Geodesic
Quiver varieties and the character ring of general linear groups over finite fields
Journal of the European Mathematical Society, Tome 15 (2013) no. 4, pp. 1375-1455

Voir la notice de l'article provenant de la source EMS Press

Given a tuple (X1​,...,Xk​) of irreducible characters of GLn​(Fq​) we define a star-shaped quiver Γ together with a dimension vector v. Assume that (X1​,...,Xk​) is generic. Our first result is a formula which expresses the multiplicity of the trivial character in the tensor product X1​⊗⋯⊗Xk​ as the trace of the action of some Weyl group on the intersection cohomology of some (non-affine) quiver varieties associated to (Γ,v). The existence of such a quiver variety is subject to some condition. Assuming that this condition is satisfied, we prove our second result: The multiplicity 〈X1​⊗⋯⊗Xk​,1〉 is non-zero if and only if v is a root of the Kac–Moody algebra associated with Γ. This is somehow similar to the connection between Horn's problem and the representation theory of GLn​(C).
DOI : 10.4171/jems/395
Classification : 20-XX, 00-XX
Keywords: Quiver varieties, tensor products of irreducible characters 
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     author = {Emmanuel Letellier},
     title = {Quiver varieties and the character ring of general linear groups over finite fields},
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Emmanuel Letellier. Quiver varieties and the character ring of general linear groups over finite fields. Journal of the European Mathematical Society, Tome 15 (2013) no. 4, pp. 1375-1455. doi: 10.4171/jems/395

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