More Like this

1. A raising operator formula for Macdonald polynomials

   https://doi.org/10.1017/fms.2025.8  

   Blasiak, J; Haiman, M; Morse, J; Pun, A; Seelinger, G H (January 2025, Forum of Mathematics, Sigma)

   Abstract We give an explicit raising operator formula for the modified Macdonald polynomials$$\tilde {H}_{\mu }(X;q,t)$$, which follows from our recent formula for$$\nabla $$on an LLT polynomial and the Haglund-Haiman-Loehr formula expressing modified Macdonald polynomials as sums of LLT polynomials. Our method just as easily yields a formula for a family of symmetric functions$$\tilde {H}^{1,n}(X;q,t)$$that we call$$1,n$$-Macdonald polynomials, which reduce to a scalar multiple of$$\tilde {H}_{\mu }(X;q,t)$$when$$n=1$$. We conjecture that the coefficients of$$1,n$$-Macdonald polynomials in terms of Schur functions belong to$${\mathbb N}[q,t]$$, generalizing Macdonald positivity. 

   more » « less
2. Wreath Macdonald operators

   https://doi.org/10.1017/fms.2025.10061  

   Orr, Daniel; Shimozono, Mark; Wen, Joshua (January 2025, Forum of Mathematics, Sigma)

   Abstract We construct a novel family of difference-permutation operators and prove that they are diagonalized by the wreath MacdonaldP-polynomials; the eigenvalues are written in terms of elementary symmetric polynomials of arbitrary degree. Our operators arise from integral formulas for the action of the horizontal Heisenberg subalgebra in the vertex representation of the corresponding quantum toroidal algebra. 

   more » « less
3. Principal Specialization of Dual Characters of Flagged Weyl Modules

   https://doi.org/10.37236/10460  

   Mészáros, Karola; St. Dizier, Avery; Tanjaya, Arthur (October 2021, The Electronic Journal of Combinatorics)

   Schur polynomials are special cases of Schubert polynomials, which in turn are special cases of dual characters of flagged Weyl modules. The principal specialization of Schur and Schubert polynomials has a long history, with Macdonald famously expressing the principal specialization of any Schubert polynomial in terms of reduced words. We prove a lower bound on the principal specialization of dual characters of flagged Weyl modules. Our result yields an alternative proof of a conjecture of Stanley about  the principal specialization of Schubert polynomials, originally proved by Weigandt. 

   more » « less
4. Wreath Macdonald polynomials as eigenstates

   https://doi.org/10.1007/s00029-025-01059-0  

   Wen, Joshua Jeishing (July 2025, Selecta Mathematica)

   Abstract We show that the wreath Macdonald polynomials for$$\mathbb {Z}/\ell \mathbb {Z}\wr \Sigma _n$$ $Z/\ell Z\wr {\Sigma }_n$, when naturally viewed as elements in the vertex representation of the quantum toroidal algebra$$U_{\mathfrak {q},\mathfrak {d}}(\ddot{\mathfrak {sl}}_\ell )$$ $U_{q,d}\left({\ddot{\mathrm{sl}}}_{\ell }\right)$, diagonalize its horizontal Heisenberg subalgebra. Our proof makes heavy use of shuffle algebra methods, and we also obtain a new proof of existence of wreath Macdonald polynomials. 

   more » « less
5. Capped vertex functions for $${\text {Hilb}}^n(\mathbb {C}^2)$$

   https://doi.org/10.1007/s11005-025-01933-0  

   Ayers, Jeffrey; Smirnov, Andrey (June 2025, Letters in Mathematical Physics)

   We obtain explicit formulas for the K-theoretic capped descendent vertex functions of $${\text {Hilb}}^n(\mathbb {C}^2)$$ for descendents given by the exterior algebra of the tautological bundle. This formula provides a one-parametric deformation of the generating function for normalized Macdonald polynomials. In particular, we show that the capped vertex functions are rational functions of the quantum parameter. 

   more » « less