Coxeter Matroids

Borovik, Alexandre; Gelfand, Israel M.; White, Neil

doi:10.1007/978-1-4612-2066-4

Cited by 101 publications

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“…k), and the brick polytope of Q is defined to be the convex hull of the brick vectors of all facets of the subword complex SC(Q), B(Q) = conv B(I) : I facet of SC(Q) .It is shown in[PS15a] that the brick polytope B(Q) is the Minkowski sum of a Coxeter matroid polytope in the sense of[BGW03].…”

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confidence: 99%

Towards a uniform subword complex description of acyclic finite type cluster algebras

Brodsky¹,

Stump²

2018

Algebraic Combinatorics

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It has been established in recent years how to approach acyclic cluster algebras of finite type using subword complexes. In this paper, we continue this study by describing the c-and g-vectors, and by providing a conjectured description of the Newton polytopes of the F -polynomials. In particular, we show that this conjectured description would imply that finite type cluster complexes are realized by the duals of the Minkowski sums of the Newton polytopes of either the F -polynomials, or of the cluster variables, respectively. Date

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confidence: 99%

Towards a uniform subword complex description of acyclic finite type cluster algebras

Brodsky¹,

Stump²

2018

Algebraic Combinatorics

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“…. 0, n even: σ 1,2 ( n−1 i=3 σ i,i+1 )σ n,3 and σ 1,3 σ 3,4 σ 4,2 ( n−1 i=5 σ i,i+1 )σ n, 5 give that c ∅ = 0 and d…”

Section: Quadratic Relationsmentioning

confidence: 99%

The Geometry of Gaussoids

et al. 2018

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A gaussoid is a combinatorial structure that encodes independence in probability and statistics, just like matroids encode independence in linear algebra. The gaussoid axioms of Lněnička and Matúš are equivalent to compatibility with certain quadratic relations among principal and almost-principal minors of a symmetric matrix. We develop the geometric theory of gaussoids, based on the Lagrangian Grassmannian and its symmetries. We introduce oriented gaussoids and valuated gaussoids, thus connecting to real and tropical geometry. We classify small realizable and non-realizable gaussoids. Positive gaussoids are as nice as positroids: they are all realizable via graphical models. a 12|3 maps to σ 12 σ 33 − σ 13 σ 23 whereas a 13|2 maps to −(σ 12 σ 23 − σ 13 σ 22 ). Maintaining this sign convention is important to keep the algebra consistent with its statistical interpretation.Let J n denote the ideal generated by all homogeneous polynomials in the kernel of the map above. This defines an irreducible variety V (J n ) of dimension n+1 2 in the projective space P 2 n−2 (4+( n 2 ))−1 whose coordinates are P ∪ A. There is a natural projection from LGr(n, 2n) onto V (J n ), obtained by deleting all minors that are neither principal nor almost-principal. This is analogous to [30, Observation III.12], where the focus was on principal minors p I .Proposition 1.1. The degree of the projective variety of principal and almost-principal minors coincides with the degree of the Lagrangian Grassmannian. For n ≥ 2, it equals degree(V (J n )) = degree(LGr(n, 2n)) = n+1 2 ! 1 n · 3 n−1 · 5 n−2 · · · (2n − 1).

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“…The symbols N, Z, R, and C refer to the natural numbers (including 0), the integers, the real numbers, and the complex numbers. is the total number of hyperplanes separating A from B-see Theorem 5.1.4 in [BGW03]. I also use two extensions of this distance function d:…”

Section: Notation and Setupmentioning

confidence: 99%

Conjugacy classes of non-translations in affine Weyl groups and applications to Hecke algebras

Rostami

2015

Trans. Amer. Math. Soc.

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Abstract. Let W = Λ W• be an Iwahori-Weyl group of a connected reductive group G over a non-archimedean local field. The subgroup W• is a finite Weyl group and the subgroup Λ is a finitely-generated abelian group (possibly containing torsion) which acts on a certain real affine space by translations.I prove that if w ∈ W and w / ∈ Λ then one can apply to w a sequence of conjugations by simple reflections, each of which is length-preserving, resulting in an element w for which there exists a simple reflection s such that

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Coxeter Matroids

Cited by 101 publications

References 0 publications

Towards a uniform subword complex description of acyclic finite type cluster algebras

Towards a uniform subword complex description of acyclic finite type cluster algebras

The Geometry of Gaussoids

Conjugacy classes of non-translations in affine Weyl groups and applications to Hecke algebras

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