Algebraic and enumerative combinatorics seminar-Zeus Dantas E Moura

Title: Algebraic and enumerative combinatorics seminar

Abstract:

Permuted-basement Macdonald polynomials E_α^σ(x_1, ..., x_n; q, t) are nonsymmetric generalizations of symmetric Macdonald polynomials indexed by a composition α and a permutation σ. They can be described combinatorially as generating functions over augmented fillings of shape α and basement σ.

We construct deterministic and probabilistic bijections on fillings that prove identities relating

E_α^σ, E_α^{σ s_i}, E_{s_i α}^σ, and E_{s_i α}^{σ s_i}.

These identities arise from two operations on the shape and basement: swapping adjacent parts of the shape, which expands

E_α^σ intoE_{s_i α}^σ and E_{s_i α}^{σ s_i}; and swapping adjacent basement entries,

which gives E_α^σ = E_α^{σ s_i} when α_i = α_{i+1}.

There will be a pre-seminar presenting relevant background at the beginning graduate level starting at 1:30pm.