The Canonical Join Complex

Barnard, Emily

doi:10.37236/7866

Cited by 26 publications

(46 citation statements)

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“…For example, Figure 7 depicts the rook R (3,2) for a = 5 and b = 4. In this picture, the number in the northwest corner of a box (i ′ , j ′ ) is the coefficient of T + (i ′ ,j ′ ) in the rook, and the number in the southeast corner is the coefficient of T − (i ′ ,j ′ ) .…”

Section: Balanced Shapes In This Subsection We Review the Results Anmentioning

confidence: 99%

“…We think of the elements (i, j) ∈ D of a diagram as being "boxes" placed at those coordinates. We use "matrix coordinates" where the box at (1, 1) is northwest of the one at (2,2). A row of a diagram D is the set of boxes (i, j) ∈ D for some fixed i ∈ Z; similarly, a column of D is the set of boxes (i, j) ∈ D for some fixed j ∈ Z.…”

Section: 2mentioning

confidence: 99%

“…We say that w ∈ S n is skew vexillary of shape σ if its Rothe diagram can be transformed to σ via some permutation of rows and columns. 2 We say that w is skew vexillary if it is skew vexillary of some shape. Example 3.2.…”

Section: Skew Vexillary Permutationsmentioning

confidence: 99%

“…Example 3.15: the initial weak order intervals corresponding to vexillary permutations of shape(2,1).This implies that w ∈ S k × S n−k . Because w has no initial or terminal fixed points, both the upper k × k and the lower (n − k) × (n − k) square of [n] × [n] must have some boxes of the Rothe diagram of w in them.…”

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confidence: 99%

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The CDE property for skew vexillary permutations

Hopkins

2019

Journal of Combinatorial Theory, Series A

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We prove a conjecture of Reiner, Tenner, and Yong which says that the initial weak order intervals corresponding to certain vexillary permutations have the coincidental down-degree expectations (CDE) property. Actually our theorem applies more generally to certain "skew vexillary" permutations (a notion we introduce), and shows that these posets are in fact "toggle CDE." As a corollary we obtain a homomesy result for rowmotion acting on semidistributive lattices in the sense of Barnard and of Thomas and Williams.

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Section: Balanced Shapes In This Subsection We Review the Results Anmentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

Section: Skew Vexillary Permutationsmentioning

confidence: 99%

mentioning

confidence: 99%

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The CDE property for skew vexillary permutations

Hopkins

2019

Journal of Combinatorial Theory, Series A

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“…In this complex, the faces are therefore indexed by the elements of the lattice. The canonical join complex was thoroughly studied in [2].…”

Section: Introductionmentioning

confidence: 99%

Distributive lattices have the intersection property

Mühle¹

2019

Math.Boh.

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Distributive lattices form an important, well-behaved class of lattices. They are instances of two larger classes of lattices: congruence-uniform and semidistributive lattices. Congruence-uniform lattices allow for a remarkable second order of their elements: the core label order; semidistributive lattices naturally possess an associated flag simplicial complex: the canonical join complex. In this article we present a characterization of finite distributive lattices in terms of the core label order and the canonical join complex, and we show that the core label order of a finite distributive lattice is always a meet-semilattice.2010 Mathematics Subject Classification. 06D05.

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The core label order of a congruence-uniform lattice

Mühle

2019

Algebra Univers.

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We investigate the alternate order on a congruence-uniform lattice L as introduced by N. Reading, which we dub the core label order of L. When L can be realized as a poset of regions of a simplicial hyperplane arrangement, the core label order is always a lattice. For general L, however, this fails. We provide an equivalent characterization for the core label order to be a lattice. As a consequence we show that the property of the core label order being a lattice is inherited to lattice quotients. We use the core label order to characterize the congruence-uniform lattices that are Boolean lattices, and we investigate the connection between congruence-uniform lattices whose core label orders are lattices and congruence-uniform lattices of biclosed sets.2010 Mathematics Subject Classification. 06B05 (primary), and 06A07 (secondary).

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The Canonical Join Complex

Cited by 26 publications

References 20 publications

The CDE property for skew vexillary permutations

The CDE property for skew vexillary permutations

Distributive lattices have the intersection property

The core label order of a congruence-uniform lattice

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