Shape changing identities for permuted-basement nonsymmetric Macdonald polynomials

Abstract

Permuted-basement Macdonald polynomials $E_α^σ(\mathbf{x};q,t)$ are nonsymmetric generalizations of symmetric Macdonald polynomials that form a basis for the polynomial ring $\mathbb{Q}(q,t)[\mathbf{x}]$ for each fixed $σ$. 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 $E_α^σ$, $E_α^{σs_i}$, $E_{s_iα}^σ$, and $E_{s_iα}^{σs_i}$. 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 $E_α^σ$ in the polynomials $\{E_{s_iα}^τ\}_τ$.