Strict Partitions Research Articles

Young walls and graded dimension formulas for finite quiver Hecke algebras of type $$A^{(2)}_{2\ell }$$ A 2 ℓ ( 2 ) and $$D^{(2)}_{\ell +1}$$ D ℓ + 1 ( 2 )

We study graded dimension formulas for finite quiver Hecke algebras $$R^{\Lambda _0}(\beta )$$ R ? 0 ( β ) of type $$A^{(2)}_{2\ell }$$ A 2 l ( 2 ) and $$D^{(2)}_{\ell +1}$$ D l + 1 ( 2 ) using combinatorics of Young walls. We introduce the notion of standard tableaux for proper Young walls and show that the standard tableaux form a graded poset with lattice structure. We next investigate Laurent polynomials associated with proper Young walls and their standard tableaux arising from the Fock space representations consisting of proper Young walls. Then, we prove the graded dimension formulas described in terms of the Laurent polynomials. When evaluating at $$q=1$$ q = 1 , the graded dimension formulas recover the dimension formulas for $$R^{\Lambda _0}(\beta )$$ R ? 0 ( β ) described in terms of standard tableaux of strict partitions.

The Gaussian free field and strict plane partitions

We study height fluctuations around the limit shape of a measure on strict plane partitions. It was shown in our earlier work that this measure is a Pfaffian process. We show that the height fluctuations converge to a pullback of the Green's function for the Laplace operator with Dirichlet boundary conditions on the first quadrant. We use a Pfaffian formula for higher moments to show that the height fluctuations are governed by the Gaussian free field. The results follow from the correlation kernel asymptotics which is obtained by the steepest descent method.

Ising limit of a Heisenberg XXZ magnet and some temperature correlation functions

We consider the Heisenberg spin-1/2 XXZ magnet in the case where the anisotropy parameter tends to infinity (the so-called Ising limit). We find the temperature correlation function of a ferromagnetic string above the ground state. Our approach to calculating correlation functions is based on expressing the wave function in the considered limit in terms of Schur symmetric functions. We show that the asymptotic amplitude of the above correlation function at low temperatures is proportional to the squared number of strict plane partitions in a box.

Multiple sums and integrals as neutral BKP tau functions

We consider multiple sums and multi-integrals as tau functions of the BKP hierarchy using neutral fermions as the simplest tool for deriving these. The sums are over projective Schur functions $Q_\alpha$ for strict partitions $\alpha$. We consider two types of such sums: weighted sums of $Q_\alpha$ over strict partitions $\alpha$ and sums over products $Q_\alpha Q_\gamma$. In this way we obtain discrete analogues of the beta-ensembles ($\beta=1,2,4$). Continuous versions are represented as multiple integrals. Such sums and integrals are of interest in a number of problems in mathematics and physics.

The correlation functions of the $XXZ$ Heisenberg chain in the case of zero or infinite anisotropy, and random walks of vicious walkers

The XXZ Heisenberg chain is considered for two specific limits of the anisotropy parameter: $\Dl\to 0$ and $\Dl\to -\infty$. The corresponding wave functions are expressed by means of the symmetric Schur functions. Certain expectation values and thermal correlation functions of the ferromagnetic string operators are calculated over the base of N-particle Bethe states. The thermal correlator of the ferromagnetic string is expressed through the generating function of the lattice paths of random walks of vicious walkers. A relationship between the expectation values obtained and the generating functions of strict plane partitions in a box is discussed. Asymptotic estimate of the thermal correlator of the ferromagnetic string is obtained in the limit of zero temperature. It is shown that its amplitude is related to the number of plane partitions.

Pfaffian Stochastic Dynamics of Strict Partitions

We study a family of continuous time Markov jump processes on strict partitions (partitions with distinct parts) preserving the distributions introduced by Borodin (1997) in connection with projective representations of the infinite symmetric group. The one-dimensional distributions of the processes (i.e., the Borodin's measures) have determinantal structure. We express the dynamical correlation functions of the processes in terms of certain Pfaffians and give explicit formulas for both the static and dynamical correlation kernels using the Gauss hypergeometric function. Moreover, we are able to express our correlation kernels (both static and dynamical) through those of the z-measures on partitions obtained previously by Borodin and Olshanski in a series of papers. The results about the fixed time case were announced in the note [El. Comm. Probab., 15 (2010), 162-175]. A part of the present paper contains proofs of those results.

