22,803 research outputs found
Fixed Point Polynomials of Permutation Groups
In this paper we study, given a group of permutations of a finite set, the so-called fixed point polynomial , where is the number of permutations in which have exactly fixed points. In particular, we investigate how root location relates to properties of the permutation group. We show that for a large family of such groups most roots are close to the unit circle and roughly uniformly distributed round it. We prove that many families of such polynomials have few real roots. We show that many of these polynomials are irreducible when the group acts transitively. We close by indicating some future directions of this research. A corrigendum was appended to this paper on 10th October 2014. </jats:p
Recognizing Graph Theoretic Properties with Polynomial Ideals
Many hard combinatorial problems can be modeled by a system of polynomial
equations. N. Alon coined the term polynomial method to describe the use of
nonlinear polynomials when solving combinatorial problems. We continue the
exploration of the polynomial method and show how the algorithmic theory of
polynomial ideals can be used to detect k-colorability, unique Hamiltonicity,
and automorphism rigidity of graphs. Our techniques are diverse and involve
Nullstellensatz certificates, linear algebra over finite fields, Groebner
bases, toric algebra, convex programming, and real algebraic geometry.Comment: 20 pages, 3 figure
Billey's formula in combinatorics, geometry, and topology
In this expository paper we describe a powerful combinatorial formula and its
implications in geometry, topology, and algebra. This formula first appeared in
the appendix of a book by Andersen, Jantzen, and Soergel. Sara Billey
discovered it independently five years later, and it played a prominent role in
her work to evaluate certain polynomials closely related to Schubert
polynomials.
Billey's formula relates many pieces of Schubert calculus: the geometry of
Schubert varieties, the action of the torus on the flag variety, combinatorial
data about permutations, the cohomology of the flag variety and of the Schubert
varieties, and the combinatorics of root systems (generalizing inversions of a
permutation). Combinatorially, Billey's formula describes an invariant of pairs
of elements of a Weyl group. On its face, this formula is a combination of
roots built from subwords of a fixed word. As we will see, it has deeper
geometric and topological meaning as well: (1) It tells us about the tangent
spaces at each permutation flag in each Schubert variety. (2) It tells us about
singular points in Schubert varieties. (3) It tells us about the values of
Kostant polynomials. Billey's formula also reflects an aspect of GKM theory,
which is a way of describing the torus-equivariant cohomology of a variety just
from information about the torus-fixed points in the variety.
This paper will also describe some applications of Billey's formula,
including concrete combinatorial descriptions of Billey's formula in special
cases, and ways to bootstrap Billey's formula to describe the equivariant
cohomology of subvarieties of the flag variety to which GKM theory does not
apply.Comment: 14 pages, presented at the International Summer School and Workshop
on Schubert Calculus in Osaka, Japan, 201
Bivariate polynomial mappings associated with simple complex Lie algebras
There are three families of bivariate polynomial maps associated with the
rank- simple complex Lie algebras and . It is
known that the bivariate polynomial map associated with induces a
permutation of if and only if for . In this paper, we give similar criteria for the other two families. As an
application, a counterexample is given to a conjecture posed by Lidl and Wells
about the generalized Schur's problem.Comment: 16 pages, 6 figure
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.
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