Twisted filtrations of Soergel bimodules and linear Rouquier complexes

Abstract

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.

The Hodge theory of the Hecke category

“…Remark 4.7. The reader is referred to [15] for a list of further positivity properties and to [21] for some recent results.…”

Dual Garside structures and Coxeter sortable elements

“…[20] in type A n and in unpublished work of Dyer [24] for abritrary Coxeter groups. A detailed study was initiated in [19], [28]. Mikado braids are elements of an Artin-Tits group attached to an arbitrary Coxeter group.…”

“…The proofs there use either topological realizations of the braid groups (i.e., in terms of Artin braids) or categorification techniques. There are indeed indications that Mikado braids play an important role in the categorifications of the braid groups (see [28], [34]). Another motivation consists of searching for a construction of the dual braid monoids which would avoid using the classification.…”

“…Mikado braids have interesting homological properties in the categorical incarnations of the braid groups (see [28], [34]). For instance, the fact that they lie in the heart of the canonical t-structure on the bounded homotopy category of Soergel bimodules can be used to derive positivity properties of their images in the Iwahori-Hecke algebra of the Coxeter system (see Section 5.1 below for more details).…”

Simple dual braids, noncrossing partitions and Mikado braids of type Dn

“…For example, they satisfy an analogue of Matsumoto's lemma in Coxeter groups [, Section 9]. We refer the reader to [, Section 4; , Section 9] (there the Mikado braids are called rational permutation braids , while the terminology Mikado braids rather refers to braids viewed topologically; it is shown however in that both are equivalent) or [, Section 3.2] for more on the topic. Another important property is that their images in the Iwahori–Hecke algebra of the Coxeter system have positivity properties; let us be more precise.…”