Schur Polynomials Research Articles

Orbit averaging coherent states: holographic three-point functions of AdS giant gravitons

We study correlation functions of two AdS giant gravitons in AdS5 × S5 and a BPS supergravity mode using holography. In the gauge theory these are described by BPS correlators of Schur polynomials of fully-symmetric representations and a single trace operator. We find full agreement between the semiclassical gravity and gauge theory computations at large N, for both diagonal and off-diagonal structure constants. Our analysis in mathcal{N} = 4 SYM provides a simpler derivation to the results in the literature, and it can be readily generalized to operators describing bound states of AdS giant gravitons as well as bubbling geometries.

General rogue waves in the AB system

In this work, general rogue wave solutions in the AB system are constructed by means of the Kadomtsev–Petviashvili hierarchy reduction method and explicit representations of these solutions are explored in terms of determinants whose matrix elements are fundamental Schur polynomials. Moreover, the dynamics of these rogue waves are investigated graphically by different choices of the free parameters. In particular, we observe that the rogue wave in the second component B is actually a four peaky-shaped rogue wave, different form the eye-shaped rogue wave in NLSE.

Unitary matrix integrals, symmetric polynomials, and long-range random walks

Unitary matrix integrals over symmetric polynomials play an important role in a wide variety of applications, including random matrix theory, gauge theory, number theory, and enumerative combinatorics. We derive novel results on such integrals and apply these and other identities to correlation functions of long-range random walks (LRRW) consisting of hard-core bosons. We generalize an identity due to Diaconis and Shahshahani which computes unitary matrix integrals over products of power sum polynomials. This allows us to derive two expressions for unitary matrix integrals over Schur polynomials, which can be directly applied to LRRW correlation functions. We then demonstrate a duality between distinct LRRW models, which we refer to as quasi-local particle-hole duality. We note a relation between the multiplication properties of power sum polynomials of degree n and fermionic particles hopping by n sites. This allows us to compute LRRW correlation functions in terms of auxiliary fermionic rather than hard-core bosonic systems. Inverting this reasoning leads to various results on long-range fermionic models as well. In principle, all results derived in this work can be implemented in experimental setups such as trapped ion systems, where LRRW models appear as an effective description. We further suggest specific correlation functions which may be applied to the benchmarking of such experimental setups.

Tableau formulas for skew Schubert polynomials

AbstractThe skew Schubert polynomials are those that are indexed by skew elements of the Weyl group, in the sense of Tamvakis [J. reine angew. Math. 652 (2011), 207–244]. We obtain tableau formulas for the double versions of these polynomials in all four classical Lie types, where the tableaux used are fillings of the associated skew Young diagram. These are the first such theorems for symplectic and orthogonal Schubert polynomials, even in the single case. We also deduce tableau formulas for double Schur, double theta, and double eta polynomials, in their specializations as double Grassmannian Schubert polynomials. The latter results generalize the tableau formulas for symmetric (and single) Schubert polynomials due to Littlewood (in type A) and the author (in types B, C, and D).

A special two-dimensional lattice by Blaszak and Szum: Solitons, breathers, lump solutions, and their interactions and dynamics

In this paper, we study a special two-dimensional lattice equation proposed by Blaszak and Szum. Upon utilizing the bilinear method, we derive solitons, breathers and rational solutions to the lattice equation both on the constant and periodic background. These solutions are given in terms of determinants whose matrix elements have simple algebraic expressions. In particular, we find three types of breather solutions, including Kuznetsov-Ma breather, Akhmediev breather and general one. By introducing two differential operators applied to the soliton solutions, we obtain rational solutions in terms of Schur polynomials. We demonstrate that rational solutions can exhibit algebraic solitons and lump solitons. We explore the asymptotic analysis to the algebraic solitons and find asymptotic algebraic solitons localized in the straight. By taking higher-order differential operators, we present multiple and higher-order rational solutions. The dynamical behaviors of these obtained solutions are illustrated and analyzed.

Rogue wave patterns associated with Okamoto polynomial hierarchies

AbstractWe show that new types of rogue wave patterns exist in integrable systems, and these rogue patterns are described by root structures of Okamoto polynomial hierarchies. These rogue patterns arise when the τ functions of rogue wave solutions are determinants of Schur polynomials with index jumps of three, and an internal free parameter in these rogue waves gets large. We demonstrate these new rogue patterns in the Manakov system and the three‐wave resonant interaction system. For each system, we derive asymptotic predictions of its rogue patterns under a large internal parameter through Okamoto polynomial hierarchies. Unlike the previously reported rogue patterns associated with the Yablonskii–Vorob'ev hierarchy, a new feature in the present rogue patterns is that the mapping from the root structure of Okamoto‐hierarchy polynomials to the shape of the rogue pattern is linear only to the leading order, but becomes nonlinear to the next order. As a consequence, the current rogue patterns are often deformed, sometimes strongly deformed, from Okamoto‐hierarchy root structures, unless the underlying internal parameter is very large. Our analytical predictions of rogue patterns are compared to true solutions, and excellent agreement is observed, even when rogue patterns are strongly deformed from Okamoto‐hierarchy root structures.

