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Younggwang Cho

Publications and source records attributed to Younggwang Cho.

4 recordsLinked to original sources
arXiv2026

Askey--Wilson polynomials with ASEP parameters

Koornwinder moments generalize the Askey--Wilson moments arising in the asymmetric simple exclusion process. Rains conjectured that, under the specialization $t=q$, the minimal numerators of Koornwinder moments have nonnegative integer coefficients. While the one-row case of this conjecture was previously proved by Corteel, Mandelshtam, and Williams using rhombic staircase tableaux, its dual counterpart, the one-column case, has remained open. In this paper, we derive a closed, manifestly positive combinatorial formula for the normalized numerators of the coefficients of the rescaled Askey--Wilson polynomials. This proves Rains' conjecture for one-column partitions, thereby providing the exact dual counterpart to the previous result.

math.CO↗
arXiv2026

Haglund--Haiman--Loehr formula via Carlsson--Mellit Algebra

We prove a Haglund--Haiman--Loehr type combinatorial formula for the torus fixed point classes $I_{μ,w}$ in the equivariant $K$-theory $K_{\mathbb{C}^{*}\times\mathbb{C}^{*}}(\operatorname{PFH}_{n,n-k})$ of parabolic flag Hilbert schemes. Under the identification of Bechtloff Weising and Orr, our result gives an HHL type formula for modified partially symmetric Macdonald functions in terms of weighted partial Dyck paths. When $w=\emptyset$, it specializes to the classical HHL formula. The proof relies essentially on the Carlsson--Gorsky--Mellit action of the Carlsson--Mellit algebra $\mathbb{A}_{q,t}$.

math.CO↗
arXiv2026

An explicit formula for Koornwinder moments and Rains' positivity conjecture

The asymmetric simple exclusion process (ASEP) is an important particle model with deep connections to orthogonal polynomials. Motivated by this connection, Corteel and Williams introduced the Koornwinder moments $M^{Z}_λ$ at $ t=q $, which generalize the moments of Askey--Wilson polynomials. They showed that the partition function of the two-species ASEP is equal to $M^{Z}_λ$ for a one-row partition $ λ$. In this paper, we investigate a conjecture of Rains on the positivity of the minimal numerator of the Koornwinder moment $M^{Z}_λ$. We derive the first explicit formula for this moment, thereby obtaining a precise formulation of the conjecture by determining the minimal denominator of $M^{Z}_λ$. We also propose a generalization of the conjecture for the more general Koornwinder moments $M^{Z}_{λ,μ}$ indexed by two partitions at special parameter values. We prove the generalized Rains' conjecture in two special cases: $(ξ,q)=(1,0)$ and $(ξ,q)=(1,1)$. For $(ξ,q)=(1,0)$, we construct a lattice path model and obtain a combinatorial formula for $M^{Z}_{λ,μ}$ in terms of non-intersecting lattice paths. For $(ξ,q)=(1,1)$, we establish an explicit product formula for $M^{Z}_{λ,μ}$ and give a combinatorial interpretation using lecture hall tableaux.

math.CO↗