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Double Bruhat cells and total positivity. (English) Zbl 0913.22011
This paper continues the study of total positivity in semisimple algebraic groups begun by A. Berenstein and the authors [Adv. Math. 122, 49–149 (1996; Zbl 0966.17011)] and by A. Berenstein and A. Zelevinsky [Comment. Math. Helv. 72, 128–166 (1997; Zbl 0891.20030)]. For any reductive algebraic group \(G\), Lusztig has defined its totally nonnegative variety \(G_{\geq 0}\), which generalizes the classically studied variety of real matrices all of whose minors are nonnegative; he was motivated by some surprising connections between this variety and his theory of canonical bases for quantum groups. In the papers cited above, the main focus was the structure of the intersection of \(G_{\geq 0}\) and a maximal unipotent subgroup \(N\) of \(G\). In this paper, the entire variety \(G_{\geq 0}\) is studied. The authors first consider the decomposition of \(G\) into its double Bruhat cells \(G^{u,v}\), which are intersections of Bruhat cells \(BuB,B^- vB^-\) for a pair of opposite Borel subgroups \(B,B^-\), giving an isomorphism between each such double cell and a Zariski-open subset of affine space. Actually they obtain a number of such isomorphisms, one for every reduced factorization of \(u\) and \(v\) as products of simple reflections. The main result is then an explicit formula for the image of each such isomorphism applied to a generic element of \(G^{u,v}\). Next they study the intersections of the nonnegative variety \(G_{\geq 0}\) and the double cells \(G^{u,v}\), giving explicit criteria in terms of matrix minors for a typical \(x\in G^{u,v}\) to belong to this intersection. More precisely, they obtain one such criterion for every choice of isomorphism between \(G^{u,v}\) and an open subset of affine space, which is given in terms of the coordinate maps in this isomorphism. Some classical determinantal identities dating from the last century are applied to these matrix minors. Finally, the general theory is specialized to \(\text{GL}_n\). The authors recover the (by now) classical result that an \(n\times n\) real matrix has all minors nonnegative if and only if a certain choice of \(n^2\) of its minors are nonnegative and they in fact give a number of ways to choose the \(n^2\) minors. The combinatorial theory of this special case is self-contained and quite interesting in its own right.

MSC:
22E46 Semisimple Lie groups and their representations
15A23 Factorization of matrices
05E15 Combinatorial aspects of groups and algebras (MSC2010)
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