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Journal of Lie Theory 28 (2018), No. 4, 1095--1118
Copyright Heldermann Verlag 2018

Complex Groups and Root Subgroup Factorization

Doug Pickrell
Mathematics Department, University of Arizona, Tucson, AZ 85721, U.S.A.

Root subgroup factorization is a refinement of triangular (or LDU) factorization. For a complex reductive Lie group, and a choice of reduced factorization of the longest Weyl group element, the forward map from root subgroup coordinates to triangular coordinates is polynomial. We show that the forward map is injective on its set of regular points and that the inverse is rational. There is an algorithm for the inverse (involving LDU factorization), and a related explicit formula for Haar measure in root subgroup coordinates. In classical cases there are preferred reduced factorizations of the longest Weyl group elements, and conjecturally in these cases there are closed form expressions for root subgroup coordinates.

Keywords: Complex reductive group, triangular factorization, root subgroup factorization

MSC: 22E67

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