22 research outputs found
Discrete series representations and K multiplicities for U(p,q). User's guide
This document is a companion for the Maple program : Discrete series and
K-types for U(p,q) available on:http://www.math.jussieu.fr/~vergne We explain
an algorithm to compute the multiplicities of an irreducible representation of
U(p)x U(q) in a discrete series of U(p,q). It is based on Blattner's formula.
We recall the general mathematical background to compute Kostant partition
functions via multidimensional residues, and we outline our algorithm. We also
point out some properties of the piecewise polynomial functions describing
multiplicities based on Paradan's results.Comment: 51 page
Multiple Bernoulli series and volumes of moduli spaces of flat bundles over surfaces
Using Szenes formula for multiple Bernoulli series we explain how to compute
Witten series associated to classical Lie algebras. Particular instances of
these series compute volumes of moduli spaces of flat bundles over surfaces,
and also certain multiple zeta values.Comment: 51 pages, 3 figures; formula in Proposition 3.1 for the Lie group of
type G_2 is corrected; new references adde
Horn conditions for Schubert positions of general quiver subrepresentations
We give inductive conditions that characterize the Schubert positions of
subrepresentations of a general quiver representation. Our results generalize
Horn's criterion for the intersection of Schubert varieties in Grassmannians
and refine Schofield's characterization of the dimension vectors of general
subrepresentations. Our proofs are inspired by Schofield's argument as well as
Belkale's geometric proof of the saturation conjecture.Comment: 34 pages, contains detailed proofs of results previously announced in
arXiv:1804.0043
Summing a polynomial function over integral points of a polygon. User's guide
This document is a companion for the Maple program \textbf{Summing a
polynomial function over integral points of a polygon}. It contains two parts.
First, we see what this programs does. In the second part, we briefly recall
the mathematical background
Computation of the highest coefficients of weighted Ehrhart quasi-polynomials of rational polyhedra
This article concerns the computational problem of counting the lattice
points inside convex polytopes, when each point must be counted with a weight
associated to it. We describe an efficient algorithm for computing the highest
degree coefficients of the weighted Ehrhart quasi-polynomial for a rational
simple polytope in varying dimension, when the weights of the lattice points
are given by a polynomial function h. Our technique is based on a refinement of
an algorithm of A. Barvinok [Computing the Ehrhart quasi-polynomial of a
rational simplex, Math. Comp. 75 (2006), pp. 1449--1466] in the unweighted case
(i.e., h = 1). In contrast to Barvinok's method, our method is local, obtains
an approximation on the level of generating functions, handles the general
weighted case, and provides the coefficients in closed form as step polynomials
of the dilation. To demonstrate the practicality of our approach we report on
computational experiments which show even our simple implementation can compete
with state of the art software.Comment: 34 pages, 2 figure
How to Integrate a Polynomial over a Simplex
This paper settles the computational complexity of the problem of integrating
a polynomial function f over a rational simplex. We prove that the problem is
NP-hard for arbitrary polynomials via a generalization of a theorem of Motzkin
and Straus. On the other hand, if the polynomial depends only on a fixed number
of variables, while its degree and the dimension of the simplex are allowed to
vary, we prove that integration can be done in polynomial time. As a
consequence, for polynomials of fixed total degree, there is a polynomial time
algorithm as well. We conclude the article with extensions to other polytopes,
discussion of other available methods and experimental results.Comment: Tables added with new experimental results. References adde
Coefficients of Sylvester's Denumerant
For a given sequence of positive integers, we consider
the combinatorial function that counts the nonnegative
integer solutions of the equation , where the right-hand side is a varying
nonnegative integer. It is well-known that is a
quasi-polynomial function in the variable of degree . In combinatorial
number theory this function is known as Sylvester's denumerant.
Our main result is a new algorithm that, for every fixed number , computes
in polynomial time the highest coefficients of the quasi-polynomial
as step polynomials of (a simpler and more explicit
representation). Our algorithm is a consequence of a nice poset structure on
the poles of the associated rational generating function for
and the geometric reinterpretation of some rational
generating functions in terms of lattice points in polyhedral cones. Our
algorithm also uses Barvinok's fundamental fast decomposition of a polyhedral
cone into unimodular cones. This paper also presents a simple algorithm to
predict the first non-constant coefficient and concludes with a report of
several computational experiments using an implementation of our algorithm in
LattE integrale. We compare it with various Maple programs for partial or full
computation of the denumerant.Comment: minor revision, 28 page
INTERMEDIATE SUMS ON POLYHEDRA II:BIDEGREE AND POISSON FORMULA
Abstract. We continue our study of intermediate sums over polyhedra,interpolating between integrals and discrete sums, whichwere introduced by A. Barvinok [Computing the Ehrhart quasi-polynomial of a rational simplex, Math. Comp. 75 (2006), 1449–1466]. By well-known decompositions, it is sufficient to considerthe case of affine cones s+c, where s is an arbitrary real vertex andc is a rational polyhedral cone. For a given rational subspace L,we integrate a given polynomial function h over all lattice slicesof the affine cone s + c parallel to the subspace L and sum up theintegrals. We study these intermediate sums by means of the intermediategenerating functions SL(s+c)(ξ), and expose the bidegreestructure in parameters s and ξ, which was implicitly used in thealgorithms in our papers [Computation of the highest coefficients ofweighted Ehrhart quasi-polynomials of rational polyhedra, Found.Comput. Math. 12 (2012), 435–469] and [Intermediate sums onpolyhedra: Computation and real Ehrhart theory, Mathematika 59(2013), 1–22]. The bidegree structure is key to a new proof for theBaldoni–Berline–Vergne approximation theorem for discrete generatingfunctions [Local Euler–Maclaurin expansion of Barvinokvaluations and Ehrhart coefficients of rational polytopes, Contemp.Math. 452 (2008), 15–33], using the Fourier analysis with respectto the parameter s and a continuity argument. Our study alsoenables a forthcoming paper, in which we study intermediate sumsover multi-parameter families of polytopes