89 research outputs found
Enumeration of Standard Young Tableaux
A survey paper, to appear as a chapter in a forthcoming Handbook on
Enumeration.Comment: 65 pages, small correction
Set partitions, tableaux, and subspace profiles under regular split semisimple matrices
We introduce a family of univariate polynomials indexed by integer
partitions. At prime powers, they count the number of subspaces in a finite
vector space that transform under a regular diagonal matrix in a specified
manner. At 1, they count set partitions with specified block sizes. At 0, they
count standard tableaux of specified shape. At -1, they count standard shifted
tableaux of a specified shape. These polynomials are generated by a new
statistic on set partitions (called the interlacing number) as well as a
polynomial statistic on standard tableaux. They allow us to express q-Stirling
numbers of the second kind as sums over standard tableaux and as sums over set
partitions.
For partitions whose parts are at most two, these polynomials are the
non-zero entries of the Catalan triangle associated to the q-Hermite orthogonal
polynomial sequence. In particular, when all parts are equal to two, they
coincide with the polynomials defined by Touchard that enumerate chord diagrams
by number of crossings.Comment: 28 pages, minor change
Generalized tableaux over arbitrary digraphs and their associated differential equations
We revisit the concepts of acyclic orderings and number of acyclic orderings
of acyclic digraphs in terms of dispositions and counters for arbitrary
multidigraphs. We prove that when we add a sequence of nested directed paths to
a directed graph there is a unique polynomial such that the generatrix function
of the family of counters is the product of the polynomial and the exponential
function. We give an application, by considering a kind of digraphs arranged in
rows introduced by the authors in a previous paper, called dispositional
digraphs, in the particular case in which the digraph has two rows, to obtain
new families of linear differential equations of small order whose coefficients
are polynomials of small degree which admit polynomial solutions. In
particular, we obtain a new differential equation associated to Catalan
numbers, and the corresponding associated polynomials, which are solution of
this differential equation; we term them Catalan differencial equation and
Catalan polynomials, respectively. We prove that the Catalan polynomials
obtained when we connect the directed path to the second vertex of the lower
row of the digraph are orthogonal polynomials for an appropriate weight
function. We characterize the digraphs that maximize the counter of connected
dispositional digraphs and we find a new differential equation associated to
these digraphs. We introduce also dispositions and counters in any multidigraph
with non-strict inequalities in the dispositions, and we find new differential
equations associated to some of them
Physical Combinatorics and Quasiparticles
We consider the physical combinatorics of critical lattice models and their
associated conformal field theories arising in the continuum scaling limit. As
examples, we consider A-type unitary minimal models and the level-1 sl(2)
Wess-Zumino-Witten (WZW) model. The Hamiltonian of the WZW model is the
invariant XXX spin chain. For simplicity, we consider these
theories only in their vacuum sectors on the strip. Combinatorially, fermionic
particles are introduced as certain features of RSOS paths. They are composites
of dual-particles and exhibit the properties of quasiparticles. The particles
and dual-particles are identified, through an energy preserving bijection, with
patterns of zeros of the eigenvalues of the fused transfer matrices in their
analyticity strips. The associated (m,n) systems arise as geometric packing
constraints on the particles. The analyticity encoded in the patterns of zeros
is the key to the analytic calculation of the excitation energies through the
Thermodynamic Bethe Ansatz (TBA). As a by-product of our study, in the case of
the WZW or XXX model, we find a relation between the location of the Bethe root
strings and the location of the transfer matrix 2-strings.Comment: 57 pages, in version 2: typos corrected, some sentences clarified,
one appendix remove
-Schur functions and affine Schubert calculus
This book is an exposition of the current state of research of affine
Schubert calculus and -Schur functions. This text is based on a series of
lectures given at a workshop titled "Affine Schubert Calculus" that took place
in July 2010 at the Fields Institute in Toronto, Ontario. The story of this
research is told in three parts: 1. Primer on -Schur Functions 2. Stanley
symmetric functions and Peterson algebras 3. Affine Schubert calculusComment: 213 pages; conference website:
http://www.fields.utoronto.ca/programs/scientific/10-11/schubert/, updates
and corrections since v1. This material is based upon work supported by the
National Science Foundation under Grant No. DMS-065264
Combinatorial Structures in Random Matrix Theory Predictions for -Functions
Our results can be viewed as applications of algebraic combinatorics in
random matrix theory. These applications are motivated by the predictive power
of random matrix theory for the statistical behavior of the celebrated Riemann
-function (and -functions in general), which was discovered by
Montgomery (with regard to zeros of -functions) and by Keating and Snaith
(with regard to values of -functions).
The first results revolve around a new operation on partitions, which we call
overlap. We prove two overlap identities for so-called Littlewood-Schur
functions. The first overlap identity represents the Littlewood-Schur function
as a sum over subsets of , while the second overlap
identity essentially represents as a sum over pairs of
partitions whose overlap equals . Both identities are derived by
applying Laplace expansion to a determinantal formula for Littlewood-Schur
functions due to Moens and Van der Jeugt. In addition, we give two visual
characterizations for the set of all pairs of partitions whose overlap is equal
to a partition .
The second result is an asymptotic formula for averages of mixed ratios of
characteristic polynomials over the unitary group, where mixed ratios are
products of ratios and/or logarithmic derivatives. Our proof of this formula is
a generalization of Bump and Gamburd's elegant combinatorial proof of Conrey,
Forrester and Snaith's formula for averages of ratios of characteristic
polynomials over the unitary group. The generalization relies on three
combinatorial results, namely the first overlap identity, a new variant of the
Murnaghan-Nakayama rule and an idea from vertex operator formalism. We conclude
this thesis by explaining how this approach might lead to new number theoretic
proofs.Comment: 155 pages, PhD thesi
T-systems and Y-systems in integrable systems
The T and Y-systems are ubiquitous structures in classical and quantum
integrable systems. They are difference equations having a variety of aspects
related to commuting transfer matrices in solvable lattice models, q-characters
of Kirillov-Reshetikhin modules of quantum affine algebras, cluster algebras
with coefficients, periodicity conjectures of Zamolodchikov and others,
dilogarithm identities in conformal field theory, difference analogue of
L-operators in KP hierarchy, Stokes phenomena in 1d Schr\"odinger problem,
AdS/CFT correspondence, Toda field equations on discrete space-time, Laplace
sequence in discrete geometry, Fermionic character formulas and combinatorial
completeness of Bethe ansatz, Q-system and ideal gas with exclusion statistics,
analytic and thermodynamic Bethe ans\"atze, quantum transfer matrix method and
so forth. This review article is a collection of short reviews on these topics
which can be read more or less independently.Comment: 156 pages. Minor corrections including the last paragraph of sec.3.5,
eqs.(4.1), (5.28), (9.37) and (13.54). The published version (JPA topical
review) also needs these correction
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