[Paper Review] On the Schur expansion of Jack polynomials

Per Alexandersson, James Haglund+1 more|arXiv (Cornell University)|May 1, 2018

Advanced Combinatorial Mathematics9 references3 citations

TL;DR

This paper proposes and provides evidence for two conjectures on the Schur expansion of modified Jack polynomials $\tilde{J}_{\mu}^{(\alpha)}$, showing that coefficients in binomial bases $\binom{\alpha+k}{n}$ and $\binom{\alpha}{k}k!$ are nonnegative integers, with real-rooted generating polynomials. The results connect to Eulerian numbers, Stirling numbers, quasi-Yamanouchi tableaux, and rook boards, offering new combinatorial interpretations and suggesting deeper algebraic structures in symmetric function theory.

ABSTRACT

We present positivity conjectures for the Schur expansion of Jack symmetric functions in two bases given by binomial coefficients. Partial results suggest that there are rich combinatorics to be found in these bases, including Eulerian numbers, Stirling numbers, quasi-Yamanouchi tableaux, and rook boards. These results also lead to further conjectures about the fundamental quasisymmetric expansions of these bases, which we prove for special cases.

Motivation & Objective

• To investigate the Schur expansion of modified Jack polynomials $\tilde{J}_{\mu}^{(\alpha)}$ in two binomial bases involving $\alpha$.
• To establish combinatorial positivity in the coefficients of these expansions, conjecturing they are nonnegative integers.
• To explore connections between the coefficients and combinatorial objects such as Eulerian numbers, Stirling numbers, quasi-Yamanouchi tableaux, and rook polynomials.
• To propose and prove special cases of conjectures on the fundamental quasisymmetric expansion of these bases.
• To provide evidence for real-rootedness of generating polynomials associated with the coefficients.

Proposed method

• Define $\tilde{J}_{\mu}^{(\alpha)}(X) = \alpha^n J_{\mu}^{(1/\alpha)}(X)$ to stabilize the Schur expansion under $\alpha \to 1/\alpha$.
• Express the Schur coefficient $\langle \tilde{J}_{\mu}^{(\alpha)}, s_\lambda \rangle$ in the basis $\binom{\alpha+k}{n}$ and $\binom{\alpha}{k}k!$.
• Use the combinatorial formula of Knop and Sahi for the monomial expansion of Jack polynomials to derive expressions for $\langle \tilde{J}_{\mu}^{(\alpha)}, s_\mu \rangle$ in hook shapes.
• Apply rook polynomial theory to Ferrers boards, showing that hit and rook polynomials have only real zeros, which implies real-rootedness of coefficient generating polynomials.
• Use Yoo’s results on Macdonald polynomials and limits as $q \to 1$ to derive explicit formulas for $\tilde{J}_{(n)}^{(\alpha)}$ in terms of Schur functions and binomial coefficients.
• Leverage the identity $\binom{\alpha+k}{n} = \sum_i \binom{\alpha}{i} \binom{k}{n-i}$ to relate the two bases and show that Conjecture 1 implies integrality in Conjecture 2.

Experimental results

Research questions

• RQ1 Are the coefficients of $\tilde{J}_{\mu}^{(\alpha)}$ in the Schur basis, when expanded in the basis $\binom{\alpha+k}{n}$, always nonnegative integers?
• RQ2 Do the generating polynomials $\sum_k a_k(\mu,\lambda) z^k$ for these coefficients have only real roots?
• RQ3 Can the coefficients in the basis $\binom{\alpha}{k}k!$ be interpreted combinatorially, and are they also nonnegative integers?
• RQ4 What is the connection between the Schur expansion coefficients and combinatorial objects like quasi-Yamanouchi tableaux, Eulerian numbers, and rook boards?
• RQ5 Can the fundamental quasisymmetric expansions of these bases be characterized, and do they support Schur positivity?

Key findings

• For $\mu = (n)$, the Schur coefficient $\langle \tilde{J}_{(n)}^{(\alpha)}, s_\lambda \rangle$ is given by $\sum_k K_{\lambda,1^n} h_k(B(c_1,\ldots,c_n)) \binom{\alpha+k}{n}$, where $h_k$ is the $k$-th elementary symmetric function of a rook board.
• The same coefficient is also expressed as $\sum_k K_{\lambda,1^n} r_{n-k}(B(c_1,\ldots,c_n)) \binom{\alpha}{k}k!$, linking to rook polynomial statistics.
• For hook-shaped $\mu = (n-\ell,1^\ell)$, the Schur coefficient $\langle \tilde{J}_{\mu}^{(\alpha)}, s_\mu \rangle$ is a sum of two terms involving $\ell!$ and $\ell \cdot \ell!$, each multiplied by symmetric functions of rook boards with specific cell weights.
• The generating polynomials $\sum_k a_k(\mu,\lambda) z^k$ and $\sum_k b_{n-k}(\mu,\lambda) z^k$ are shown to have only real roots in special cases, due to the real-rootedness of rook and hit polynomials of Ferrers boards.
• The coefficient $K_{\lambda,1^n}$, which counts standard tableaux of shape $\lambda$ with content $1^n$, appears as a multiplicative factor in the expansion for $\mu = (n)$.
• The results suggest a potential bijection between rook board interpretations and tableau interpretations for the $\binom{\alpha}{k}k!$ basis, which could yield a full combinatorial interpretation.