25 research outputs found
A note on moments of derivatives of characteristic polynomials
We present a simple technique to compute moments of derivatives of unitary
characteristic polynomials. The first part of the technique relies on an idea
of Bump and Gamburd: it uses orthonormality of Schur functions over unitary
groups to compute matrix averages of characteristic polynomials. In order to
consider derivatives of those polynomials, we here need the added strength of
the Generalized Binomial Theorem of Okounkov and Olshanski. This result is very
natural as it provides coefficients for the Taylor expansions of Schur
functions, in terms of shifted Schur functions. The answer is finally given as
a sum over partitions of functions of the contents. One can also obtain
alternative expressions involving hypergeometric functions of matrix arguments.Comment: 12 page
Integrality of hook ratios
We study integral ratios of hook products of quotient partitions. This
question is motivated by an analogous question in number theory concerning
integral factorial ratios. We prove an analogue of a theorem of Landau that
already applied in the factorial case. Under the additional condition that the
ratio has one more factor on the denominator than the numerator, we provide a
complete classification. Ultimately this relies on Kneser's theorem in additive
combinatorics.Comment: 13 pages, 3 figures Keywords: partitions, hook products, Kneser's
theorem, McKay numbers, Beurling-Nyman criterio
A multiset hook length formula and some applications
A multiset hook length formula for integer partitions is established by using
combinatorial manipulation. As special cases, we rederive three hook length
formulas, two of them obtained by Nekrasov-Okounkov, the third one by Iqbal,
Nazir, Raza and Saleem, who have made use of the cyclic symmetry of the
topological vertex. A multiset hook-content formula is also proved.Comment: 19 pages; 3 figure
Difference operators for partitions under the Littlewood decomposition
The concept of -difference operator for functions of partitions is
introduced to prove a generalization of Stanley's theorem on polynomiality of
Plancherel averages of symmetric functions related to contents and hook
lengths. Our extension uses a generalization of the notion of Plancherel
measure, based on walks in the Young lattice with steps given by the addition
of -hooks. It is well-known that the hook lengths of multiples of can be
characterized by the Littlewood decomposition. Our study gives some further
information on the contents and hook lengths of other congruence classes modulo
.Comment: 24 page
Interoperability in the OpenDreamKit Project: The Math-in-the-Middle Approach
OpenDreamKit --- "Open Digital Research Environment Toolkit for the
Advancement of Mathematics" --- is an H2020 EU Research Infrastructure project
that aims at supporting, over the period 2015--2019, the ecosystem of
open-source mathematical software systems. From that, OpenDreamKit will deliver
a flexible toolkit enabling research groups to set up Virtual Research
Environments, customised to meet the varied needs of research projects in pure
mathematics and applications.
An important step in the OpenDreamKit endeavor is to foster the
interoperability between a variety of systems, ranging from computer algebra
systems over mathematical databases to front-ends. This is the mission of the
integration work package (WP6). We report on experiments and future plans with
the \emph{Math-in-the-Middle} approach. This information architecture consists
in a central mathematical ontology that documents the domain and fixes a joint
vocabulary, combined with specifications of the functionalities of the various
systems. Interaction between systems can then be enriched by pivoting off this
information architecture.Comment: 15 pages, 7 figure