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].
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)
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. |