Sets of reflections defining twisted Bruhat orders

Edgar, Tom

doi:10.1007/s10801-007-0060-9

Cited by 5 publications

(13 citation statements)

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“…The second part of (b) follows since the formula in (a) implies that x · (Φ + \ Γ) = Φ + \ (x · Γ). Part (c) is proved by reducing to the easily checked case of dihedral groups by considering the intersections of Γ with the maximal dihedral reflection subgroups (see 6.3) of W ; see [19,Proposition 2.6]. Part (d) is proved in [26] for the standard reflection representation of [4] or [22] by a straightforward modification of a well known argument for finite Weyl groups.…”

Section: 1mentioning

confidence: 99%

On the Weak Order of Coxeter Groups

Dyer¹

2019

Can. J. Math.-J. Can. Math.

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This paper provides some evidence for conjectural relations between extensions of (right) weak order on Coxeter groups, closure operators on root systems, and Bruhat order. The conjecture focused upon here refines an earlier question as to whether the set of initial sections of reflection orders, ordered by inclusion, forms a complete lattice. Meet and join in weak order are described in terms of a suitable closure operator. Galois connections are defined from the power set of W to itself, under which maximal subgroups of certain groupoids correspond to certain complete meet subsemilattices of weak order. An analogue of weak order for standard parabolic subsets of any rank of the root system is defined, reducing to the usual weak order in rank zero, and having some analogous properties in rank one (and conjecturally in general). t t t t t t t t t t t J J J J J J J J J J J

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Section: 1mentioning

confidence: 99%

On the Weak Order of Coxeter Groups

Dyer¹

2019

Can. J. Math.-J. Can. Math.

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“…w m−2 , w m−1 } are s-stable since our total order is (A, s)-compatible and m is even. Now since R is a free right R smodule of rank two we have that the functor B s ⊗ − : R → R is exact hence tensoring the short exact sequence above we get another short exact sequence (14) 0…”

Section: 2mentioning

confidence: 99%

“…Hence B s (Γ A ≥wm B) has a filtration with subquotients given by (shifts) of R w j for j ≥ m, while B s (B/Γ A ≥wm B) has a filtration with subquotients given by (shifts) of R w j for j < m. But since B s B ∈ F A ∆ and any nonzero element in R w j generates a subbimodule isomorphic to R w j as ungraded bimodule, hence has support Gr(w j ), it follows that B s (Γ A ≥wm B) is exactly the subbimodule of B s B containing the elements whose support is included in j≥m Gr(w j ). The short exact sequence (14) can therefore be rewritten as (1). Now considering the exact sequence…”

Section: 2mentioning

confidence: 99%

“…Denoting by T x,A the image of x A in the Hecke algebra, the set {T x,A } x∈W yields a basis of H. For A = ∅ it is the standard basis, while for A = T it is the costandard basis. Dyer [8] and Edgar [14] attached to any biclosed set A a Bruhat-like order < A and a Z-valued length function ℓ A . We use these orders to generalize Soergel's standard filtration and interpret structure constants of the canonical basis in the basis {T x,A } as graded multiplicities (Theorem 4.10):…”

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confidence: 99%

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Twisted filtrations of Soergel bimodules and linear Rouquier complexes

Gobet

2017

Journal of Algebra

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We consider twisted standard filtrations of Soergel bimodules associated to arbitrary Coxeter groups and show that the graded multiplicities in these filtrations can be interpreted as structure constants in the Hecke algebra. This corresponds to the positivity of the polynomials occurring when expressing an element of the canonical basis in a generalized standard basis twisted by a biclosed set of roots in the sense of Dyer, and comes as a corollary of Soergel's conjecture. We then show the positivity of the corresponding inverse polynomials in case the biclosed set is an inversion set of an element or its complement by generalizing a result of Elias and Williamson on the linearity of the Rouquier complexes associated to lifts of these basis elements in the Artin-Tits group. These lifts turn out to be generalizations of Mikado braids as introduced in a joint work with Digne. This second positivity property generalizes a result of Dyer and Lehrer from finite to arbitrary Coxeter groups.

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“…Initial sections of reflection orders, which are certain subsets A ⊂ T , lead to define partial pre-orders ≤ A on W that are similar to Bruhat order. The twisted length function plays an important role to determine which are the subsets A ⊂ T such that ≤ A is a partial order [1,7]. The collection of subsets of T satisfying this property is exactly the set of biclosed subsets of T , which is denoted by B(T ) (we recall this notion in Section 1.1).…”

Section: Introductionmentioning

confidence: 99%