Positive expressions for skew divided difference operators

For permutations v , w ∈ S n , Macdonald defines the skew divided difference operators ∂ w / v as the unique linear operators satisfying ∂ w ( P Q ) = ∑ v v ( ∂ w / v P ) · ∂ v Q for all polynomials P and Q . We prove that ∂ w / v has a positive expression in terms of divided difference operators ∂...

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Bibliographic Details
Published in:Journal of algebraic combinatorics 2015-11, Vol.42 (3), p.861-874
Main Author: Liu, Ricky Ini
Format: Article
Language:English
Subjects:
Combinatorics
Computer Science
Convex and Discrete Geometry
Group Theory and Generalizations
Lattices
Mathematics
Mathematics and Statistics
Order
Ordered Algebraic Structures
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Summary:For permutations v , w ∈ S n , Macdonald defines the skew divided difference operators ∂ w / v as the unique linear operators satisfying ∂ w ( P Q ) = ∑ v v ( ∂ w / v P ) · ∂ v Q for all polynomials P and Q . We prove that ∂ w / v has a positive expression in terms of divided difference operators ∂ i j for i < j . In fact, we prove that the analogous result holds in the Fomin–Kirillov algebra E n , which settles a conjecture of Kirillov.
ISSN:0925-9899
1572-9192
DOI:10.1007/s10801-015-0606-1