A Partial Order on the Orthogonal Group

Brady, Thomas A.; Watt, Colum

doi:10.1081/agb-120005817

Cited by 34 publications

(72 citation statements)

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“…We note that the partial order on W is the restriction to W of a partial order on the orthogonal group O(n) which is introduced in [9] and investigated further in [10]. Some of the notation and results from those papers will be used here.…”

Section: Motivationmentioning

confidence: 99%

Non-crossing partition lattices in finite real reflection groups

Brady

Watt²

2007

Trans. Amer. Math. Soc.

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193

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Abstract. For a finite real reflection group W with Coxeter element γ we give a case-free proof that the closed interval, [I, γ], forms a lattice in the partial order on W induced by reflection length. Key to this is the construction of an isomorphic lattice of spherical simplicial complexes. We also prove that the greatest element in this latter lattice embeds in the type W simplicial generalised associahedron, and we use this fact to give a new proof that the geometric realisation of this associahedron is a sphere.

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Section: Motivationmentioning

confidence: 99%

Non-crossing partition lattices in finite real reflection groups

Brady

Watt²

2007

Trans. Amer. Math. Soc.

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193

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“…The c-noncrossing partitions are certain elements of W . Brady and Watt showed [8,Lemma 5] that a c-noncrossing partition can be recovered (among the set of all c-noncrossing partitions) from its fixed point set. The fixed point set of a c-noncrossing partition is called a c-noncrossing subspace.…”

Section: Clusters and Noncrossing Partitionsmentioning

confidence: 99%

Cambrian fans

Reading¹,

Speyer²

2009

J. Eur. Math. Soc.

156

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Abstract. For a finite Coxeter group W and a Coxeter element c of W, the c-Cambrian fan is a coarsening of the fan defined by the reflecting hyperplanes of W. Its maximal cones are naturally indexed by the c-sortable elements of W. The main result of this paper is that the known bijection cl c between c-sortable elements and c-clusters induces a combinatorial isomorphism of fans. In particular, the c-Cambrian fan is combinatorially isomorphic to the normal fan of the generalized associahedron for W. The rays of the c-Cambrian fan are generated by certain vectors in the Worbit of the fundamental weights, while the rays of the c-cluster fan are generated by certain roots. For particular ("bipartite") choices of c, we show that the c-Cambrian fan is linearly isomorphic to the c-cluster fan. We characterize, in terms of the combinatorics of clusters, the partial order induced, via the map cl c , on c-clusters by the c-Cambrian lattice. We give a simple bijection from c-clusters to c-noncrossing partitions that respects the refined (Narayana) enumeration. We relate the Cambrian fan to well-known objects in the theory of cluster algebras, providing a geometric context for g-vectors and quasi-Cartan companions.

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“…Analogous results for spherical isometries already exist and are easy to state: when w is an orthogonal linear transformation of R n only fixing the origin, for example, there is a natural bijection between minimal length factorizations of w into reflections fixing the origin and complete flags of linear subspaces in R n [BW02]. In other words, the structure of all such factorizations is encoded in the lattice of linear subspaces of R n with one factorization for each maximal chain.…”

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confidence: 90%

Factoring Euclidean isometries

Brady

McCammond

2015

Int. J. Algebra Comput.

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Abstract. Every isometry of a finite dimensional euclidean space is a product of reflections and the minimum length of a reflection factorization defines a metric on its full isometry group. In this article we identify the structure of intervals in this metric space by constructing, for each isometry, an explicit combinatorial model encoding all of its minimal length reflection factorizations. The model is largely independent of the isometry chosen in that it only depends on whether or not some point is fixed and the dimension of the space of directions that points are moved.Every good geometry book proves that each isometry of euclidean nspace is a product of at most n+1 reflections and several more-advanced sources include Scherk's theorem which identifies the minimal length of such a reflection factorization from the basic geometric attributes of the isometry under consideration [Sch50, Die71, ST89, Tay92]. The structure of the full set of minimal length reflection factorizations, on the other hand, does not appear to have been given an elementary treatment in the literature even though the proof only requires basic geometric tools.1 In this article we construct, for each isometry, an explicit combinatorial model encoding all of its minimal length reflection factorizations. The model is largely independent of the isometry chosen in that it only depends on whether or not some point is fixed and the dimension of the space of directions that points are moved.Analogous results for spherical isometries already exist and are easy to state: when w is an orthogonal linear transformation of R n only fixing the origin, for example, there is a natural bijection between minimal length factorizations of w into reflections fixing the origin and complete flags of linear subspaces in R n [BW02]. In other words, the structure

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A Partial Order on the Orthogonal Group

Cited by 34 publications

References 1 publication

Non-crossing partition lattices in finite real reflection groups

Non-crossing partition lattices in finite real reflection groups

Cambrian fans

Factoring Euclidean isometries

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