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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].

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