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A172379 The unique d=3 cycle arising in higher dimensional cluster combinatorics and representation theory 0
1357, 1358, 1368, 1468, 2468, 2469, 2479, 2579, 3579 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Oppermann, p. 43, Computer experiments have not detected any similar phenomena when d = 2. Higher Auslander algebras were introduced by Iyama generalizing classical concepts from representation theory of finite dimensional algebras. Recently these higher analogs of classical representation theory have been increasingly studied. Cyclic polytopes are classical objects of study in convex geometry. In particular, their triangulations have been studied with a view towards generalizing the rich combinatorial structure of triangulations of polygons. In this paper, we demonstrate a connection between these two seemingly unrelated subjects.

We study triangulations of even-dimensional cyclic polytopes and tilting modules for higher Auslander algebras of linearly oriented type A which are summands of the cluster tilting module. We show that such tilting modules correspond bijectively to triangulations. Moreover mutations of tilting modules correspond to bistellar flips of triangulations. For any d-representation finite algebra we introduce a certain d-dimensional cluster category and study its cluster tilting objects. For higher Auslander algebras of linearly oriented type A we obtain a similar correspondence between cluster tilting objects and triangulations of a certain cyclic polytope. Finally we study certain functions on generalized laminations in cyclic polytopes, and show that they satisfy analogs of tropical cluster exchange relations. Moreover we observe that the terms of these exchange relations are closely related to the terms occuring in the mutation of cluster tilting objects.

LINKS

Table of n, a(n) for n=1..9.

Steffen Oppermann, Hugh Thomas, Higher dimensional cluster combinatorics and representation theory, Jan 30, 2010.

CROSSREFS

Sequence in context: A056103 A259249 A215865 * A192319 A125270 A316680

Adjacent sequences:  A172376 A172377 A172378 * A172380 A172381 A172382

KEYWORD

fini,full,nonn

AUTHOR

Jonathan Vos Post, Feb 01 2010

STATUS

approved

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Last modified February 23 21:20 EST 2020. Contains 332195 sequences. (Running on oeis4.)