Guilherme Zeus
Dantas e Moura

Recent Publications (all)

• Shape changing identities for permuted-basement nonsymmetric Macdonald polynomials (2026)

  Guilherme Zeus Dantas e Moura Olya Mandelshtam

  
  Accepted to the Proceedings of the 38th Conference on Formal Power Series and Algebraic Combinatorics (Seattle)
  Abstract

  Permuted-basement Macdonald polynomials $E^\sigma_\alpha(\mathbf{x}; q, t)$ are nonsymmetric generalizations of symmetric Macdonald polynomials that form a basis for the polynomial ring $\mathbb{Q}(q, t)[\mathbf{x}]$ for each fixed $\sigma$. There are combinatorial formulas for them as generating functions over composition-shaped non-attacking fillings. In this extended abstract, we bijectively prove identities for the relationship between $E^\sigma_\alpha, E^{\sigma s_i}_\alpha, E^\sigma_{s_i\alpha}$, and $E^{\sigma s_i}_{s_i\alpha}$. These identities correspond to two combinatorial operations on non-attacking fillings: (1) swapping adjacent entries in the basement, generalizing a result of Alexandersson (2019), and (2) swapping adjacent parts in the shape, which yields a straightening rule for expanding $E^\sigma_\alpha$ in the polynomials $\{E^\tau_{s_i\alpha}\}_\tau$.

• Deterministic and Probabilistic Bijective Combinatorics for Macdonald Polynomials (2025)

  thesis corrections

  Guilherme Zeus Dantas e Moura

  
  Master's thesis for a degree in Combinatorics and Optimization at the University of Waterloo
  Abstract

  Permuted-basement Macdonald polynomials $E^\sigma_\alpha(\mathbf{x}; q, t)$ are nonsymmetric generalizations of symmetric Macdonald polynomials indexed by a composition $\alpha$ and a permutation $\sigma$. They form a basis for the polynomial ring $\mathbb{Q}(q, t)[\mathbf{x}]$ for each fixed permutation $\sigma$. They can be described combinatorially as generating functions over augmented fillings of composition shape $\alpha$ with a basement permutation $\sigma$.

  We construct deterministic bijections and probabilistic bijections on fillings that prove identities relating $E^\sigma_\alpha$, $E^{\sigma s_i}_\alpha$, $E^\sigma_{s_i \alpha}$, and $E^{\sigma s_i}_{s_i \alpha}$. These identities correspond to two combinatorial operations on the shape and basement of the fillings: swapping adjacent parts in the shape, which expands $E^\sigma_\alpha$ in terms of $E^\sigma_{s_i \alpha}$ and $E^{\sigma s_i}_{s_i \alpha}$; and swapping adjacent entries in the basement, which gives $E^\sigma_\alpha = E^{\sigma s_i}_\alpha$ when $\alpha_i = \alpha_{i+1}$.

• Probabilistic Entry Swapping Bijections for Non-Attacking Fillings (2025)

  arXiv

  Guilherme Zeus Dantas e Moura Olya Mandelshtam

  
  
  Abstract

  Non-attacking fillings are combinatorial objects central to the theory of Macdonald polynomials. A probabilistic bijection for partition-shaped non-attacking fillings was introduced by Mandelshtam (2024) to prove a compact formula for symmetric Macdonald polynomials. In this work, we generalize this probabilistic bijection to composition-shaped non-attacking fillings. As an application, we provide a bijective proof to extend a symmetry theorem for permuted-basement Macdonald polynomials established by Alexandersson (2019), proving a version with fewer assumptions.

• Cluster monomials in graph Laurent phenomenon algebras (2025)

  journal arXiv

  Ramanuja Charyulu Telekicherla Kandalam Guilherme Zeus Dantas e Moura Dora Woodruff

  
  Algebraic Combinatorics, Volume 8 (2025) no. 4, pp. 997–1019
  Abstract

  Laurent Phenomenon algebras, first introduced by Lam and Pylyavskyy, are a generalization of cluster algebras that still possess many salient features of cluster algebras. Graph Laurent Phenomenon algebras, defined by Lam and Pylyavskyy, are a subclass of Laurent Phenomenon algebras whose structure is given by the data of a directed graph. The main result of this paper is that the cluster monomials of a graph Laurent Phenomenon algebra form a linear basis, conjectured by Lam and Pylyavskyy and analogous to a result for cluster algebras by Caldero and Keller.