Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.12326

On the Nazarov-Sklyanin Lax operator for Macdonald functions

Ryan Mickler
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study the Lax operator for Macdonald symmetric functions introduced by Nazarov and Sklyanin in 2015, extending the spectral theory developed by Mickler-Moll in the Jack case. We extend the NS operator to a Lax operator acting on symmetric functions with coefficients in polynomials in an auxiliary variable, and prove it has an orthogonal eigenbasis indexed by partitions and their addable corners, with the corner contents as eigenvalues, and leading terms given by Macdonald symmetric functions. We obtain the Macdonald analogue of the author's earlier Jack box-sum identity for Littlewood-Richardson coefficients. We then strengthen this box-sum identity to a two-parameter formula, proved using a finite-rank product formula of Warnaar. Its residues give explicit linear relations among Macdonald Littlewood-Richardson coefficients.

Explore related subjects

math.COmath.QA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Ryan Mickler. 2026-10-08. On the Nazarov-Sklyanin Lax operator for Macdonald functions. https://arxiv.org/abs/2610.12326

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Hecke Triangle Groups and Dessins d'Enfants

We study finite-index subgroups of the Hecke triangle group $H_q$ of signature $(2,q,\infty)$ through the finite-edge tessellation, special polygons, and the associated dessins. The quotient finite-edge graph is a connected bipartite ribbon graph in which even vertices have degree $1$ or $2$ and odd vertices may have any degree $d\mid q$; geometrically, degree $d$ records a $d$-cluster and an elliptic stabilizer of order $q/d$. We determine two finite ambiguities that arise when one passes from this quotient graph to a special polygon. First, all tree diagrams obtained from a fixed ribbon graph are classified by the possible spanning-tree cuts modulo graph automorphisms. Second, at a proper cluster one must record which cyclic gap contains the omitted branches of the universal $q$-star; every such choice develops to a special polygon. Combining the two gives an explicit finite description of all polygon orbits over a fixed dessin, together with a separate count of normalized realizations in the fixed tessellation. Concrete examples for $H_5$, $H_6$, and $H_7$ illustrate the geometry, and two arithmetic corollaries concern level-$2$ cusp labels and Galois orbits of principal congruence dessins.

math.CO↗
arXiv2026

A new simple family of non-periodic tilings with square tiles

We define a new family of non-periodic tilings with square tiles that is mutually locally derivable with some family of tilings with isosceles right triangles. Both families are defined by simple local rules, and the proof of their non-periodicity is as simple as that of the non-periodicity of Robinson's tilings. We also relate the construction to the classical chair tiling: forgetting the decoration, our tilings map almost one-to-one onto the chair, our substitution is essentially a square root of the chair substitution, and our local rules are perfect, defining exactly the family of substitution tilings.

math.CO↗
arXiv2026

Partitioning perfect graphs into comparability graphs

We study how many comparability subgraphs are needed to partition the edge set of a perfect graph. We show that many classes of perfect graphs can be partitioned into (at most) two comparability subgraphs and this holds for almost all perfect graphs. On the other hand, we prove that for interval graphs an arbitrarily large number of comparability subgraphs might be necessary.

math.CO↗