The Shifted Schur Process and Asymptotics of Large Random Strict Plane Partitions

The Shifted Schur Process and Asymptotics of Large Random Strict Plane Partitions

Random walks on strict partitions

We construct a diffusion process in the infinite-dimensional simplex consisting of all nonincreasing infinite sequences of nonnegative numbers with sum less than or equal to one. The process is constructed as a limit of a certain sequence of Markov chains. The state space of the nth chain is the set of all strict partitions of n (that is, partitions with distinct parts). As n →∞, these random walks converge to a continuous-time strong Markov process in the infinite-dimensional simplex. The process has continuous sample paths. The main result about the limit process is an expression for its pre-generator as a formal second-order differential operator in a polynomial algebra. Bibliography: 29 titles.

Random Strict Partitions and Determinantal Point Processes

We present new examples of determinantal point processes with infinitely many particles. The particles live on the half-lattice $\{1,2,\dots\}$ or on the open half-line $(0,+\infty)$. The main result is the computation of the correlation kernels. They have integrable form and are expressed through the Euler gamma function (the lattice case) and the classical Whittaker functions (the continuous case). Our processes are obtained via a limit transition from a model of random strict partitions introduced by Borodin (1997) in connection with the problem of harmonic analysis for projective characters of the infinite symmetric group.

Generalized Q-Functions and UC Hierarchy of B-Type

A generalization of the Schur Q-function is introduced. This generalization, called generalized Q-function, is indexed by any pair of strict partitions, and can be expressed by the Pfaffian. A connection to the theory of integrable systems is clarified. Firstly, the bilinear identities satisfied by the generalized Q-functions are given and proved to be equivalent to a system of partial differential equations of infinite order. This system is called the UC hierarchy of B-type (BUC hierarchy). Secondly, the algebraic structure of the BUC hierarchy is investigated from the representation theoretic viewpoint. Some new kind of the boson-fermion correspondence is established, and a representation of an infinite dimensional Lie algebra, denoted by $\mathfrak{go}_{2\infty}$, is obtained. The bilinear identities are translated to the language of neutral fermions, which turn out to characterize a $\boldsymbol{G}$-orbit of the vacuum vector, where $\boldsymbol{G}$ is the group corresponding to $\mathfrak{go}_{2\infty}$.

A generalization of MacMahon’s formula

We generalize the generating formula for plane partitions known as MacMahon’s formula as well as its analog for strict plane partitions. We give a 2-parameter generalization of these formulas related to Macdonald’s symmetric functions. The formula is especially simple in the Hall-Littlewood case. We also give a bijective proof of the analog of MacMahon’s formula for strict plane partitions.

On free fermions and plane partitions

We use free fermion methods to re-derive a result of Okounkov and Reshetikhin relating charged fermions to random plane partitions, and to extend it to relate neutral fermions to strict plane partitions.

The Andrews–Stanley partition function and Al-Salam–Chihara polynomials

For any partition λ let ω ( λ ) denote the four parameter weight ω ( λ ) = a ∑ i ≥ 1 ⌈ λ 2 i − 1 / 2 ⌉ b ∑ i ≥ 1 ⌊ λ 2 i − 1 / 2 ⌋ c ∑ i ≥ 1 ⌈ λ 2 i / 2 ⌉ d ∑ i ≥ 1 ⌊ λ 2 i / 2 ⌋ , and let ℓ ( λ ) be the length of λ . We show that the generating function ∑ ω ( λ ) z ℓ ( λ ) , where the sum runs over all ordinary (resp. strict) partitions with parts each ≤ N , can be expressed by the Al-Salam–Chihara polynomials. As a corollary we derive Andrews’ result by specializing some parameters and Boulet’s results by letting N → + ∞ . In the last section we prove a Pfaffian formula for the weighted sum ∑ ω ( λ ) z ℓ ( λ ) P λ ( x ) where P λ ( x ) is Schur’s P -function and the sum runs over all strict partitions.