General high-order lump solutions and their dynamics in the Levi equations

General high-order lump solutions are derived for the Levi equations based on the Hirota bilinear method and Kadomtsev-Petviashvili (KP) hierarchy reduction technique. These lump solutions are given in terms of Gram determinants whose matrix elements are connected to Schur polynomials. Thus, our solutions have explicit algebraic expressions. Their dynamic behaviors are analyzed by using density maps. It is shown that when the absolute value of one group of these internal parameters in the lump solutions is very large, lump solutions exhibit obvious geometric structures. Interestingly, we have shown that their initial and middle state solutions possess various exciting geometric patterns, including hexagon, decagon, tetradecagon, etc and other quasi-structures in addition to the standard triangle, pentagon type patterns. Because the internal parameters are not complex conjugates of each other, the dynamic behaviors of solutions are richer. These results make several contributions to the current literature and have a number of important implications for further analysis of fluid dynamics in non-homogeneous media.

Exotic Lagrangian tori in Grassmannians

We describe an iterative construction of Lagrangian tori in the complex Grassmannian \operatorname{Gr}(k,n) , based on the cluster algebra structure of the coordinate ring of a mirror Landau–Ginzburg model proposed by Marsh and Rietsch (2020). Each torus comes with a Laurent polynomial, and local systems controlled by the k -variables Schur polynomials at the n -th roots of unity. We use this data to give examples of monotone Lagrangian tori that are neither displaceable nor Hamiltonian isotopic to each other, and that support nonzero objects in different summands of the spectral decomposition of the Fukaya category over \mathbb{C} .

Bilinear character correlators in superintegrable theory

We continue investigating the superintegrability property of matrix models, i.e. factorization of the matrix model averages of characters. This paper focuses on the Gaussian Hermitian example, where the role of characters is played by the Schur functions. We find a new intriguing corollary of superintegrability: factorization of an infinite set of correlators bilinear in the Schur functions. More exactly, these are correlators of products of the Schur functions and polynomials K_Delta that form a complete basis in the space of invariant matrix polynomials. Factorization of these correlators with a small subset of these K_Delta follow from the fact that the Schur functions are eigenfunctions of the generalized cut-an-join operators, but the full set of K_Delta is generated by another infinite commutative set of operators, which we manifestly describe.

Flagged Schur Polynomial Duality via a Lattice Path Bijection

This paper proves an identity between flagged Schur polynomials, giving a duality between row flags and column flags. This identity generalises both the binomial determinant duality theorem due to Gessel and Viennot and the symmetric function duality theorem due to Aitken. As corollaries we obtain the lifts of the binomial determinant duality theorem to \(q\)-binomial coefficients and to symmetric polynomials. Our method is a path counting argument on a novel lattice generalising that used by Gessel and Viennot.

General high-order rogue waves in the Hirota equation

In this work, we focus on the construction of Nth-order rogue waves for the Hirota equation by utilizing the bilinear method and Kadomtsev–Petviashvili (KP) hierarchy reduction technique. The formula can be represented in terms of determinants whose matrix elements are related to Schur polynomials. In addition, some interesting dynamic patterns of rogue waves are exhibited analytically and graphically.

Skew RSK dynamics: Greene invariants, affine crystals and applications toq-Whittaker polynomials

AbstractIterating the skew RSK correspondence discovered by Sagan and Stanley in the late 1980s, we define deterministic dynamics on the space of pairs of skew Young tableaux$(P,Q)$. We find that these skew RSK dynamics display conservation laws which, in the picture of Viennot’s shadow line construction, identify generalizations of Greene invariants. The introduction of a novel realization of$0$-th Kashiwara operators reveals that the skew RSK dynamics possess symmetries induced by an affine bicrystal structure, which, combined with connectedness properties of Demazure crystals, leads to the linearization of the time evolution. Studying asymptotic evolution of the dynamics started from a pair of skew tableaux$(P,Q)$, we discover a new bijection$\Upsilon : (P,Q) \mapsto (V,W; \kappa , \nu )$. Here,$(V,W)$is a pair of vertically strict tableaux, that is, column strict fillings of Young diagrams with no condition on rows, with the shape prescribed by the Greene invariant,$\kappa $is an array of nonnegative weights and$\nu $is a partition. An application of this construction is the first bijective proof of Cauchy and Littlewood identities involvingq-Whittaker polynomials. New identities relating sums ofq-Whittaker and Schur polynomials are also presented.

