An Order on Circular Permutations

Motivated by the study of affine Weyl groups, a ranked poset structure is defined on the set of circular permutations in $S_n$ (that is, $n$-cycles). It is isomorphic to the poset of so-called admitted vectors, and to an interval in the affine symmetric group $\tilde S_n$ with the weak order. The poset is a semidistributive lattice, and the rank function, whose range is cubic in $n$, is computed by some special formula involving inversions. We prove also some links with Eulerian numbers, triangulations of an $n$-gon, and Young's lattice.

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Noncrossing Arc Diagrams, Tamari Lattices, and Parabolic Quotients of the Symmetric Group

Annals of Combinatorics ◽

10.1007/s00026-021-00532-9 ◽

2021 ◽

Author(s):

Henri Mühle

Keyword(s):

Symmetric Group ◽

Partial Order ◽

Structural Investigation ◽

Canonical Representation ◽

Weak Order ◽

Noncrossing Partitions ◽

Tamari Lattice ◽

Semidistributive Lattice ◽

Tamari Lattices

AbstractOrdering permutations by containment of inversion sets yields a fascinating partial order on the symmetric group: the weak order. This partial order is, among other things, a semidistributive lattice. As a consequence, every permutation has a canonical representation as a join of other permutations. Combinatorially, these canonical join representations can be modeled in terms of arc diagrams. Moreover, these arc diagrams also serve as a model to understand quotient lattices of the weak order. A particularly well-behaved quotient lattice of the weak order is the well-known Tamari lattice, which appears in many seemingly unrelated areas of mathematics. The arc diagrams representing the members of the Tamari lattices are better known as noncrossing partitions. Recently, the Tamari lattices were generalized to parabolic quotients of the symmetric group. In this article, we undertake a structural investigation of these parabolic Tamari lattices, and explain how modified arc diagrams aid the understanding of these lattices.

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Affine Weyl Groups as Infinite Permutations

The Electronic Journal of Combinatorics ◽

10.37236/1356 ◽

1998 ◽

Vol 5 (1) ◽

Cited By ~ 13

Author(s):

Henrik Eriksson ◽

Kimmo Eriksson

Keyword(s):

Poincaré Series ◽

Weak Order ◽

Length Function ◽

Weyl Groups ◽

Unified Theory ◽

Affine Weyl Groups ◽

Poincare Series ◽

Permutation Models ◽

Refined Version

We present a unified theory for permutation models of all the infinite families of finite and affine Weyl groups, including interpretations of the length function and the weak order. We also give new combinatorial proofs of Bott's formula (in the refined version of Macdonald) for the Poincaré series of these affine Weyl groups.

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Solution of tetrahedron equation and cluster algebras

Journal of High Energy Physics ◽

10.1007/jhep05(2021)103 ◽

2021 ◽

Vol 2021 (5) ◽

Author(s):

P. Gavrylenko ◽

M. Semenyakin ◽

Y. Zenkevich

Keyword(s):

Integrable System ◽

Cluster Algebra ◽

Newton Polygon ◽

Cluster Algebras ◽

Weyl Groups ◽

Newton Polygons ◽

Tetrahedron Equation ◽

Affine Weyl Groups ◽

New Formalism ◽

Bruhat Cell

Abstract We notice a remarkable connection between the Bazhanov-Sergeev solution of Zamolodchikov tetrahedron equation and certain well-known cluster algebra expression. The tetrahedron transformation is then identified with a sequence of four mutations. As an application of the new formalism, we show how to construct an integrable system with the spectral curve with arbitrary symmetric Newton polygon. Finally, we embed this integrable system into the double Bruhat cell of a Poisson-Lie group, show how triangular decomposition can be used to extend our approach to the general non-symmetric Newton polygons, and prove the Lemma which classifies conjugacy classes in double affine Weyl groups of A-type by decorated Newton polygons.

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Bijective projections on parabolic quotients of affine Weyl groups

Journal of Algebraic Combinatorics ◽

10.1007/s10801-014-0559-9 ◽

2014 ◽

Vol 41 (4) ◽

pp. 911-948 ◽

Cited By ~ 1

Author(s):

Elizabeth Beazley ◽

Margaret Nichols ◽

Min Hae Park ◽

XiaoLin Shi ◽

Alexander Youcis

Keyword(s):

Weyl Groups ◽

Affine Weyl Groups

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Tight Quotients and Double Quotients in the Bruhat Order

The Electronic Journal of Combinatorics ◽

10.37236/1871 ◽

2005 ◽

Vol 11 (2) ◽

Cited By ~ 5

Author(s):

John R. Stembridge

Keyword(s):

Coxeter Groups ◽

Upper Bounds ◽

Bruhat Order ◽

Weyl Groups ◽

Order Dimension ◽

Parabolic Subgroups ◽

Affine Weyl Groups ◽

Positive Side ◽

Maximal Parabolic Subgroups

It is a well-known theorem of Deodhar that the Bruhat ordering of a Coxeter group is the conjunction of its projections onto quotients by maximal parabolic subgroups. Similarly, the Bruhat order is also the conjunction of a larger number of simpler quotients obtained by projecting onto two-sided (i.e., "double") quotients by pairs of maximal parabolic subgroups. Each one-sided quotient may be represented as an orbit in the reflection representation, and each double quotient corresponds to the portion of an orbit on the positive side of certain hyperplanes. In some cases, these orbit representations are "tight" in the sense that the root system induces an ordering on the orbit that yields effective coordinates for the Bruhat order, and hence also provides upper bounds for the order dimension. In this paper, we (1) provide a general characterization of tightness for one-sided quotients, (2) classify all tight one-sided quotients of finite Coxeter groups, and (3) classify all tight double quotients of affine Weyl groups.

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