Related papers: Infinite wedge and random partitions

We prove Fredholm determinants build out from generalizations of Schur measures, or equivalently, arbitrary multiplicative statistics of the original Schur measures are tau-functions of the 2D Toda lattice hierarchy. Our result apply to…

We establish necessary and sufficient conditions for convergence (in the sense of finite dimensional distributions) of multiplicative measures on the set of partitions. We show that this convergence is equivalent to asymptotic independence…

We study the $t$-Schur measure on partitions, defined by $ \mathbb{P}(\lambda)=Z^{-1}S_\lambda(x;t)s_\lambda(y) $, where $S_\lambda(x;t)$ denotes the $t$-Schur symmetric functions and $s_\lambda(y)$ the ordinary Schur functions, and $Z$ is…

The shifted Schur measure introduced by Tracy and Widom is a measure on the set of all strict partitions, which is defined by Schur $Q$-functions. The main aim of this paper is to calculate the correlation function of this measure, which is…

We prove a conjecture of H.Widom stated in [W] (math/0108008) about the reality of eigenvalues of certain infinite matrices arising in asymptotic analysis of large Toeplitz determinants. As a byproduct we obtain a new proof of A.Okounkov's…

We prove a conjecture of Cuttler et al.~[2011] [A. Cuttler, C. Greene, and M. Skandera; \emph{Inequalities for symmetric means}. European J. Combinatorics, 32(2011), 745--761] on the monotonicity of \emph{normalized Schur functions} under…

Considering Schur positivity of differences of plethysms of homogeneous symmetric functions, we introduce a new relation on integer partitions. This relation is conjectured to be a partial order, with its restriction to one part partitions…

We consider Schur function expansion for the partition function of the model of normal matrices. We show that this expansion coincides with Takasaki expansion \cite{Tinit} for tau functions of Toda lattice hierarchy. We show that the…

We define a measure on tuples of partitions, called the symplectic Schur process, that should be regarded as the right analogue of the Schur process of Okounkov-Reshetikhin for the Cartan type C. The weights of our measure include factors…

The Macdonald symmetric functions are used to define measures on the set of all partitions of all integers. Probabilistic algorithms are given for growing partitions according to these measures. The case of Hall-Littlewood polynomials is…

We consider an operator of Bernstein for symmetric functions, and give an explicit formula for its action on an arbitrary Schur function. This formula is given in a remarkably simple form when written in terms of some notation based on the…

We establish the universal edge scaling limit of random partitions with the infinite-parameter distribution called the Schur measure. We explore the asymptotic behavior of the wave function, which is a building block of the corresponding…

We introduce the edge Schur functions $E^{\lambda}$ that are defined as a generating series over edge labeled tableaux. We formulate $E^{\lambda}$ as the partition function for a solvable lattice model, which we use to show they are…

We introduce the Plancherel measure on the set of partition collections, which parameterize irreducible representations of order n general linear group over a finite field. We prove that as n goes to infinity, the random partitions from the…

We introduce the new combinatorial approach of plethystic type of tableaux, as a method to understand coefficients of Schur functions appearing in plethysms $s_\nu[h_\lambda]$ and $s_{\nu}[e_{\lambda}]$, for any partitions $\lambda$ and…

We study a linear map on symmetric functions that ``divides'' a partition by a positive integer $k$, sending a Schur function indexed by a partition of $kn$ to a symmetric function indexed by partitions of $n$. We determine its Schur…

We introduce multicritical Schur measures, which are probability laws on integer partitions which give rise to non-generic fluctuations at their edge. They are in the same universality classes as one-dimensional momentum-space models of…

Elementary proofs are given for sums of Schur functions over partitions into at most n parts each less than or equal to m for which i) all parts are even, ii) all parts of the conjugate partition are even. Also, an elementary proof of a…

We review and present new studies on the relation between the partition functions of integrable lattice models and symmetric polynomials, and its combinatorial representation theory based on the correspondence, including our work. In…

The plethysm product of Schur functions corresponds to composing polynomial representations of infinite general linear groups. Finding the plethysm coefficients $\langle s_\nu \circ s_\mu, s_\lambda\rangle$ that express an arbitrary…