5,437 research outputs found
Left cells containing a fully commutative element
AbstractLet W be a finite or an affine Coxeter group and Wc the set of all the fully commutative elements in W. For any left cell L of W containing some fully commutative element, our main result of the paper is to prove that there exists a unique element (say wL) in L∩Wc such that any z∈L has the form z=xwL with ℓ(z)=ℓ(x)+ℓ(wL) for some x∈W. This implies that L is left connected, verifying a conjecture of Lusztig in our case
Combinatorics of fully commutative involutions in classical Coxeter groups
An element of a Coxeter group is fully commutative if any two of its
reduced decompositions are related by a series of transpositions of adjacent
commuting generators. In the present work, we focus on fully commutative
involutions, which are characterized in terms of Viennot's heaps. By encoding
the latter by Dyck-type lattice walks, we enumerate fully commutative
involutions according to their length, for all classical finite and affine
Coxeter groups. In the finite cases, we also find explicit expressions for
their generating functions with respect to the major index. Finally in affine
type , we connect our results to Fan--Green's cell structure of the
corresponding Temperley--Lieb algebra.Comment: 25 page
The enumeration of fully commutative affine permutations
We give a generating function for the fully commutative affine permutations
enumerated by rank and Coxeter length, extending formulas due to Stembridge and
Barcucci--Del Lungo--Pergola--Pinzani. For fixed rank, the length generating
functions have coefficients that are periodic with period dividing the rank. In
the course of proving these formulas, we obtain results that elucidate the
structure of the fully commutative affine permutations.Comment: 18 pages; final versio
Kazhdan-Lusztig cells in planar hyperbolic Coxeter groups and automata
Let C be a one- or two-sided Kazhdan--Lusztig cell in a Coxeter group (W,S),
and let Reduced(C) denote the set of reduced expressions of all w in C,
regarded as a language over the alphabet S. Casselman has conjectured that
Reduced(C) is regular. In this paper we give a conjectural description of the
cells when W is the group corresponding to a hyperbolic polygon, and show that
our conjectures imply Casselman's.Comment: 16 pages, 6 figures, v2: revised following referee's comment
Noncommutative localization in noncommutative geometry
The aim of these notes is to collect and motivate the basic localization
toolbox for the geometric study of ``spaces'', locally described by
noncommutative rings and their categories of one-sided modules.
We present the basics of Ore localization of rings and modules in much
detail. Common practical techniques are studied as well. We also describe a
counterexample for a folklore test principle. Localization in negatively
filtered rings arising in deformation theory is presented. A new notion of the
differential Ore condition is introduced in the study of localization of
differential calculi.
To aid the geometrical viewpoint, localization is studied with emphasis on
descent formalism, flatness, abelian categories of quasicoherent sheaves and
generalizations, and natural pairs of adjoint functors for sheaf and module
categories. The key motivational theorems from the seminal works of Gabriel on
localization, abelian categories and schemes are quoted without proof, as well
as the related statements of Popescu, Watts, Deligne and Rosenberg.
The Cohn universal localization does not have good flatness properties, but
it is determined by the localization map already at the ring level. Cohn
localization is here related to the quasideterminants of Gelfand and Retakh;
and this may help understanding both subjects.Comment: 93 pages; (including index: use makeindex); introductory survey, but
with few smaller new result
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