3,2,12,5)$ . This is the maximal length element in the double coset

\[S_{(1,3)} w S_{(2,1,1)}=\left\{(2, 3, 12, 5), (2, 4, 11, 5),\right.\]

\[\left.(3, 2, 12, 5), (3, 4, 10, 5), (4, 2, 11, 5), (4, 3, 10, 5)\right\}.\]

Now, for $k=1$ , we have

\[E_{4}(w)=\left\{(2, 3, 12, 5), (3, 2, 9, 8), (4, 2, 11, 5), (3, 4, 10, 5)\right\}\]

so that $\mathrm{dinv}_{4}(w)=6-4=2$ . On the other hand, $({\mathbf{m}},{\mathbf{a}})$ has three attacking pairs, $\left\{(2,3),(2,4),(3,4)\right\}$ . Since (2,3) and (2,4) are the pairs for which $b_i>b_j$ , we see that $\mathrm{dinv}_1({\mathbf{m}},{\mathbf{a}},{\mathbf{b}})=2$ , in agreement with Proposition 5.4.

Example 5.6 Let us compute the Dyck path for the terms ${\mathbf{m}},{\mathbf{a}}$ from Example 2.5, using Proposition 5.4. Then we have that

\[w_{\min}=(19, 20, 5, 16, 21, 6), \quad w_{\max}=(24, 23, 2, 15, 22, 1)\]

are the minimal and maximal representatives of the left coset of $\mathrm{aff}({\mathbf{m}},{\mathbf{a}})\in S_n\backslash W^+_n$ . Then

\[a({\mathbf w}_{12}(w_{\min}))=(0, 2, 4, 4, 1, 2), \quad a({\mathbf w}_{12}(w_{\max}))=(0, 1, 2, 1, 0, 1),\]

and the unique (n,n)-Dyck path whose area sequence is the difference (0, 1, 2, 3, 1, 1) is the expected one from Figure 1.

In this language, we have a corollary of Theorem A.

Corollary 5.7. The coefficient of the monomial symmetric functions associated to $\lambda=\mu=(1^n)$ is given by

(41) \begin{equation} [{X}_{1^n} Y_{1^n}]\nabla^k e_n\left[ \frac{XY}{(1-q)(1-t)}\right]=\sum_{w\in W^+_n} t^{\mathrm{area}(w)} q^{\mathrm{dinv}_{kn}(w)}.\end{equation}

5.2 Polygraphs and the Hilbert scheme

If M is a representation of $S_n\times \cdots \times S_n$ with k factors, we will denote the Frobenius character by

\[\mathcal{F}_{X_1,\ldots ,X_k} M \in \mathbb{C}[x_{i,j}]^{S_n\times \cdots \times S_n},\]

which is a function in k sets of variables, $X_i=(x_{i,1},x_{i,2},\ldots )$ , individually symmetric in each one. For doubly graded modules, the Frobenius character encodes the degrees with the q,t variables, namely

\[\mathcal{F} M=\sum_{i,j} q^it^j \mathcal{F} M^{(i,j)}\]

where $M^{(i,j)}$ is the homogeneous component of the bigrading.

In Haiman’s theory [Reference HaimanHai01b], the expression in Theorem A is the equivariant index of a sheaf on the Hilbert scheme of points in the complex plane $\mathrm{Hilb}_n \mathbb{C}^2$ , with respect to the usual torus action . Let P be the Procesi bundle of rank $n!$ whose fibers carry an action of $S_n$ isomorphic to the regular representation. The modified Macdonald polynomial is the Frobenius character $\tilde{H}_\lambda=\mathcal{F} P\big|_{\lambda}$ of the fiber of P at a monomial ideal, which are the torus-fixed points of $\mathrm{Hilb}_n \mathbb{C}^2$ . Explicitly, they are given by the Garsia–Haiman module, as in Haiman’s proof of the $n!$ conjecture [Reference HaimanHai01a]. Then by a noncompact version of the localization theorem due to Nakajima of [Reference Nakajima and YoshiokaNY03, Proposition 4.1], we have

