696 research outputs found
Conjectures about certain parabolic Kazhdan--Lusztig polynomials
Irreducibility results for parabolic induction of representations of the
general linear group over a local non-archimedean field can be formulated in
terms of Kazhdan--Lusztig polynomials of type . Spurred by these results and
some computer calculations, we conjecture that certain alternating sums of
Kazhdan--Lusztig polynomials known as parabolic Kazhdan--Lusztig polynomials
satisfy properties analogous to those of the ordinary ones.Comment: final versio
Path representation of maximal parabolic Kazhdan-Lusztig polynomials
We provide simple rules for the computation of Kazhdan--Lusztig polynomials
in the maximal parabolic case. They are obtained by filling regions delimited
by paths with "Dyck strips" obeying certain rules. We compare our results with
those of Lascoux and Sch\"utzenberger.Comment: v3: fixed proof of lemma
The Robinson-Schensted Correspondence and -web Bases
We study natural bases for two constructions of the irreducible
representation of the symmetric group corresponding to : the {\em
reduced web} basis associated to Kuperberg's combinatorial description of the
spider category; and the {\em left cell basis} for the left cell construction
of Kazhdan and Lusztig. In the case of , the spider category is the
Temperley-Lieb category; reduced webs correspond to planar matchings, which are
equivalent to left cell bases. This paper compares the images of these bases
under classical maps: the {\em Robinson-Schensted algorithm} between
permutations and Young tableaux and {\em Khovanov-Kuperberg's bijection}
between Young tableaux and reduced webs.
One main result uses Vogan's generalized -invariant to uncover a close
structural relationship between the web basis and the left cell basis.
Intuitively, generalized -invariants refine the data of the inversion set
of a permutation. We define generalized -invariants intrinsically for
Kazhdan-Lusztig left cell basis elements and for webs. We then show that the
generalized -invariant is preserved by these classical maps. Thus, our
result allows one to interpret Khovanov-Kuperberg's bijection as an analogue of
the Robinson-Schensted correspondence.
Despite all of this, our second main result proves that the reduced web and
left cell bases are inequivalent; that is, these bijections are not
-equivariant maps.Comment: 34 pages, 23 figures, minor corrections and revisions in version
Mask formulas for cograssmannian Kazhdan-Lusztig polynomials
We give two contructions of sets of masks on cograssmannian permutations that
can be used in Deodhar's formula for Kazhdan-Lusztig basis elements of the
Iwahori-Hecke algebra. The constructions are respectively based on a formula of
Lascoux-Schutzenberger and its geometric interpretation by Zelevinsky. The
first construction relies on a basis of the Hecke algebra constructed from
principal lower order ideals in Bruhat order and a translation of this basis
into sets of masks. The second construction relies on an interpretation of
masks as cells of the Bott-Samelson resolution. These constructions give
distinct answers to a question of Deodhar.Comment: 43 page
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
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