11 research outputs found
A closed character formula for symmetric powers of irreducible representations
We prove a closed character formula for the symmetric powers
of a fixed irreducible representation of a complex semi-simple Lie
algebra by means of partial fraction decomposition. The formula
involves rational functions in rank of many variables which are
easier to determine than the weight multiplicities of
themselves. We compute those rational functions in some interesting cases.
Furthermore, we introduce a residue-type generating function for the weight
multiplicities of and explain the connections between our
character formula, vector partition functions and iterated partial fraction
decomposition.Comment: 14 pages, 1 figure, published versio
Extension of the Bernoulli and Eulerian Polynomials of Higher Order and Vector Partition Function
Following the ideas of L. Carlitz we introduce a generalization of the
Bernoulli and Eulerian polynomials of higher order to vectorial index and
argument. These polynomials are used for computation of the vector partition
function , i.e., a number of integer solutions to a linear
system . It is shown that can be expressed through the vector Bernoulli polynomials of higher order.Comment: 18 page
Presburger arithmetic, rational generating functions, and quasi-polynomials
Presburger arithmetic is the first-order theory of the natural numbers with
addition (but no multiplication). We characterize sets that can be defined by a
Presburger formula as exactly the sets whose characteristic functions can be
represented by rational generating functions; a geometric characterization of
such sets is also given. In addition, if p=(p_1,...,p_n) are a subset of the
free variables in a Presburger formula, we can define a counting function g(p)
to be the number of solutions to the formula, for a given p. We show that every
counting function obtained in this way may be represented as, equivalently,
either a piecewise quasi-polynomial or a rational generating function. Finally,
we translate known computational complexity results into this setting and
discuss open directions.Comment: revised, including significant additions explaining computational
complexity results. To appear in Journal of Symbolic Logic. Extended abstract
in ICALP 2013. 17 page
On polynomials counting essentially irreducible maps
We consider maps on genus- surfaces with (labeled) faces of prescribed
even degrees. It is known since work of Norbury that, if one disallows vertices
of degree one, the enumeration of such maps is related to the counting of
lattice point in the moduli space of genus- curves with labeled points
and is given by a symmetric polynomial in the
face degrees . We generalize this by restricting to
genus- maps that are essentially -irreducible for , which
loosely speaking means that they are not allowed to possess contractible cycles
of length less than and each such cycle of length is required to
bound a face of degree . The enumeration of such maps is shown to be again
given by a symmetric polynomial in
the face degrees with a polynomial dependence on . These polynomials satisfy
(generalized) string and dilaton equations, which for uniquely
determine them. The proofs rely heavily on a substitution approach by Bouttier
and Guitter and the enumeration of planar maps on genus- surfaces.Comment: 37 pages, 5 figure
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From amplitudes to contact cosmological correlators
Abstract: Our understanding of quantum correlators in cosmological spacetimes, including those that we can observe in cosmological surveys, has improved qualitatively in the past few years. Now we know many constraints that these objects must satisfy as consequences of general physical principles, such as symmetries, unitarity and locality. Using this new understanding, we derive the most general scalar four-point correlator, i.e., the trispectrum, to all orders in derivatives for manifestly local contact interactions. To obtain this result we use techniques from commutative algebra to write down all possible scalar four-particle amplitudes without assuming invariance under Lorentz boosts. We then input these amplitudes into a contact reconstruction formula that generates a contact cosmological correlator in de Sitter spacetime from a contact scalar or graviton amplitude. We also show how the same procedure can be used to derive higher-point contact cosmological correlators. Our results further extend the reach of the boostless cosmological bootstrap and build a new connection between flat and curved spacetime physics
From amplitudes to contact cosmological correlators
Abstract: Our understanding of quantum correlators in cosmological spacetimes, including those that we can observe in cosmological surveys, has improved qualitatively in the past few years. Now we know many constraints that these objects must satisfy as consequences of general physical principles, such as symmetries, unitarity and locality. Using this new understanding, we derive the most general scalar four-point correlator, i.e., the trispectrum, to all orders in derivatives for manifestly local contact interactions. To obtain this result we use techniques from commutative algebra to write down all possible scalar four-particle amplitudes without assuming invariance under Lorentz boosts. We then input these amplitudes into a contact reconstruction formula that generates a contact cosmological correlator in de Sitter spacetime from a contact scalar or graviton amplitude. We also show how the same procedure can be used to derive higher-point contact cosmological correlators. Our results further extend the reach of the boostless cosmological bootstrap and build a new connection between flat and curved spacetime physics
The partial-fractions method for counting solutions to integral linear systems
Dedicated to Lou Billera on the occasion of his sixtieth birthday. Abstract. We present a new tool to compute the number φA(b) of integer solutions to the linear system x≥0, Ax = b, where the coefficients of A and b are integral. φA(b) is often described as a vector partition function. Our methods use partial fraction expansions of Euler’s generating function for φA(b). A special class of vector partition functions are Ehrhart (quasi-)polynomials counting integer points in dilated polytopes. 1. Euler’s generating function We are interested in computing the number of integer solutions of the linear system x ∈ R d ≥0, Ax = b, where A is a nonnegative (m × d)-integral matrix and b ∈ Z m. We think of A as fixed and study the number of solutions φA(b) as a function of b. (Strictly speaking, this function should only be defined for those b which lie in the nonnegative linear span of the columns of A.) The function φA(b), often called a vector partition function, appears in a wealth of mathematica