Higher Bruhat orders, related to the symmetric group

Manin, Yu. I.; Shekhtman, V. V.

doi:10.1007/bf01077278

Cited by 29 publications

(19 citation statements)

References 1 publication

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…If that fixed vertex is chosen to be the tiling shown in Fig. 8 (n=5), then the poset we obtain is the higher Bruhat order B(n, 2), described in [10] and [16] .., [a 1 + } } } +a m&1 +1, ..., n], while preserving the consecutiveness and the order of the elements within each block. Then _ can be thought of as a (special kind of) permutation of a multiset with a i copies of i, for i=1, ..., m. Since the different copies of the same i are not permuted among themselves, we can make the edges corresponding to them parallel.…”

Section: T(_) As a Graph And The Case _=w 0 (N)mentioning

confidence: 99%

See 1 more Smart Citation

Rhombic Tilings of Polygons and Classes of Reduced Words in Coxeter Groups

Elnitsky

1997

Journal of Combinatorial Theory, Series A

View full text Add to dashboard Cite

Section: T(_) As a Graph And The Case _=w 0 (N)mentioning

confidence: 99%

“…In Section 3, we will see that this graph is connected and bipartite. For the special case of w 0 , this graph (also considered as a ranked poset) has occurred in the study of quantum groups (Berenstein and Zelevinsky, [2]), as well as that of the higher Bruhat orders (Ziegler,[16]; Manin and Schechtman, [10]; and others).…”

Section: Introductionmentioning

confidence: 99%

Rhombic Tilings of Polygons and Classes of Reduced Words in Coxeter Groups

Elnitsky

1997

Journal of Combinatorial Theory, Series A

View full text Add to dashboard Cite

“…Since then significant work has been done on the "higher Bruhat orders" [9,10,16,5]. Some of these results are central to our approach, so we present them as we give the following definitions.…”

mentioning

confidence: 99%

Acyclic sets of linear orders via the Bruhat orders

Galambos

Reiner

2007

Soc Choice Welfare

View full text Add to dashboard Cite

We use Ziegler's results [16] on the higher Bruhat orders to show that Abello's acyclic sets of linear orders [1] can be described as the permutations of equivalence classes of maximal reduced decompositions. This allows us to strengthen Abello's structural result: we show that acyclic sets arising from this construction are distributive sublattices of the weak Bruhat order. Fishburn's "alternating scheme" is the set of permutations of one such equivalence class of maximal reduced decompositions. Any acyclic set that arises in this way can be represented by an arrangement of pseudolines, and we use this representation to derive a formula enumerating the "alternating scheme."

show abstract

“…следствие 3.8) следует, что потолок нашей камеры состоит из един-ственного ребра (цвет которого обозначим через j) и за этим ребром распола-гается (противочасовая) циклическая камера Z. Локальная картинка вблизи этого ребра выглядит как на рис. 27. Итак, мы имеем три нециклические ка-меры Z − k, S p = Z − j и Z − i вайринга W .…”

Section: пользуясь отношением наunclassified