arXiv subjects

Paul Zinn-Justin

Publications and source records attributed to Paul Zinn-Justin.

At least 37 records · Page 2Linked to original sources

arXiv2020

Honeycombs for Hall polynomials

We propose a new formulation of Hall polynomials in terms of honeycombs, which were previously introduced in the context of the Littlewood--Richardson rule. We prove a Pieri rule and associativity for our honeycomb formula, thus showing equality with Hall polynomials. Our proofs are linear algebraic in nature, extending nontrivially the corresponding bijective results for ordinary Littlewood--Richardson coefficients [A. Knutson, T. Tao, C. Woodward, 2004].

math.CO↗

arXiv2019

Restricting Schubert classes to symplectic Grassmannians using self-dual puzzles

Given a Schubert class on $Gr(k,V)$ where $V$ is a symplectic vector space of dimension $2n$, we consider its restriction to the symplectic Grassmannian $SpGr(k,V)$ of isotropic subspaces. Pragacz gave tableau formulae for positively computing the expansion of these $H^*(Gr(k,V))$ classes into Schubert classes of the target when $k=n$, which corresponds to expanding Schur polynomials into $Q$-Schur polynomials. Coskun described an algorithm for their expansion when $k\leq n$. We give a puzzle-based formula for these expansions, while extending them to equivariant cohomology. We make use of a new observation that usual Grassmannian puzzle pieces are already enough to do some $2$-step Schubert calculus, and apply techniques from quantum integrable systems (``scattering diagrams'').

math.RT↗

arXiv2019

Eulerian orientations and the six-vertex model on planar map

We address the enumeration of planar 4-valent maps equipped with an Eulerian orientation by two different methods, and compare the solutions we thus obtain. With the first method we enumerate these orientations as well as a restricted class which we show to be in bijection with general Eulerian orientations. The second method, based on the work of Kostov, allows us to enumerate these 4-valent orientations with a weight on some vertices, corresponding to the six vertex model. We prove that this result generalises both results obtained using the first method, although the equivalence is not immediately clear.

math.CO↗

arXiv2018

Loop Models and $K$-Theory

This is a review/announcement of results concerning the connection between certain exactly solvable two-dimensional models of statistical mechanics, namely loop models, and the equivariant $K$-theory of the cotangent bundle of the Grassmannian. We interpret various concepts from integrable systems ($R$-matrix, partition function on a finite domain) in geometric terms. As a byproduct, we provide explicit formulae for $K$-classes of various coherent sheaves, including structure and (conjecturally) square roots of canonical sheaves and canonical sheaves of conormal varieties of Schubert varieties.

math.AG↗

arXiv2017

1324-avoiding permutations revisited

We give an improved algorithm for counting the number of $1324$-avoiding permutations, resulting in $14$ further terms of the generating function, which is now known for all patterns of length $\le 50$. We re-analyse the generating function and find additional evidence for our earlier conclusion that unlike other classical length-$4$ pattern-avoiding permutations, the generating function does not have a simple power-law singularity, but rather, the number of $1324$-avoiding permutations of length $n$ behaves as \[ B\cdot μ^n \cdot μ_1^{\sqrt{n}} \cdot n^g. \] We estimate $μ=11.600 \pm 0.003$, $μ_1 = 0.0400 \pm 0.0005$, $g = -1.1 \pm 0.1$ while the estimate of $B$ depends sensitively on the precise value of $μ$, $μ_1$ and $g$. This reanalysis provides substantially more compelling arguments for the presence of the stretched exponential term $μ_1^{\sqrt{n}}$.

math.CO↗

arXiv2016

Littlewood-Richardson coefficients for Grothendieck polynomials from integrability

We study the Littlewood-Richardson coefficients of double Grothendieck polynomials indexed by Grassmannian permutations. Geometrically, these are the structure constants of the equivariant $K$-theory ring of Grassmannians. Representing the double Grothendieck polynomials as partition functions of an integrable vertex model, we use its Yang-Baxter equation to derive a series of product rules for the former polynomials and their duals. The Littlewood-Richardson coefficients that arise can all be expressed in terms of puzzles without gashes, which generalize previous puzzles obtained by Knutson-Tao and Vakil.

math.CO↗

arXiv2016

Type $\hat{\mathrm C}$ Brauer loop schemes and loop model with boundaries

In this paper we study the Brauer loop model on a strip and the associated quantum Knizhnik--Zamolodchikov (qKZ) equation. We show that the minimal degree solution of the Brauer qKZ equation with one of four different possible boundary conditions, gives the multidegrees of the irreducible components of generalizations of the Brauer loop scheme of [Knutson--Zinn-Justin '07] with one of four kinds of symplectic-type symmetry. This is accomplished by studying these irreducible components, which are indexed by link patterns, and describing the geometric action of Brauer generators on them. We also provide recurrence relations for the multidegrees and compute the sum rules (multidegrees of the whole schemes).