Maximal Projective Degrees for Strict Partitions

Let $\lambda$ be a partition, and denote by $f^\lambda$ the number of standard tableaux of shape $\lambda$. The asymptotic shape of $\lambda$ maximizing $f^\lambda$ was determined in the classical work of Logan and Shepp and, independently, of Vershik and Kerov. The analogue problem, where the number of parts of $\lambda$ is bounded by a fixed number, was done by Askey and Regev – though some steps in this work were assumed without a proof. Here these steps are proved rigorously. When $\lambda$ is strict, we denote by $g^\lambda$ the number of standard tableau of shifted shape $\lambda$. We determine the partition $\lambda$ maximizing $g^\lambda$ in the strip. In addition we give a conjecture related to the maximizing of $g^\lambda$ without any length restrictions.

Strict partitions and discrete dynamical systems

We prove that the set of partitions with distinct parts of a given positive integer under dominance ordering can be considered as a configuration space of a discrete dynamical model with two transition rules and with the initial configuration being the singleton partition. This allows us to characterize its lattice structure, fixed point, and longest chains as well as their length, using Chip Firing Game theory. Finally, we study the recursive structure of infinite extension of the lattice of strict partitions.

Mullineux involution and twisted affine Lie algebras

We use Naito and Sagaki's work [S. Naito, D. Sagaki, Lakshmibai–Seshadri paths fixed by a diagram automorphism, J. Algebra 245 (2001) 395–412; S. Naito, D. Sagaki, Standard paths and standard monomials fixed by a diagram automorphism, J. Algebra 251 (2002) 461–474] on Lakshmibai–Seshadri paths fixed by diagram automorphisms to study the partitions fixed by Mullineux involution. We characterize the set of Mullineux-fixed partitions in terms of crystal graphs of basic representations of twisted affine Lie algebras of type A 2 ℓ ( 2 ) and of type D ℓ + 1 ( 2 ) . We set up bijections between the set of symmetric partitions and the set of partitions into distinct parts. We propose a notion of double restricted strict partitions. Bijections between the set of restricted strict partitions (respectively, the set of double restricted strict partitions) and the set of Mullineux-fixed partitions in the odd case (respectively, in the even case) are obtained.

Correlation functions of the shifted Schur measure

The shifted Schur measure introduced in [TW2] 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 given by a pfaffian. As an application, we prove that a limit distribution of parts of partitions with respect to a shifted version of the Plancherel measure for symmetric groups is identical with the corresponding distribution of the original Plancherel measure. In particular, we obtain a limit distribution of the length of the longest ascent pair for a random permutation. Further we give expressions of the mean value and the variance of the size of partitions with respect to the measure defined by Hall-Littlewood functions.

The q-Variations of Sylvester’s Bijection Between Odd and Strict Partitions

The q-identities corresponding to Sylvester’s bijection between odd and strict partitions are investigated. In particular, we show that Sylvester’s bijection implies the Rogers-Fine identity and give a simple proof of a partition theorem of Fine, which does not follow directly from Sylvester’s bijection. Finally, the so-called (m, c)-analogues of Sylvester’s bijection are also discussed.

Correlation Functions of Strict Partitions and Twisted Fock Spaces

Using twisted Fock spaces, we formulate and study two twisted versions of the n-point correlation functions of Bloch–Okounkov, and then identify them with q-expectation values of certain functions on the set of (odd) strict partitions. We find closed formulas for the 1-point functions in both cases in terms of Jacobi θ-functions. These correlation functions afford several distinct interpretations.

Sylvester's bijection between strict and odd partitions

We show that Sylvester's bijection between strict partitions and odd ones can be obtained by an appropriate coding of partitions.