Abstract
For permutations \(v,w \in \mathfrak S_n\), Macdonald defines the skew divided difference operators \(\partial _{w/v}\) as the unique linear operators satisfying \(\partial _w(PQ) = \sum _v v(\partial _{w/v}P) \cdot \partial _vQ\) 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 \(i<j\). In fact, we prove that the analogous result holds in the Fomin–Kirillov algebra \({\mathcal {E}}_n\), which settles a conjecture of Kirillov.
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Liu, R.I. Positive expressions for skew divided difference operators. J Algebr Comb 42, 861–874 (2015). https://doi.org/10.1007/s10801-015-0606-1
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DOI: https://doi.org/10.1007/s10801-015-0606-1