Schur positivity of the $1,n$-Macdonald polynomials and the positive $\mathrm{GL}_l$-character expansion of the Macdonald series $\mathbf{H}_\mu$

Blasiak, Haiman, Morse, Pun, and Seelinger (2023) proved an explicit raising operator formula for the modified Macdonald polynomials $\tilde{H}_\mu(X;q,t)$ and, en route, defined for every coprime pair $(m,n)$ a family of symmetric functions $\tilde{H}^{m,n}_\mu(X;q,t)$, called $m,n$-Macdonald polynomials, via the action of the Burban–Schiffmann elliptic Hall algebra on symmetric functions. They conjectured (Conjecture 5.2.2) that every $1,n$-Macdonald polynomial is Schur positive, i.e. expands in the Schur basis with coefficients in $\mathbb{N}[q,t]$, and proved this equivalent (Conjecture 3.4.1) to the statement that the Macdonald series $\mathbf{H}_\mu(\mathbf{z};q,t)$ — a Weyl symmetrization of an explicit rational function — is a nonnegative combination of irreducible $\mathrm{GL}_l$ characters. This strictly generalizes the classical Macdonald positivity theorem (the case $n=1$). The conjecture remains open.

Background

A partition $\mu$ is a weakly decreasing sequence of positive integers; partitions index the most important bases of the ring of symmetric functions, among them the Schur functions $s_\lambda$, which are the irreducible characters of the general linear groups, and the modified Macdonald polynomials $\tilde{H}_\mu(X;q,t)$, a two-parameter ($q$ and $t$) basis central to algebraic combinatorics. A symmetric function is called Schur positive if its expansion in the Schur basis has all coefficients in $\mathbb{N}[q,t]$. The Macdonald positivity theorem — proved by Haiman through the geometry of Hilbert schemes (parameter spaces of zero-dimensional subschemes of the plane) and his $n!$ theorem — states that the $q,t$-Kostka coefficients $\tilde{K}_{\lambda\mu}(q,t)$ in $\tilde{H}_\mu=\sum_\lambda \tilde{K}_{\lambda\mu}(q,t)s_\lambda$ lie in $\mathbb{N}[q,t]$; it is one of the deepest results in the field. Positivity statements of this kind often conceal a combinatorial or representation-theoretic interpretation (for instance, the shuffle theorem of Haglund–Haiman–Loehr–Remmel–Ulyanov, proved by Carlsson and Mellit, exhibits $\nabla e_n$, an operator-deformed symmetric function built from Macdonald theory, as a weighted sum over Dyck paths).

The source paper works with two related objects. First, the elliptic Hall algebra $\mathcal{E}$, introduced by Burban and Schiffmann from Hall algebras of coherent sheaves on elliptic curves, acts (via Schiffmann–Vasserot) on symmetric functions and contains distinguished subalgebras $\Lambda(X^{m,n})$, one for each pair of coprime integers $(m,n)$, each isomorphic to the algebra of symmetric functions. For coprime positive $m,n$, the $m,n$-Macdonald polynomial is defined by $\tilde{H}^{m,n}_\mu=\tilde{H}_\mu[-(1-q)(1-t)X^{m,n}]\cdot 1$ (square brackets denote plethystic substitution, in which a monomial-sum expression is substituted for the alphabet of a symmetric function); the case $(m,1)$ reduces to a scalar multiple of $\tilde{H}_\mu$ itself. Second, for a partition $\mu$ of $l$, the Weyl symmetrization operator $\boldsymbol\sigma$ for $\mathrm{GL}_l$ sends a Laurent polynomial $f(z_1,\dots,z_l)$ to $\sum_{w\in S_l} w\big(f/\prod_{i<j}(1-z_j/z_i)\big)$, converting Laurent monomials into $\mathrm{GL}_l$ character combinations; applying $\boldsymbol\sigma$ to an explicit rational function built from the positive roots of $\mathrm{GL}_l$ and the arm and leg statistics of the boxes of $\mu$ (the numbers of boxes strictly to the right of, respectively strictly below, a given box in the Young diagram of $\mu$) yields the Macdonald series $\mathbf{H}_\mu(\mathbf{z};q,t)$. The authors' Theorem 5.2.1 shows that the $1,n$-Macdonald polynomials carry exactly the same data as $\mathbf{H}_\mu$: $\tilde{H}^{1,n}_\mu$ is a polynomial truncation of $(z_1\cdots z_l)^n\mathbf{H}_\mu$, and $\mathbf{H}_\mu$ is recovered as the $n\to\infty$ limit. The polynomial truncation of $\mathbf{H}_\mu$ recovers the classical positivity theorem, but positivity of the full series, equivalently of the whole $1,n$ family, is conjectural. Computationally the authors found Schur positivity for all $1,n$-Macdonald polynomials they checked, while for $(m,n)$ with $m\neq 1$ and $n\neq 1$ the $m,n$-Macdonald polynomials are typically not Schur positive (Remark 5.2.3), so the $1,n$ line is a genuinely special boundary of the theory.

Problem Statement

Prove or disprove Conjecture 5.2.2 of Blasiak–Haiman–Morse–Pun–Seelinger (arXiv:2307.06517), in the equivalent form of their Conjecture 3.4.1: for every partition $\mu$ of $l$ and every positive integer $n$, the $1,n$-Macdonald polynomial $\tilde{H}^{1,n}_\mu(X;q,t)$ — defined by $\tilde{H}_\mu[-(1-q)(1-t)X^{1,n}]\cdot 1$ via the action of the elliptic Hall algebra on symmetric functions, and computable by the paper's explicit raising operator formula — is Schur positive, i.e. its Schur expansion coefficients lie in $\mathbb{N}[q,t]$; equivalently, the Macdonald series $\mathbf{H}_\mu(\mathbf{z};q,t)$ expands as $\mathbf{H}_\mu(\mathbf{z};q,t)=\sum_\nu \mathbf{K}_{\nu,\mu}(q,t)\,\chi_\nu$ with $\mathbf{K}_{\nu,\mu}(q,t)\in\mathbb{N}[q,t]$, where $\chi_\nu$ ranges over irreducible $\mathrm{GL}_l$ characters. The two formulations are equivalent by a theorem of the source paper, so the positivity statements in the two forms stand or fall together: the statement is either true for every partition $\mu$ of $l$ and every positive integer $n$, or false for some such pair, in which case it is refuted by a single counterexample.

References

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