TY - UNPB

T1 - New interpretations of the higher Stasheff--Tamari orders

AU - Williams, Nicholas J.

N1 - 41 pages, 10 figures; v2: fixed typos and added references; v3: fixed typos, added references, other minor revisions; v4: added references, changed convention for multiplying arrows in path algebra

PY - 2020/7/24

Y1 - 2020/7/24

N2 - In 1996, Edelman and Reiner defined the two higher Stasheff--Tamari orders on triangulations of cyclic polytopes and conjectured them to coincide. We open up an algebraic angle for approaching this conjecture by showing how these orders arise naturally in the representation theory of the higher Auslander algebras of type $A$, denoted $A_{n}^{d}$. For this we give new combinatorial interpretations of the orders, making them comparable. We then translate these combinatorial interpretations into the algebraic framework. We also show how triangulations of odd-dimensional cyclic polytopes arise in the representation theory of $A_{n}^{d}$, namely as equivalence classes of maximal green sequences. We furthermore give the odd-dimensional counterpart to the known description of $2d$-dimensional triangulations as sets of non-intersecting $d$-simplices of a maximal size. This consists in a definition of two new properties which imply that a set of $d$-simplices produces a $(2d+1)$-dimensional triangulation.

AB - In 1996, Edelman and Reiner defined the two higher Stasheff--Tamari orders on triangulations of cyclic polytopes and conjectured them to coincide. We open up an algebraic angle for approaching this conjecture by showing how these orders arise naturally in the representation theory of the higher Auslander algebras of type $A$, denoted $A_{n}^{d}$. For this we give new combinatorial interpretations of the orders, making them comparable. We then translate these combinatorial interpretations into the algebraic framework. We also show how triangulations of odd-dimensional cyclic polytopes arise in the representation theory of $A_{n}^{d}$, namely as equivalence classes of maximal green sequences. We furthermore give the odd-dimensional counterpart to the known description of $2d$-dimensional triangulations as sets of non-intersecting $d$-simplices of a maximal size. This consists in a definition of two new properties which imply that a set of $d$-simplices produces a $(2d+1)$-dimensional triangulation.

KW - math.CO

KW - math.RT

KW - Primary: 05E10, Secondary: 06A07, 52B11

M3 - Preprint

BT - New interpretations of the higher Stasheff--Tamari orders

ER -