Generalized quivers associated to reductive groups
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)math.mit.edu
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