19 research outputs found
The anti-spherical category
We study a diagrammatic categorification (the "anti-spherical category") of
the anti-spherical module for any Coxeter group. We deduce that Deodhar's
(sign) parabolic Kazhdan-Lusztig polynomials have non-negative coefficients,
and that a monotonicity conjecture of Brenti's holds. The main technical
observation is a localisation procedure for the anti-spherical category, from
which we construct a "light leaves" basis of morphisms. Our techniques may be
used to calculate many new elements of the -canonical basis in the
anti-spherical module.Comment: Best viewed in colo
-Jones-Wenzl idempotents
For a prime number and any natural number we introduce, by giving an
explicit recursive formula, the -Jones-Wenzl projector
, an element of the Temperley-Lieb algebra
with coefficients in . We prove that these projectors give the
indecomposable objects in the -Hecke category over , or equivalently, they give the projector in
to the top tilting module. The way in which we find these
projectors is by categorifying the fractal appearing in the expression of the
-canonical basis in terms of the Kazhdan-Lusztig basis for .Comment: 15 pages, 21 figures. Many minor changes. Major change of notation.
Final versio