95 research outputs found
Transitive factorizations of permutations and geometry
We give an account of our work on transitive factorizations of permutations.
The work has had impact upon other areas of mathematics such as the enumeration
of graph embeddings, random matrices, branched covers, and the moduli spaces of
curves. Aspects of these seemingly unrelated areas are seen to be related in a
unifying view from the perspective of algebraic combinatorics. At several
points this work has intertwined with Richard Stanley's in significant ways.Comment: 12 pages, dedicated to Richard Stanley on the occasion of his 70th
birthda
Stein's Method, Jack Measure, and the Metropolis Algorithm
The one parameter family of Jack(alpha) measures on partitions is an
important discrete analog of Dyson's beta ensembles of random matrix theory.
Except for special values of alpha=1/2,1,2 which have group theoretic
interpretations, the Jack(alpha) measure has been difficult if not intractable
to analyze. This paper proves a central limit theorem (with an error term) for
Jack(alpha) measure which works for arbitrary values of alpha. For alpha=1 we
recover a known central limit theorem on the distribution of character ratios
of random representations of the symmetric group on transpositions. The case
alpha=2 gives a new central limit theorem for random spherical functions of a
Gelfand pair. The proof uses Stein's method and has interesting ingredients: an
intruiging construction of an exchangeable pair, properties of Jack
polynomials, and work of Hanlon relating Jack polynomials to the Metropolis
algorithm.Comment: very minor revisions; fix a few misprints and update bibliograph
Jack polynomials and orientability generating series of maps
We study Jack characters, which are the coefficients of the power-sum
expansion of Jack symmetric functions with a suitable normalization. These
quantities have been introduced by Lassalle who formulated some challenging
conjectures about them. We conjecture existence of a weight on non-oriented
maps (i.e., graphs drawn on non-oriented surfaces) which allows to express any
given Jack character as a weighted sum of some simple functions indexed by
maps. We provide a candidate for this weight which gives a positive answer to
our conjecture in some, but unfortunately not all, cases. In particular, it
gives a positive answer for Jack characters specialized on Young diagrams of
rectangular shape. This candidate weight attempts to measure, in a sense, the
non-orientability of a given map.Comment: v2: change of title, substantial changes of the content v3:
substantial changes in the presentatio
Matchings and Representation Theory
In this thesis we investigate the algebraic properties of matchings via representation theory. We identify three scenarios in different areas of combinatorial mathematics where the algebraic structure of matchings gives keen insight into the combinatorial problem at hand. In particular, we prove tight conditional lower bounds on the computational complexity of counting Hamiltonian cycles, resolve an asymptotic version of a conjecture of Godsil and Meagher in Erdos-Ko-Rado combinatorics, and shed light on the algebraic structure of symmetric semidefinite relaxations of the perfect matching proble
Combinatorial and Algebraic Enumeration: a survey of the work of Ian P. Goulden and David M. Jackson
Non-orientable branched coverings, -Hurwitz numbers, and positivity for multiparametric Jack expansions
We introduce a one-parameter deformation of the 2-Toda tau-function of
(weighted) Hurwitz numbers, obtained by deforming Schur functions into Jack
symmetric functions. We show that its coefficients are polynomials in the
deformation parameter with nonnegative integer coefficients. These
coefficients count generalized branched coverings of the sphere by an arbitrary
surface, orientable or not, with an appropriate -weighting that "measures"
in some sense their non-orientability. Notable special cases include
non-orientable dessins d'enfants for which we prove the most general result so
far towards the Matching-Jack conjecture and the "-conjecture" of Goulden
and Jackson from 1996, expansions of the -ensemble matrix model,
deformations of the HCIZ integral, and -Hurwitz numbers that we introduce
here and that are -deformations of classical (single or double) Hurwitz
numbers obtained for . A key role in our proof is played by a
combinatorial model of non-orientable constellations equipped with a suitable
-weighting, whose partition function satisfies an infinite set of PDEs.
These PDEs have two definitions, one given by Lax equations, the other one
following an explicit combinatorial decomposition.Comment: 56 pages, 6 figures; v2: definition of generalized branched covers
fixed; combinatorial decomposition and corresponding equations now presented
for connected objects and duality introduced; proof of piecewise
polynomiality changed accordingly; v3: minor correction
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