Unitary and Euclidean representations of a quiver

Abstract

A unitary (Euclidean) representation of a quiver is given by assigning to each vertex a unitary (Euclidean) vector space and to each arrow a linear mapping of the corresponding vector spaces. We recall an algorithm for reducing the matrices of a unitary representation to canonical form, give a certain description of the representations in canonical form, and reduce the problem of classifying Euclidean representations to the problem of classifying unitary representations. We also describe the set of dimensions of all indecomposable unitary (Euclidean) representations of a quiver and establish the number of parameters in an indecomposable unitary representation of a given dimension.

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• Complexity of matrix problems
• Locally-scalar representations of graphs in the category of Hilbert spaces
• Systems of subspaces of a unitary space
• Canonical Forms for Unitary Congruence and *Congruence
• Canonical matrices of isometric operators on indefinite inner product spaces
• Specht's criterion for systems of linear mappings
• A canonical form for nonderogatory matrices under unitary similarity
• Littlewood's algorithm and quaternion matrices
• Simultaneous unitary equivalences
• Criterion of unitary similarity for upper triangular matrices in general position
• Topological classification of chains of linear mappings
• Operators on positive semidefinite inner product spaces
• Classification of squared normal operators on unitary and Euclidean spaces