(42) \begin{equation} \nabla^k e_n\left[\frac{XY}{(1-q)(1-t)}\right]= \sum_{i}(-1)^i \mathcal{F}_{Y,X} R^i\Gamma(P\otimes P^{*}\otimes \mathcal{L}^k).\end{equation}

We next define a variant of Haiman’s polygraph modules [Reference HaimanHai01a]. Fix n and let ${\mathbf{x}}$ denote the set of variables $(x_1,\ldots ,x_n)$ , for some variable x. Let $\mathbb{C}[{\mathbf{x}},{\mathbf{y}}]\cdot S_n$ denote the free left $\mathbb{C}[{\mathbf{x}},{\mathbf{y}}]$ -module with one free generator for each permutation $\tau\in S_n$ . Consider the following variant of Haiman’s map from [Reference HaimanHai01a] equation (152):

(43) \begin{equation} \phi : \mathbb{C}[{\mathbf{x}},{\mathbf{y}},{\mathbf{z}},{\mathbf{w}}]\rightarrow \mathbb{C}[{\mathbf{x}},{\mathbf{y}}]\cdot S_n,\quad g({\mathbf{x}},{\mathbf{y}},{\mathbf{z}},{\mathbf{w}})\mapsto \sum_{\tau \in S_n} g({\mathbf{x}},{\mathbf{y}},\tau({\mathbf{x}}),\tau({\mathbf{y}}))\tau.\end{equation}

We define a module M as the image of $\phi$ , as a $\mathbb{C}[{\mathbf{x}},{\mathbf{y}}]$ -module. We have the usual biograding on M compatible with the grading on the ring $\mathbb{C}[{\mathbf{x}},{\mathbf{y}},{\mathbf{z}},{\mathbf{w}}]$ , in which the degree of the ${\mathbf{x}},{\mathbf{z}}$ variables are (1,0), and the ${\mathbf{y}},{\mathbf{w}}$ variables have degree (0,1). Note that ${\mathbf{x}},{\mathbf{y}}$ have nothing to do with the symmetric function variables X,Y.

There is an action of $S_n\times S_n$ on M, which may also be interpreted as a commuting left and right action by

\[(\sigma_1,\sigma_2)\cdot f({\mathbf{x}},{\mathbf{y}})\tau=\sigma_1 \cdot f({\mathbf{x}},{\mathbf{y}})\tau \cdot \sigma_2^{-1}= f(\sigma_1({\mathbf{x}}),\sigma_1({\mathbf{y}})) \left(\sigma_1 \tau \sigma_2^{-1}\right).\]

Then $\phi$ intertwines this action with the one where the first factor simultaneously permutes ${\mathbf{x}},{\mathbf{y}}$ , and the second factor permutes ${\mathbf{z}},{\mathbf{w}}$ . Notice that the left $S_n$ -action on M is compatible with the action on the ground ring $\mathbb{C}[{\mathbf{x}},{\mathbf{y}}]$ by simultaneously permuting the indices of the variables, whereas the right $S_n$ -action does not act on the variables. Another way to say this is that M is a bigraded module over the smash product $\mathbb{C}[{\mathbf{x}},{\mathbf{y}}]\rtimes S_n$ , which is the noncommutative ring by adjoining a generator for each $\sigma \in S_n$ with the relation

\[\sigma x_i= x_{\sigma_i} \sigma,\quad\sigma y_i= y_{\sigma_i},\]

and that the right action of $S_n$ acts by automorphisms of M.

The following conjecture was proved in [Reference Alvarez and LosevAL24].

Conjecture 5.8. As a module over $\mathbb{C}[{\mathbf{x}},{\mathbf{y}}] \rtimes S_n$ , M is the image of the Procesi bundle P under the Haiman–Bridgeland–King–Reid isomorphism

\[F \mapsto R\Gamma_{\mathrm{Hilb}_n}(P\otimes F).\]

The higher derived functors $R^i\Gamma(P\otimes P)$ vanish, and so $M\cong \Gamma_{\mathrm{Hilb}_n}(P\otimes P)$ . Moreover, we have that M is free when regarded as a module over $\mathbb{C}[{\mathbf{x}}]$ , in other words forgetting the $\mathbb{C}[{\mathbf{y}}]$ -action.

