215 research outputs found
Kazhdan-Lusztig polynomials and drift configurations
The coefficients of the Kazhdan-Lusztig polynomials are
nonnegative integers that are upper semicontinuous on Bruhat order.
Conjecturally, the same properties hold for -polynomials of
local rings of Schubert varieties. This suggests a parallel between the two
families of polynomials. We prove our conjectures for Grassmannians, and more
generally, covexillary Schubert varieties in complete flag varieties, by
deriving a combinatorial formula for . We introduce \emph{drift
configurations} to formulate a new and compatible combinatorial rule for
. From our rules we deduce, for these cases, the coefficient-wise
inequality .Comment: 26 pages. To appear in Algebra & Number Theor
Inequalities on Bruhat graphs, R- and Kazhdan-Lusztig polynomials
From a combinatorial perspective, we establish three inequalities on
coefficients of - and Kazhdan-Lusztig polynomials for crystallographic
Coxeter groups: (1) Nonnegativity of -coefficients of -polynomials,
(2) a new criterion of rational singularities of Bruhat intervals by sum of
quadratic coefficients of -polynomials, (3) existence of a certain strict
inequality (coefficientwise) of Kazhdan-Lusztig polynomials. Our main idea is
to understand Deodhar's inequality in a connection with a sum of
-polynomials and edges of Bruhat graphs.Comment: 16 page
Governing Singularities of Schubert Varieties
We present a combinatorial and computational commutative algebra methodology
for studying singularities of Schubert varieties of flag manifolds.
We define the combinatorial notion of *interval pattern avoidance*. For
"reasonable" invariants P of singularities, we geometrically prove that this
governs (1) the P-locus of a Schubert variety, and (2) which Schubert varieties
are globally not P. The prototypical case is P="singular"; classical pattern
avoidance applies admirably for this choice [Lakshmibai-Sandhya'90], but is
insufficient in general.
Our approach is analyzed for some common invariants, including
Kazhdan-Lusztig polynomials, multiplicity, factoriality, and Gorensteinness,
extending [Woo-Yong'05]; the description of the singular locus (which was
independently proved by [Billey-Warrington '03], [Cortez '03],
[Kassel-Lascoux-Reutenauer'03], [Manivel'01]) is also thus reinterpreted.
Our methods are amenable to computer experimentation, based on computing with
*Kazhdan-Lusztig ideals* (a class of generalized determinantal ideals) using
Macaulay 2. This feature is supplemented by a collection of open problems and
conjectures.Comment: 23 pages. Software available at the authors' webpages. Version 2 is
the submitted version. It has a nomenclature change: "Bruhat-restricted
pattern avoidance" is renamed "interval pattern avoidance"; the introduction
has been reorganize
- …