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Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond. (English) Zbl 0844.20027
Piatetski-Shapiro, Ilya (ed.) et al., The Schur lectures (1992). Ramat-Gan: Bar-Ilan University, Isr. Math. Conf. Proc. 8, 1-182 (1995).
The various roads to the extension of the polynomial invariant theory of linear groups $$G$$ are possible. One of the roads for reductive groups $$G$$ is developed in the essay. This road defines intention to transfer the results of classical invariant theory to general varieties. Realization of this purpose illustrates the following result. Let $$V$$ be a complex vector space; $$A$$ is a Cartan subgroup of the reductive group $$G$$. A $$G$$-module $$V$$ is weight multiplicity-free (WMF) if the $$A$$-eigenspaces are all one-dimensional. A complete list of the irreducible modules of simple groups, which are WMF, is discovered. In particular, it includes the two 27-dimensional representations of $$E_6$$, the 56-dimensional representation of $$E_7$$ and the 7-dimensional representation of $$G_2$$. This shows that WMF distinguish the Gosset polytopes $$2_{21}$$ and $$3_{21}$$ (the polytope $$4_{21}$$ is left aside here).
For the entire collection see [Zbl 0821.00011].

##### MSC:
 20G05 Representation theory for linear algebraic groups 14L30 Group actions on varieties or schemes (quotients) 13A50 Actions of groups on commutative rings; invariant theory 15A75 Exterior algebra, Grassmann algebras 15A66 Clifford algebras, spinors 15A69 Multilinear algebra, tensor calculus 15A72 Vector and tensor algebra, theory of invariants 14L24 Geometric invariant theory