Quiver varieties and the character ring of general linear groups over finite fields

Given a tuple ( X_1,, X_k) of irreducible characters of GL_n( F_q) we define a star-shaped quiver Γ together with a dimension vector v . Assume that ( X_1,, X_k) is generic . Our first result is a formula which expresses the multiplicity of the trivial character in the tensor product X_1⊗⋯⊗ X_k 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 X_1⊗⋯⊗ X_k,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 GL_n( C) .