Associahedron, Cyclohedron and Permutohedron as compactifications of configuration spaces

Lambrechts, Pascal; Turchin, Victor; Volic, Ismar

doi:10.36045/bbms/1274896208

Cited by 13 publications

(11 citation statements)

References 26 publications

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“…Associahedron, permutahedron and configuration spaces. Here we remind two well-known constructions [St,Ko3,LTV] (see also lecture notes [Me2]) which will be used later. Let Conf o n (R) = {x 1 < x 2 < .…”

Section: Appendix B Configuration Space Models For Bipermutahedra Anmentioning

confidence: 99%

An Explicit Two Step Quantization of Poisson Structures and Lie Bialgebras

Merkulov

Willwacher

2018

Commun. Math. Phys.

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We develop a new approach to deformation quantizations of Lie bialgebras and Poisson structures which goes in two steps.In the first step one associates to any Poisson (resp. Lie bialgebra) structure a so called quantizable Poisson (resp. Lie bialgebra) structure. We show explicit transcendental formulae for this correspondence.In the second step one deformation quantizes a quantizable Poisson (resp. Lie bialgebra) structure. We show again explicit transcendental formulae for this second step correspondence (as a byproduct we obtain configuration space models for biassociahedron and bipermutohedron).In the Poisson case the first step is the most non-trivial one and requires a choice of an associator while the second step quantization is essentially unique, it is independent of a choice of an associator and can be done by a trivial induction. We conjecture that similar statements hold true in the case of Lie bialgebras.The main new result is a surprisingly simple explicit universal formula (which uses only smooth differential forms) for universal quantizations of finite-dimensional Lie bialgebras.

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Section: Appendix B Configuration Space Models For Bipermutahedra Anmentioning

confidence: 99%

An Explicit Two Step Quantization of Poisson Structures and Lie Bialgebras

Merkulov

Willwacher

2018

Commun. Math. Phys.

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“…Then real and complex De Concini-Procesi models turned out to play a key role in several fields of mathematical research: subspace and toric arrangements, toric varieties (see for instance [7], [15], [36]), tropical geometry (see [14]), moduli spaces and configuration spaces (see for instance [11], [28]), box splines, vector partition functions and index theory (see [6], [2]), discrete geometry (see [12]).…”

Section: 2mentioning

confidence: 99%

On Models of the Braid Arrangement and their Hidden Symmetries

Callegaro

Gaiffi

2015

Int Math Res Notices

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The De Concini-Procesi wonderful models of the braid arrangement of type A n−1 are equipped with a natural Sn action, but only the minimal model admits an 'hidden' symmetry, i.e. an action of S n+1 that comes from its moduli space interpretation. In this paper we explain why the non minimal models don't admit this extended action: they are 'too small'. In particular we construct a supermaximal model which is the smallest model that can be projected onto the maximal model and again admits an extended S n+1 action. We give an explicit description of a basis for the integer cohomology of this supermaximal model. Furthermore, we deal with another hidden extended action of the symmetric group: we observe that the symmetric group S n+k acts by permutation on the set of k-codimensionl strata of the minimal model. Even if this happens at a purely combinatorial level, it gives rise to an interesting permutation action on the elements of a basis of the integer cohomology.

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“…The latter two polytopes may be obtained as compactifications of configuration spaces of n points on a line and on a circle, respectively; see e.g. [FM94,AS94,Kon99,Sin04,Gai03,LTV10].…”

Section: Introductionmentioning

confidence: 99%

Poset associahedra

Galashin¹

2021

Preprint

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For each poset P , we construct a polytope A (P ) called the P -associahedron. Similarly to the case of graph associahedra, the faces of A (P ) correspond to certain tubings of P . The Stasheff associahedron is a compactification of the configuration space of n points on a line, and we recover A (P ) as an analogous compactification of the space of order-preserving maps P → R. Motivated by the study of totally nonnegative critical varieties in the Grassmannian, we introduce affine poset cyclohedra and realize these polytopes as compactifications of configuration spaces of n points on a circle. For particular choices of (affine) posets, we obtain associahedra, cyclohedra, permutohedra, and type B permutohedra as special cases.

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Associahedron, Cyclohedron and Permutohedron as compactifications of configuration spaces

Cited by 13 publications

References 26 publications

An Explicit Two Step Quantization of Poisson Structures and Lie Bialgebras

An Explicit Two Step Quantization of Poisson Structures and Lie Bialgebras

On Models of the Braid Arrangement and their Hidden Symmetries

Poset associahedra

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