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arXiv · 2606.02395

Shape changing identities for permuted-basement nonsymmetric Macdonald polynomials

Guilherme Zeus Dantas e Moura·Olya Mandelshtam
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Abstract

Permuted-basement Macdonald polynomials $E_\alpha^\sigma(\mathbf{x};q,t)$ 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 $E_\alpha^\sigma$, $E_\alpha^{\sigma s_i}$, $E_{s_i\alpha}^\sigma$, and $E_{s_i\alpha}^{\sigma s_i}$. 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 $E_\alpha^\sigma$ in the polynomials $\{E_{s_i\alpha}^\tau\}_\tau$.

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Guilherme Zeus Dantas e Moura, Olya Mandelshtam. 2026-06-01. Shape changing identities for permuted-basement nonsymmetric Macdonald polynomials. https://arxiv.org/abs/2606.02395

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