On the Auslander–Reiten quiver for the category of representations of partially ordered sets with an involution

Abstract In this article, we provide a description of the Auslander–Reiten quiver for certain posets endowed with an involution, which we denote as types $$\mathfrak {U}_n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> and $$\mathfrak {U}_\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>∞</mml:mi> </mml:msub> </mml:math> . These posets appear in Zavadskij’s Differentiation III (Can Math Soc Conf Proc 11:299–323, 1991). Our approach follows the classical Auslander–Reiten theory developed by Auslander, Reiten, and Smalø (Representation Theory of Artin Algebras. Cambridge Studies in Advanced Mathematics, UK, 1992). To this end, we establish a natural exact structure for the category of representations of a partially ordered set with an involution, and subsequently delineate the projective and injective representations.