Canonically Jordan recoverable categories for modules over the path algebra of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>n</mml:mi> </
Abstract
<jats:p> Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Q</mml:mi> </mml:math> be a quiver of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> type and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>𝕂</mml:mi> </mml:math> be an algebraically closed field. A nilpotent endomorphism of a quiver representation induces a linear transformation of the vector space at each vertex. Generically, among all nilpotent endomorphisms of a fixed representation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> , there exists a well-defined Jordan form of each of these linear transformations <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo form="prefix">GenJF</mml:mo> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , called the generic Jordan form data of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> . A subcategory of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo form="prefix">rep</mml:mo> <mml:mo>(</mml:mo> <mml:mi>Q</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is Jordan recoverable if we can recover <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> up to isomorphism from its generic Jordan form data. </jats:p> <jats:p> There is a procedure that allows one to invert the map from representations to generic Jordan form data. The subcategories for which this procedure applies are called canonically Jordan recoverable. We focus on the subcategories of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo form="prefix">rep</mml:mo> <mml:mo>(</mml:mo> <mml:mi>Q</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> that are canonically Jordan recoverable, and we give a combinatorial characterization of them. </jats:p>