2010/06/22 by Riccardo Aragona, Aragona, Riccardo
Mathematics · #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA) #math.RA #math.RT
paper · pdf · doi:10.48550/arxiv.1006.4378
arxiv created 2010/06/22 · openalex publication_date 2010/06/22 · arxiv updated 2010/06/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This is my PhD thesis supervised by Professor Jerzy Weyman. A symmetric quiver (Q,σ) is a finite quiver without oriented cycles Q=(Q0,Q1) equipped with a contravariant involution σ on Q0\sqcup Q1. 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,σ) 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 cV and, in the case when matrix defining cV is skew-symmetric, by the Pfaffians pfV.