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Hammocks and the Nazarova–Roiter Algorithm
Author(s) -
Lin Yanan
Publication year - 1998
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610798005894
Subject(s) - quiver , partially ordered set , mathematics , representation (politics) , function (biology) , translation (biology) , combinatorics , partial function , pure mathematics , algorithm , algebra over a field , discrete mathematics , biology , biochemistry , evolutionary biology , politics , messenger rna , political science , gene , law
Hammocks have been considered by Brenner [ 1 ], who gave a numerical criterion for a finite translation quiver to be the Auslander–Reiten quiver of some representation‐finite algebra. Ringel and Vossieck [ 11 ] gave a combinatorial definition of left hammocks which generalised the concept of hammocks in the sense of Brenner, as a translation quiver H and an additive function h on H (called the hammock function) satisfying some conditions. They showed that a thin left hammock with finitely many projective vertices is just the preprojective component of the Auslander–Reiten quiver of the category of S‐spaces, where S is a finite partially ordered set (abbreviated as ‘poset’). An important role in the representation theory of posets is played by two differentiation algorithms. One of the algorithms was developed by Nazarova and Roiter [ 8 ], and it reduces a poset S with a maximal element a to a new poset S′= a ∂S. The second algorithm was developed by Zavadskij [ 13 ], and it reduces a poset S with a suitable pair ( a , b ) of elements a , b to a new poset S′=∂ ( a , b ) S. The main purpose of this paper is to construct new left hammocks from a given one, and to show the relationship between these new left hammocks and the Nazarova–Roiter algorithm. In a later paper [ 5 ], we discuss the relationship between hammocks and the Zavadskij algorithm.