Positive expressions for skew divided difference operators

Abstract

Abstract. 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 ∂ ij 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.

Skew divided difference operators in the Nichols algebra associated to a finite Coxeter group

“…Item (4) corresponds to [15,Proposition 2.10(a)] and can be deduced immediately from Remark 2.13. Item (5) corresponds to [15,Proposition 2.10(c)]. Ad Item (6).…”

“…Item (1), (2), (3) are obvious from the definition of the maps ρ andS. Item (4) corresponds to [15,Proposition 2.10(a)] and can be deduced immediately from Remark 2.13. Item (5) corresponds to [15,Proposition 2.10(c)].…”

“…The statement that the map ρ is an involution can be easily seen from the original definition or using induction on the Z ≥0 -degree and Item (3). The statement that the mapS is an involution corresponds to [15,Proposition 2.10(b)]. The statement that the antipode on T (V ) is invertible follows from the definition ofS and the fact that the maps ρ andS are invertible (as they are involutions as it was just explained).…”

“…Definition 3.5 and Remark 3.13). These maps already occur in [15] with the sole difference that there a nonstandard braiding of kΓ kΓ YD (different from the braiding introduced in Section 2, to be precise: its inverse) is in use. Nevertheless, the results and their proofs are analogous.…”

“…Let S m be the symmetric group on m letters. In [15], Liu proves that skew divided difference operators ∂ w/v where v, w ∈ S m can be written as polynomials with nonnegative integer coefficients in divided difference operators of the form ∂ ij where 1 ≤ i < j ≤ m. Liu actually proves the stronger and analogous statement for the Fomin-Kirillov algebra E m as introduced in [11] (i.e. he uses considerably less relations among the generators) by constructing an explicit and manifestly positive formula for the elements x w/v ∈ E m which act via ∂ w/v .…”