Vol. 102, No. 2, 1982

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On stratifying pairs of linear mappings

Christopher George Gibson and T. D. Ward

Vol. 102 (1982), No. 2, 329–345

The complex linear representations (of fixed dimension) of an oriented graph form a finite dimensional vector space M with a natural action of a product G of general linear groups. It is interesting to look for natural Whitney stratifications of M invariant under G. For the Dynkin diagrams An, Dn, E6, E7, E8 such stratifications are provided by the orbits; and for the extended Dynkin diagrams Ãn, Dn, 6, 7, 8 one might expect to obtain such stratifications by ‘neglecting moduli’, in an obvious way. This is known to be the case for Ã0. For Ã1 we show that this procedure does yield a stratification, and that at least the regular strata satisfy the Whitney conditions.

Mathematical Subject Classification 2000
Primary: 58C27
Secondary: 05C20, 05C40, 15A99, 16A64
Received: 25 July 1980
Published: 1 October 1982
Christopher George Gibson
T. D. Ward