To connect this with Theorem A, observe that combining (42) at $k=1$ with Conjecture 5.8 gives the following.

Conjecture 5.10. We have

(44) \begin{equation}\nabla e_n\left[\frac{XY}{(1-q)(1-t)}\right]=\mathcal{F}_{Y,X} M.\end{equation}

Remark 5.11. The conjecture is motivated by the following geometric picture. Recall the following commutative diagram [Reference HaimanHai01a].

The diagram is a reduced cartesian product and the space $X_n$ is the isospectral Hilbert scheme. The map $\pi$ is finite and $P=\pi_* \mathcal{O}_{X_n}$ . Thus the ring $\Gamma_{\mathrm{Hilb}_n}(P\otimes P)$ is the ring of functions on $X_n\times_{\mathrm{Hilb}_n\mathbb{C}^2} X_n$ , which is a closed subscheme of $X_n\times X_n$ . On the other hand, $X_n\times_{\mathrm{Hilb}_n\mathbb{C}^2} X_n$ is reduced, so it coincides with the reduced fiber product

\[X_n\times_{\mathrm{Hilb}_n\mathbb{C}^2} X_n = (\mathrm{Hilb}_n \times_{\mathbb{C}^{2n}/S_n} (\mathbb{C}^{2n}\times_{\mathbb{C}^{2n}/S_n} \mathbb{C}^{2n}))_{\mathrm{red}}.\]

The space $\mathbb{C}^{2n}\times_{\mathbb{C}^{2n}/S_n}\mathbb{C}^{2n}$ is the union of graphs of permutations viewed as maps $\mathbb{C}^{2n}\to \mathbb{C}^{2n}$ . This induces a covering of $X_n\times_{\mathrm{Hilb}_n\mathbb{C}^2} X_n$ by $n!$ copies of $X_n$ . Passing to the rings of functions we obtain ring homomorphisms,

\[\Gamma(\mathcal{O}_{X_n}) \otimes \Gamma(\mathcal{O}_{X_n}) \to \Gamma(\mathcal{O}_{X_n\times_{\mathrm{Hilb}_n\mathbb{C}^2} X_n}) \to \bigoplus_{\sigma\in S_n} \Gamma(\mathcal{O}_{X_n}),\]

whose composition becomes the map $\phi$ of (43) under the identification $\Gamma(\mathcal{O}_{X_n})=\Gamma_{\mathrm{Hilb}_n \mathbb{C}^{2}} (P)=\mathbb{C}[{\mathbf{x}},{\mathbf{y}}]$ , see [Reference HaimanHai01b]. The second map above is injective because the functor $\Gamma$ is left exact. If we knew that the first map is surjective, we would have

\[\Gamma_{\mathrm{Hilb}_n\mathbb{C}^2}(P\otimes P) =\Gamma(\mathcal{O}_{X_n\times_{\mathrm{Hilb}_n\mathbb{C}^2} X_n}) = \mathrm{Im}\ \phi.\]

The conjecture is then reduced to the vanishing of the higher cohomologies of the ideal sheaf of $X_n\times_{\mathrm{Hilb}_n\mathbb{C}^2} X_n$ in $X_n \times X_n$ .

5.3 Connection with affine Springer fibers

The m-restricted permutations can be interpreted as the torus-fixed points of a certain affine Springer fiber $\mathcal{B}_{n,m}=\mathcal{B}_{\gamma}$ of the type studied in [Reference Goresky, Kottwitz and MacphersonGKM03] as follows. Define $v_i \in \mathbb{C}^n((t))$ by

\[v_i=t^{-(i-i_1)/n} e_{i_1},\quad i_1=((i-1)\ {\rm mod}\ n)+1,\]

where $e_i$ is the standard basis vector in $\mathbb{C}^n$ and $i_1$ is the unique element in $\{1,\ldots ,n\}$ congruent to i modulo n. We can then describe affine permutations as invertible matrices with values in $\mathbb{C}[t,t^{-1}]$ by the action $w\cdot v_i=v_{w_i}$ . For instance, we would have

