Bases of the Intersection Cohomology of Grassmannian Schubert Varieties

Patimo, Leonardo

doi:10.48550/arxiv.1908.11606

Cited by 4 publications

(4 citation statements)

References 10 publications

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“…Independently, Kenyon and Wilson also introduced coverinclusive Dyck tilings in the study of double dimer models [7,8] (conjectures in [7] are proved in [9]). Since then, cover-inclusive Dyck tilings appear in the connection to other research fields such as Schramm-Loewner evolution [6,14,15], fully packed loops [4], and intersection cohomology of Grassmannian Schubert varieties [13].…”

Section: Introductionmentioning

confidence: 99%

Dyck tilings of type $D$

Shigechi

2021

Preprint

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We introduce and study cover-inclusive and cover-exclusive Dyck tilings of type D. It is shown that the generating functions of Dyck tilings of type D are expressed in terms of the generating function of ballot tilings of type B. We introduce link patterns of type D and plane trees for a ballot path, and construct a map from trees to Z[q]. This map gives the generating function of cover-inclusive Dyck tilings of type D associated to the ballot path.

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Section: Introductionmentioning

confidence: 99%

Dyck tilings of type $D$

Shigechi

2021

Preprint

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“…Similarly, one make use of cover-inclusive tilings to compute P + λ,µ as in [21]. Cover-inclusive Dyck tiligns also appear in research areas in mathematical physics [3,7,8,9,10,14,15,16]. There are several generalizations of Dyck tilings.…”

Section: Introductionmentioning

confidence: 99%

Rational Dyck tilings

Shigechi

2021

Preprint

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We introduce rational Dyck tilings, or (a, b)-Dyck tilings, and study them by the decomposition into (1, 1)-Dyck tilings. This decomposition allows us to make use of combinatorial models for (1, 1)-Dyck tilings such as the Hermite history and the Dyck tiling strip bijection. Together with b-Stirling permutations associated to the rational Dyck tilings, we obtain a correspondence between an (a, b)-Dyck tiling and a tuple of ab (1, 1)-Dyck tilings.

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“…Independently, Kenyon and Wilson introduced them in the study of the double-dimer model and spanning trees [6,7]. Since then, Dyck tilings has appeared in connection with different contexts such as fully packed loop models [4], multiple Schramm-Loewner evolutions [12,13], and the intersection cohomology of Grassmannian Schubert varieties [11].…”

Section: Introductionmentioning

confidence: 99%

Symmetric Dyck tilings, ballot tableaux and tree-like tableaux of shifted shapes

Shigechi¹

2020

Preprint

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Symmetric Dyck tilings and ballot tilings are certain tilings in the region surrounded by two ballot paths. We study the relations of combinatorial objects which are bijective to symmetric Dyck tilings such as labeled trees, Hermite histories, and perfect matchings. We also introduce two operations on labeled trees for symmetric Dyck tilings: symmetric Dyck tiling strip (symDTS) and symmetric Dyck tiling ribbon (symDTR). We give two definitions of Hermite histories for symmetric Dyck tilings, and show that they are equivalent by use of the correspondence between symDTS operation and an Hermite history. Since ballot tilings form a subset in the set of symmetric Dyck tilings, we construct an inclusive map from labeled trees for ballot tilings to labeled trees for symmetric Dyck tilings. By this inclusive map, the results for symmetric Dyck tilings can be applied to those of ballot tilings. We introduce and study the notions of ballot tableaux and treelike tableaux of shifted shapes, which are generalizations of Dyck tableaux and tree-like tableaux, respectively. The correspondence between ballot tableaux and tree-like tableaux of shifted shapes is given by using the symDTR operation and the structure of labeled trees for symmetric Dyck tilings.

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Bases of the Intersection Cohomology of Grassmannian Schubert Varieties

Cited by 4 publications

References 10 publications

Dyck tilings of type $D$

Dyck tilings of type $D$

Rational Dyck tilings

Symmetric Dyck tilings, ballot tableaux and tree-like tableaux of shifted shapes

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