math-ph↗

arXiv2016

Hall polynomials, inverse Kostka polynomials and puzzles

We study two different one-parameter generalizations of Littlewood--Richardson coefficients, namely Hall polynomials and generalized inverse Kostka polynomials, and derive new combinatorial formulae for them. Our combinatorial expressions are closely related to puzzles, originally introduced by Knutson and Tao in their work on the equivariant cohomology of the Grassmannian.

math-ph↗

arXiv2015

Theta function solutions of the qKZB equation for a face model

We consider the quantum Knizhnik-Zamolodchikov-Bernard equation for a face model with elliptic weights, the SOS model. We provide explicit solutions as theta functions. On the so-called combinatorial line, in which the model is equivalent to the three-colour model, these solutions are shown to be eigenvectors of the transfer matrix with periodic boundary conditions.

math-ph↗

arXiv2015

Refined Cauchy/Littlewood identities and six-vertex model partition functions: III. Deformed bosons

We study Hall-Littlewood polynomials using an integrable lattice model of $t$-deformed bosons. Working with row-to-row transfer matrices, we review the construction of Hall-Littlewood polynomials (of the $A_n$ root system) within the framework of this model. Introducing appropriate double-row transfer matrices, we extend this formalism to Hall-Littlewood polynomials based on the $BC_n$ root system, and obtain a new combinatorial formula for them. We then apply our methods to prove a series of refined Cauchy and Littlewood identities involving Hall-Littlewood polynomials. The last two of these identities are new, and relate infinite sums over hyperoctahedrally symmetric Hall-Littlewood polynomials with partition functions of the six-vertex model on finite domains.

math-ph↗

arXiv2014

Equilibrium problems for Raney densities

The Raney numbers are a class of combinatorial numbers generalising the Fuss--Catalan numbers. They are indexed by a pair of positive real numbers $(p,r)$ with $p>1$ and $0 < r \le p$, and form the moments of a probability density function. For certain $(p,r)$ the latter has the interpretation as the density of squared singular values for certain random matrix ensembles, and in this context equilibrium problems characterising the Raney densities for $(p,r) = (θ+1,1)$ and $(θ/2+1,1/2)$ have recently been proposed. Using two different techniques --- one based on the Wiener--Hopf method for the solution of integral equations and the other on an analysis of the algebraic equation satisfied by the Green's function --- we establish the validity of the equilibrium problems for general $θ> 0$ and similarly use both methods to identify the equilibrium problem for $(p,r) = (θ/q+1,1/q)$, $θ> 0$ and $q \in \mathbb Z^+$. The Wiener--Hopf method is used to extend the latter to parameters $(p,r) = (θ/q + 1, m+ 1/q)$ for $m$ a non-negative integer, and also to identify the equilibrium problem for a family of densities with moments given by certain binomial coefficients.

math-ph↗

arXiv2014

The Brauer loop scheme and orbital varieties

A. Joseph invented multidegrees in [Jo84] to study orbital varieties, which are the components of an orbital scheme, itself constructed by intersecting a nilpotent orbit with a Borel subalgebra. Their multidegrees, known as Joseph polynomials, give a basis of a (Springer) representation of the Weyl group. In the case of the nilpotent orbit $\{ M^2=0 \}$, the orbital varieties can be indexed by noncrossing chord diagrams in the disc. In this paper we study the normal cone to the orbital scheme inside this nilpotent orbit $\{ M^2 = 0 \}$. This gives a better-motivated construction of the Brauer loop scheme we introduced in [KZJ07], whose components are indexed by all chord diagrams (now possibly with crossings) in the disc. The multidegrees of its components, the Brauer loop varieties, were shown to reproduce the ground state of the Brauer loop model in statistical mechanics [DFZJ06]. Here, we reformulate and slightly generalize these multidegrees in order to express them as solutions of the rational quantum Knizhnik--Zamolodchikov equation associated to the Brauer algebra. In particular, the vector of the multidegrees satisfies two sets of equations, corresponding to the $e_i$ and $f_i$ generators of the Brauer algebra. We describe here the geometric meaning of both $e_i$ and $f_i$ equations in our slightly extended setting. We also describe the corresponding actions at the level of orbital varieties: while only the $e_i$ equations make sense directly on the Joseph polynomials, the $f_i$ equations also appear if one introduces a broader class of varieties. We explain the connection of the latter with matrix Schubert varieties.

math.AG↗

arXiv2012

A doubly-refined enumeration of alternating sign matrices and descending plane partitions

It was shown recently by the authors that, for any n, there is equality between the distributions of certain triplets of statistics on nxn alternating sign matrices (ASMs) and descending plane partitions (DPPs) with each part at most n. The statistics for an ASM A are the number of generalized inversions in A, the number of -1's in A and the number of 0's to the left of the 1 in the first row of A, and the respective statistics for a DPP D are the number of nonspecial parts in D, the number of special parts in D and the number of n's in D. Here, the result is generalized to include a fourth statistic for each type of object, where this is the number of 0's to the right of the 1 in the last row of an ASM, and the number of (n-1)'s plus the number of rows of length n-1 in a DPP. This generalization is proved using the known equality of the three-statistic generating functions, together with relations which express each four-statistic generating function in terms of its three-statistic counterpart. These relations are obtained by applying the Desnanot-Jacobi identity to determinantal expressions for the generating functions, where the determinants arise from standard methods involving the six-vertex model with domain-wall boundary conditions for ASMs, and nonintersecting lattice paths for DPPs.