$(-2,0,5) \mapsto \left( \begin {array}{ccc} t&0&0\\ 0&0&{t}^{-1}\\ 0&t&0\end {array} \right).$

Now let $\gamma=\gamma_{n,m}$ be the topologically nilpotent operator

\[\gamma (v_i)=a_{i} v_{i-m}.\]

The $a_i$ are distinct nonzero complex numbers for $1\leq i \leq d$ , and $a_{i+d}=a_i$ for $d=\gcd(n,m)$ . For $m=kn$ , we have $\gamma=\mathrm{diag}(a_1t^k,\ldots ,a_nt^k)$ , corresponding to the unramified case studied in [Reference Goresky, Kottwitz and MacPhersonGKM04]. The affine Springer fiber associated to $\gamma$ is defined by

(45) \begin{equation}\mathcal{B}_{\gamma}=\left\{wI_n \in \mathcal{Y}_{n}:w^{-1} \gamma w \in \mathrm{Lie}(I_n)\right\}\end{equation}

where $I_n$ is the Iwahori subgroup. Then an affine permutation w is m-restricted precisely when $wI_n \in \mathcal{B}_{n,m}=\mathcal{B}_{\gamma}$ , realizing w as a matrix as above. They are in fact the fixed points of $\mathcal{B}_{n,m}$ for the action of a certain restricted torus. In this case the n-dimensional torus $T \subset GL_n(C)$ acts by multiplication on the left via, as well as the extended $(n+1)$ -dimensional torus T, which includes loop rotation, both having discrete fixed points described by $W^+_{n,d}$ .

The space $\mathcal{B}_{n,m}$ has an affine paving by the results of [Reference Lusztig and SmeltLS91] in the coprime case, and [Reference Goresky, Kottwitz and MacphersonGKM03] for the general case, including $\mathcal{B}_{n,m}$ for general (n,m) in type A. In the unramified case of $m=kn$ studied in [Reference Goresky, Kottwitz and MacPhersonGKM04], every component in the paving is GKM with respect to the torus $\tilde{T}\cong (\mathbb{C}^*)^{n+1}$ consisting of the maximal torus $T\subset GL_n(\mathbb{C})\subset GL_n(\mathbb{C}((t)))$ together with the ‘loop rotation.’ The torus fixed points are the entire set $W_n$ , and the outgoing edges in the GKM graph correspond precisely to the elements of $E_{nk}(w)$ defined above. The equivariant Borel–Moore homology $H_{\tilde{T}}^*(\mathcal{B}_{n,kn})$ is a submodule of the equivariant homology of the fixed point set, which is a free $\mathbb{C}[{\mathbf{x}},\epsilon]$ -module with basis $W_n$ , where $\mathbb{C}[{\mathbf{x}},\epsilon]$ is identified with the equivariant cohomology of a point. Moreover, the fundamental classes of the closures of the elements of the affine paving are a free basis of $H_{\tilde{T}}^*(\mathcal{B}_{n,nk})$ [Reference GrahamGra01, Reference Edidin and GrahamEG96, Reference BrionBri98].

By [Reference GrahamGra01, Proposition 2.1], we have a dual basis of equivariant cohomology, which must be triangular in the Bruhat order in the opposite direction. By the GKM property, the leading terms must be

