Canonically Jordan recoverable categories for modules over the path algebra of A_n type quivers · arXivDesk
2308.16626Aug 31, 2023v2 : minor changes following reports of this article as part of my Ph.D. thesis; v3 : new section 3.3 on two ways of calculating GenRep that coïncide + minor changes; v4 : minor changes and improvement in the presentation of the technical sections (4,5 and 6)
Canonically Jordan recoverable categories for modules over the path algebra of An type quivers
Let Q be a quiver of An type and K 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 X, there exists a well-defined Jordan form of each of these linear transformations GenJF(X)
Nearby in the stack
, called the generic Jordan form data of
X
. A subcategory of
rep(Q)
is Jordan recoverable if we can recover
X
up to isomorphism from its generic Jordan form data. There is a procedure which allows one to invert the map from representations to generic Jordan form data. The subcategories for which this procedure works are called canonically Jordan recoverable. We focus on the subcategories of
rep(Q)
that are canonically Jordan recoverable, and we give a combinatorial characterization of them.