979 research outputs found
Avoiding maximal parabolic subgroups of S_k
We find an explicit expression for the generating function of the number of
permutations in S_n avoiding a subgroup of S_k generated by all but one simple
transpositions. The generating function turns out to be rational, and its
denominator is a rook polynomial for a rectangular board
Lower bounds for Kazhdan-Lusztig polynomials from patterns
We give a lower bound for the value at q=1 of a Kazhdan-Lustig polynomial in
a Weyl group W in terms of "patterns''. This is expressed by a "pattern map"
from W to W' for any parabloic subgroup W'. This notion generalizes the concept
of patterns and pattern avoidance for permutations to all Weyl groups. The main
tool of the proof is a "hyperbolic localization" on intersection cohomology;
see the related paper http://front.math.ucdavis.edu/math.AG/0202251Comment: 14 pages; updated references. Final version; will appear in
Transformation Groups vol.8, no.
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
Enumerating pattern avoidance for affine permutations
In this paper we study pattern avoidance for affine permutations. In
particular, we show that for a given pattern p, there are only finitely many
affine permutations in that avoid p if and only if p avoids
the pattern 321. We then count the number of affine permutations that avoid a
given pattern p for each p in S_3, as well as give some conjectures for the
patterns in S_4.Comment: 11 pages, 3 figures; fixed typos and proof of Proposition
Existential questions in (relatively) hyperbolic groups {\it and} Finding relative hyperbolic structures
This arXived paper has two independant parts, that are improved and corrected
versions of different parts of a single paper once named "On equations in
relatively hyperbolic groups".
The first part is entitled "Existential questions in (relatively) hyperbolic
groups". We study there the existential theory of torsion free hyperbolic and
relatively hyperbolic groups, in particular those with virtually abelian
parabolic subgroups. We show that the satisfiability of systems of equations
and inequations is decidable in these groups.
In the second part, called "Finding relative hyperbolic structures", we
provide a general algorithm that recognizes the class of groups that are
hyperbolic relative to abelian subgroups.Comment: Two independant parts 23p + 9p, revised. To appear separately in
Israel J. Math, and Bull. London Math. Soc. respectivel
On the boundary and intersection motives of genus 2 Hilbert-Siegel varieties
We study genus 2 Hilbert-Siegel varieties, i.e. Shimura varieties
corresponding to the group \mbox{GSp}_{4,F} over a totally real field ,
along with the relative Chow motives of abelian type
over obtained from irreducible representations of
\mbox{GSp}_{4,F}. We analyse the weight filtration on the degeneration of
such motives at the boundary of the Baily-Borel compactification and we find a
criterion on the highest weight which characterises the absence of
the middle weights 0 and 1 in the corresponding degeneration. Thanks to
Wildeshaus' theory, the absence of these weights allows us to construct
Hecke-equivariant Chow motives over , whose realizations equal
interior (or intersection) cohomology of with -coefficients.
We give applications to the construction of motives associated to automorphic
representations.Comment: 39 pages; comments very welcome! (v2): some typos fixed, minor
changes in the text (v3): other typos fixed, some prerequisites shortened
(now 36 pages), minor changes in the text (v4) final version, accepted for
publication in Documenta Mathematica (40 pages
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