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

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DOI

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 
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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     author = {Emmanuel Letellier},
     title = {Quiver varieties and the character ring of general linear groups over finite fields},
     journal = {Journal of the European Mathematical Society},
     pages = {1375--1455},
     year = {2013},
     volume = {15},
     number = {4},
     doi = {10.4171/jems/395},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/jems/395/}
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