We study bialgebra structures on quiver coalgebras and monoidal structures on the categories of locally nilpotent and locally finite quiver representations. It is shown that the path coalgebra of an arbitrary quiver admits natural bialgebra structures. This endows the category of locally nilpotent and locally finite representations of an arbitrary quiver with natural monoidal structures from bialgebras. We also obtain theorems of Gabriel type for pointed bialgebras and hereditary finite pointed...

Given a tuple $({𝒳}_1,...,{𝒳}_k)$ of irreducible characters of $G{L}_n\left(F_q\right)$ we define a star-shaped quiver $\Gamma$ 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 ${𝒳}_1\otimes \cdots \otimes {𝒳}_k$ as the trace of the action of some Weyl group on the intersection cohomology of some (non-affine) quiver varieties associated to $(\Gamma ,v)$. The existence of such a quiver variety is subject to some condition. Assuming that this condition is satisfied, we...