(46) \begin{equation} a_{w,k}({\mathbf{x}},\epsilon)=\prod_{t_{i,j+ln} \in E_{kn}(w)} (x_i-x_j+l\epsilon),\end{equation}

where $E^{i,j}_{m}(w)$ is the set of transpositions defined above. By restricting the torus and setting $\epsilon=0$ , we lose the GKM property but have localization, as well as the given basis. The standard description of the corresponding homology is then given as a subspace of the $\mathbb{C}({\mathbf{x}})$ -vector space with the same fixed point basis, which is different from the description of [Reference Goresky, Kottwitz and MacPhersonGKM04]. For instance, in the case of $k=\infty$ , we can compare the coefficient $a_{w,\infty}({\mathbf{x}})$ with the leading terms in Kostant and Kumar’s nil Hecke ring [Reference Kostant and KumarKK86, Reference Lam, Lapointe, Morse, Schilling, Shimozono and ZabrockiLLM⁺14], which encodes the equivariant homology of the affine flag variety

\[A_{w}=\sum_{v\leq_{bru} w} c_{v,w}({\mathbf{x}}) v,\quad c_{w,w}({\mathbf{x}})=a_{w,\infty}({\mathbf{x}})^{-1}.\]

Theorem A was discovered by attempting to embed the above module M as a submodule of the $GL_n$ version of $H_*^T(\mathcal{B}_{n,kn})$ , in which the fixed points only consist of positive permutations

\[M\subset \bigoplus_{{\mathbf{m}},\tau} \mathbb{C}[{\mathbf{x}}]y^{{\mathbf{m}}} \tau=\bigoplus_{w\in W^+} e_w,\quad w=(\tau_1+m_{\tau_1}n,\ldots ,\tau_n+m_{\tau_n}n).\]

A construction of this type was used in [Reference Carlsson and OblomkovCO18] for instance, in which the authors exhibited an isomorphism $DR_n\cong H_*(\mathcal{B}_{n,n+1})$ related to the ones studied in [Reference Oblomkov and YunOY14], and used it to study the diagonal coinvariant algebra $DR_n$ as a module over $\mathbb{C}[{\mathbf{x}}]$ . In another example, Kivinen showed that Haiman’s alternant ideal $J_n\subset \mathbb{C}[{\mathbf{x}},{\mathbf{y}}]$ in general Lie type satisfies a suitable version of the GKM relations, and therefore injects into the equivariant Borel–Moore homology of the Grassmannian version of $\mathcal{B}_{n,kn}$ . In type A, when combined with Haiman’s results, it follows that the map $J_n^k \rightarrow H^T_*(\mathcal{B}_{n,kn}^{Grass})$ is an isomorphism when the y-variables are inverted, see [Reference KivinenKiv20, Theorem 1.1].

Now let

\[b_{w,k}({\mathbf{x}})=\prod_{i<j}(x_i-x_j)^{k-\#E^{i,j}_{kn}(w)},\]

whose degree is $\mathrm{dinv}_{kn}(w)$ . Here $E^{i,j}_{kn}(w)$ is the set of transpositions $t_{a,b}\in E_{kn}(w)$ for which $\{\bar{a},\bar{b}\}=\{\bar{i},\bar{j}\}$ , where the bar is the congruence class modulo n. The following conjecture illustrates the connection with Theorem A in the case $k=1$ .

1. (1) The $A_w$ freely generate M as a $\mathbb{C}[{\mathbf{x}}]$ -module.

2. (2) The coefficients satisfy $c_{v,w}({\mathbf{x}})=0$ unless $v\leq_{bru} w$ , and the leading term is given by $c_{w,w}({\mathbf{x}})=b_{w,1}({\mathbf{x}})$ .

3. (3) For any compositions $\alpha,\beta$ , if w is the element of maximal length in $S_\alpha \backslash W^+_n / S_\beta$ , then $A_w \in M^{\alpha,\beta}$ , the invariant subspace with respect to the product of the corresponding Young subgroups.

In particular, there is the expected freeness of M over $\mathbb{C}[{\mathbf{x}}]$ . On the other hand, in light of Conjecture 5.12, we expect that $\mathcal{F}_{Y,X} M=\Omega_1[X,Y]$ with respect to the dot and star actions mentioned in the introduction. To see this in the case of the Hilbert series, we take the contribution to $\Omega_1[X,Y]$ for which ${\mathbf{a}},{\mathbf{b}}$ have all distinct entries. Then the automorphism factor is trivial, and we obtain the sum in (1), using Proposition 5.4 to relate the corresponding dinv statistics.