Lusztig, George Total positivity and canonical bases. (English) Zbl 0890.20034 Lehrer, Gus (ed.) et al., Algebraic groups and Lie groups. A volume of papers in honour of the late R. W. Richardson. Cambridge: Cambridge University Press. Aust. Math. Soc. Lect. Ser. 9, 281-295 (1997). Let \(G\) be a split reductive algebraic group over the real numbers \(\mathbb{R}\) and denote by \(G_{\geq0}\) the semi-subgroup of totally positive elements of \(G\) [as defined in: G. Lusztig, Prog. Math. 123, 531-568 (1994; Zbl 0845.20034)].The first result of the paper shows that the combinatorics which is necessary to describe the relation between the two parameterizations of the canonical basis of a \(G\)-module, coming from regarding it either as a highest or lowest weight module, also appears in the geometry of \(G_{\geq0}\) over \(\mathbb{R}(\epsilon)\), where \(\epsilon\) is an indeterminate.The second result is that \(G_{\geq0}\) can be defined by explicit inequalities, provided by certain canonical basis elements.For the entire collection see [Zbl 0857.00012]. Reviewer: St.Helmke (Kyoto) Cited in 3 ReviewsCited in 19 Documents MSC: 20G20 Linear algebraic groups over the reals, the complexes, the quaternions 14L17 Affine algebraic groups, hyperalgebra constructions 20G05 Representation theory for linear algebraic groups 20M20 Semigroups of transformations, relations, partitions, etc. Keywords:canonical bases; totally positive elements; split reductive algebraic groups; parametrizations of canonical bases; weight modules Citations:Zbl 0845.20034 PDFBibTeX XMLCite \textit{G. Lusztig}, Aust. Math. Soc. Lect. Ser. 9, 281--295 (1997; Zbl 0890.20034)