math.CO↗

arXiv2011

On the weighted enumeration of alternating sign matrices and descending plane partitions

We prove a conjecture of Mills, Robbins and Rumsey [Alternating sign matrices and descending plane partitions, J. Combin. Theory Ser. A 34 (1983), 340-359] that, for any n, k, m and p, the number of nxn alternating sign matrices (ASMs) for which the 1 of the first row is in column k+1 and there are exactly m -1's and m+p inversions is equal to the number of descending plane partitions (DPPs) for which each part is at most n and there are exactly k parts equal to n, m special parts and p nonspecial parts. The proof involves expressing the associated generating functions for ASMs and DPPs with fixed n as determinants of nxn matrices, and using elementary transformations to show that these determinants are equal. The determinants themselves are obtained by standard methods: for ASMs this involves using the Izergin-Korepin formula for the partition function of the six-vertex model with domain-wall boundary conditions, together with a bijection between ASMs and configurations of this model, and for DPPs it involves using the Lindstrom-Gessel-Viennot theorem, together with a bijection between DPPs and certain sets of nonintersecting lattice paths.

math.CO↗

arXiv2008

Punctured plane partitions and the q-deformed Knizhnik--Zamolodchikov and Hirota equations

We consider partial sum rules for the homogeneous limit of the solution of the q-deformed Knizhnik--Zamolodchikov equation with reflecting boundaries in the Dyck path representation. We show that these partial sums arise in a solution of the discrete Hirota equation, and prove that they are the generating functions of $τ^2$-weighted punctured cyclically symmetric transpose complement plane partitions where $τ=-(q+q^{-1})$. In the cases of no or minimal punctures, we prove that these generating functions coincide with $τ^2$-enumerations of vertically symmetric alternating sign matrices and modifications thereof.

math.CO↗

arXiv2006

A scheme related to the Brauer loop model

We introduce the_Brauer loop scheme_ E := {M in M_N(C) : M\cp M = 0}, where \cp is a certain degeneration of the ordinary matrix product. Its components of top dimension, floor(N^2/2), correspond to involutions πin S_N having one or no fixed points. In the case N even, this scheme contains the upper-upper scheme from [Knutson '04] as a union of (N/2)! of its components. One of those is a degeneration of the_commuting variety_ of pairs of commuting matrices. The_Brauer loop model_ is a quantum integrable stochastic process introduced in [de Gier--Nienhuis '04], and some of the entries of its Perron-Frobenius eigenvector were observed (conjecturally) to match the degrees of the components of the upper-upper scheme. We extend this, with proof, to_all_ the entries: they are the degrees of the components of the Brauer loop scheme. Our proof of this follows the program outlined in [Di Francesco--Zinn-Justin '04]. In that paper, the entries of the Perron-Frobenius eigenvector were generalized from numbers to polynomials, which allowed them to be calculated inductively using divided difference operators. We relate these polynomials to the multidegrees of the components of the Brauer loop scheme, defined using an evident torus action on E. In particular, we obtain a formula for the degree of the commuting variety, previously calculated up to 4x4 matrices.

math.AG↗

arXiv2004

On the Asymptotic Number of Plane Curves and Alternating Knots

We present a conjecture for the power-law exponent in the asymptotic number of types of plane curves as the number of self-intersections goes to infinity. In view of the description of prime alternating links as flype equivalence classes of plane curves, a similar conjecture is made for the asymptotic number of prime alternating knots. The rationale leading to these conjectures is given by quantum field theory. Plane curves are viewed as configurations of loops on a random planar lattices, that are in turn interpreted as a model of 2d quantum gravity with matter. The identification of the universality class of this model yields the conjecture. Since approximate counting or sampling planar curves with more than a few dozens of intersections is an open problem, direct confrontation with numerical data yields no convincing indication on the correctness of our conjectures. However, our physical approach yields a more general conjecture about connected systems of curves. We take advantage of this to design an original and feasible numerical test, based on recent perfect samplers for large planar maps. The numerical datas strongly support our identification with a conformal field theory recently described by Read and Saleur.

math-ph↗

arXiv2004

Algebraic Bethe Ansatz for the FPL^2 model

An exact solution of the model of fully packed loops of two colors on a square lattice has recently been proposed by Dei Cont and Nienhuis using the coordinate Bethe Ansatz approach. We point out here a simpler alternative, in which the transfer matrix is directly identified as a product of R-matrices; this allows to apply the (nested) algebraic Bethe Ansatz, which leads to the same Bethe equations. We comment on some of the applications of this result.

math-ph↗