2018/04/17 by Bärligea, Christoph · 1 citation
#05E15 (Secondary) #14N15 #20G42 (Primary) 20F55 #Combinatorics (math.CO) #FOS: Mathematics #Quantum Algebra (math.QA)
paper · doi:10.48550/arxiv.1804.06156
Let (W,S) be a finite Coxeter system with root system R and with set of positive roots R+. For α∈ R, v,w∈ W, we denote by ∂α, ∂w and ∂w/v the divided difference operators and skew divided difference operators acting on the coinvariant algebra of W. Generalizing the work of Liu, we prove that ∂w/v can be written as a polynomial with nonnegative coefficients in ∂α where α∈ R+. In fact, we prove the stronger and analogous statement in the Nichols-Woronowicz algebra model for Schubert calculus on W after Bazlov. We draw consequences of this theorem on saturated chains in the Bruhat order, and partially treat the question when ∂w/v can be written as a monomial in ∂α where α∈ R+. In an appendix, we study related combinatorics on shuffle elements and Bruhat intervals of length two.