Partial-rogue waves that come from nowhere but leave with a trace in the Sasa-Satsuma equation

Partial-rogue waves, i.e., waves that “come from nowhere but leave with a trace”, are analytically predicted and numerically confirmed in the Sasa-Satsuma equation. We show that, among a class of rational solutions in this equation that can be expressed through determinants of 3-reduced Schur polynomials, partial-rogue waves would arise if these rational solutions are of certain orders, where the associated generalized Okamoto polynomials have real but not imaginary roots, or imaginary but not real roots. We further show that, at large negative time, these partial-rogue waves approach the constant-amplitude background, but at large positive time, they split into several fundamental rational solitons, whose numbers are determined by the number of real or imaginary roots in the underlying generalized Okamoto polynomial. Our asymptotic predictions are compared to true solutions, and excellent agreement is observed.

Zero dynamics for a class of robustly stable polynomials

In this contribution we introduce two different methods to construct robustly stable Schur polynomials that depend on an uncertain parameter, by using basic properties of orthogonal polynomials on the unit circle. We also develop a procedure to describe the behavior of the zeros of a class of uncertain polynomials with respect to the uncertain parameters. Some illustrative examples are presented.

Kohnert's rule for flagged Schur modules

Flagged Schur modules generalize the irreducible representations of the general linear group under the action of the Borel subalgebra. Their characters include many important generalizations of Schur polynomials, such as Demazure characters, flagged skew Schur polynomials, and Schubert polynomials. In this paper, we prove the characters of flagged Schur modules can be computed using a simple combinatorial algorithm due to Kohnert if and only if the indexing diagram is northwest. This gives a new proof that characters of flagged Schur modules are nonnegative sums of Demazure characters and gives a representation theoretic interpretation for Kohnert polynomials.

Q-Pearson pair and moments in q-deformed ensembles

The generalisation of continuous orthogonal polynomial ensembles from random matrix theory to the q-lattice setting is considered. We take up the task of initiating a systematic study of the corresponding moments of the density from two complementary viewpoints. The first requires knowledge of the ensemble average with respect to a general Schur polynomial, from which the spectral moments follow as a corollary. In the case of little q-Laguerre weight, a particular \(_3 \phi _2\) basic hypergeometric polynomial is used to express density moments. The second approach is to study the q-Laplace transform of the un-normalised measure. Using integrability properties associated with the q-Pearson equation for the q-classical weights, a fourth-order q-difference equation is obtained, generalising a result of Ledoux in the continuous classical cases.

Skew Schur Polynomials and Cyclic Sieving Phenomenon

Let $k$ and $m$ be positive integers and $\lambda/\mu$ a skew partition. We compute the principal specialization of the skew Schur polynomials $s_{\lambda /\mu}(x_1, \ldots, x_{k})$ modulo $q^m-1$ under suitable conditions. We interpret the results thus obtained from the viewpoint of the cyclic sieving phenomenon on semistandard Young skew tableaux of shape $\lambda/\mu$. As an application, we deal with evaluations of the principal specialization of the skew Schur polynomials at roots of unity.

LLT Cumulants and Graph Coloring

The purpose of this note is to introduce a new family of quasi-symmetric functions called LLT cumulants and discuss its properties. We define LLT cumulants using the algebraic framework for conditional cumulants and we prove that the Macdonald cumulant has an explicit positive expansion in terms of LLT cumulants of ribbon shapes, generalizing the classical decomposition of Macdonald polynomials. We also find a natural combinatorial interpretation of the LLT cumulant of a given directed graph as a weighted generating function of colorings of its subgraphs.We use this graph theoretical framework to prove various positivity results. This includes monomial positivity, positivity in fundamental quasisymmetric functions and related positivity of the coefficients of Schur polynomials indexed by hook shapes. We also prove $e$-positivity for vertical-shape LLT cumulants, after the shift of variable $q \to q+1$, which refines a recent result of Alexandersson and Sulzgruber. All these results give evidence towards Schur-positivity of LLT cumulants, which we conjecture here. We prove that this conjecture implies Schur-positivity of Macdonald cumulants, and we give more evidence by proving the conjecture for LLT cumulants of melting lollipops that refines a recent result of Huh, Nam and Yoo.

Hirschman–Widder densities

Hirschman and Widder introduced a class of P\'olya frequency functions given by linear combinations of one-sided exponential functions. The members of this class are probability densities, and the class is closed under convolution but not under pointwise multiplication. We show that, generically, a polynomial function of such a density is a P\'olya frequency function only if the polynomial is a homothety, and also identify a subclass for which each positive-integer power is a P\'olya frequency function. We further demonstrate connections between the Maclaurin coefficients, the moments of these densities, and the recovery of the density from finitely many moments, via Schur polynomials.

Hodge–Riemann Relations for Schur Classes in the Linear and Kähler Cases

Abstract We prove a version of the Hodge–Riemann bilinear relations for Schur polynomials of Kähler forms and for Schur polynomials of positive forms on a complex vector space.