Semi-invariants of Symmetric Quivers

Details Semi-invariants of Symmetric Quivers Semi-invariants of Symmetric Quivers

2010-06-22

arxiv.org/abs/1006.4378v1

Mathematics

Representation Theory

Scientific paper

This is my PhD thesis supervised by Professor Jerzy Weyman. A symmetric quiver $(Q,\sigma)$ is a finite quiver without oriented cycles $Q=(Q_0,Q_1)$ equipped with a contravariant involution $\sigma$ on $Q_0\sqcup Q_1$. The involution allows us to define a nondegenerate bilinear form $<,>$ on a representation $V$ of $Q$. We shall say that $V$ is orthogonal if $<,>$ is symmetric and symplectic if $<,>$ is skew-symmetric. Moreover we define an action of products of classical groups on the space of orthogonal representations and on the space of symplectic representations. So we prove that if $(Q,\sigma)$ is a symmetric quiver of finite type or of tame type then the rings of semi-invariants for this action are spanned by the semi-invariants of determinantal type $c^V$ and, in the case when matrix defining $c^V$ is skew-symmetric, by the Pfaffians $pf^V$.

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