Generalized quivers associated to reductive groups

Harm Derksen


October 28,
refreshments at 3:45pm


Suppose we are given a quiver (directed graph) S, for example o--->o--->o. A representation of this quiver is triple (V,W,X) of vector spaces together with a pair (f,g) where f:V-->W and g:W-->X are linear maps. If we fix the dimensions, say (dim V,dim W, dim X)=(a,b,c), then the isomorphism classes of all representations of S with these dimensions correspond to GL(a)xGL(b)xGL(c)-orbits in the direct sum of Hom(k^a,k^b) and Hom(k^b,k^c). So very informally: "Quiver are orbits of a products of GL_n's in certain nice representations". This perspective will motivate us to give a definition of quiver representations for arbitrary reductive groups. It turns out to make sense: for orthogonal and symplectic groups our generalization has a nice interpretation, namely, they correspond to so-called orthogonal and symplectic representations of quivers with an arrow-inverting automorphism (called symmetric quivers). Among other things we will classify the symmetric quivers of finite and tame type, and their indecomposable representations. This is joint work with Jerzy Weyman at Northeastern.

Speaker's Contact Info: hderksen(at-sign)

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