Geodesic
Positive expressions for skew divided difference operators
Journal of Algebraic Combinatorics, Tome 42 (2015) no. 3, pp. 861-874.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

For permutations $v,w\in\frak S_n$, Macdonald defines the skew divided difference operators $\partial_{w/v}$ as the unique linear operators satisfying $\partial_w(PQ)=\sum_vv(\partial_{w/v}P)\cdot\partial_vQ$ for all polynomials $P$ and $Q$. We prove that $\partial_{w/v}$ has a positive expression in terms of divided difference operators $\partial_{ij}$ for $i$. In fact, we prove that the analogous result holds in the Fomin-Kirillov algebra $\mathcal E_n$, which settles a conjecture of A. N. Kirillov [SIGMA, Symmetry Integrability Geom. Methods Appl. 3, Paper 072, 14 p. (2007; Zbl 1146.05053)].
Classification : 05E15, 16T05, 33D80
Keywords: skew divided difference operator, Fomin-Kirillov algebra, braided Hopf algebra, Bruhat order 
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Liu, Ricky Ini. Positive expressions for skew divided difference operators. Journal of Algebraic Combinatorics, Tome 42 (2015) no. 3, pp. 861-874. http://geodesic.mathdoc.fr/item/JAC_2015__42_3_a1/