Rectangular Young Diagrams Research Articles

A Multiple Hook Removing Game Whose Starting Position is a Rectangular Young Diagram with Unimodal Numbering

A Multiple Hook Removing Game Whose Starting Position is a Rectangular Young Diagram with Unimodal Numbering

A Quantized Analogue of the Markov–Krein Correspondence

Abstract We study a family of measures originating from the signatures of the irreducible components of representations of the unitary group, as the size of the group goes to infinity. Given a random signature $\lambda $ of length $N$ with counting measure $\textbf {m}$, we obtain a random signature $\mu $ of length $N-1$ through projection onto a unitary group of lower dimension. The signature $\mu $ interlaces with the signature $\lambda $, and we record the data of $\mu ,\lambda $ in a random rectangular Young diagram $w$. We show that under a certain set of conditions on $\lambda $, both $\textbf {m}$ and $w$ converge as $N\to \infty $. We provide an explicit moment-generating function relationship between the limiting objects. We further show that the moment-generating function relationship induces a bijection between bounded measures and certain continual Young diagrams, which can be viewed as a quantized analogue of the Markov–Krein correspondence.

Cyclotomic expansions of HOMFLY-PT colored by rectangular Young diagrams

We conjecture a closed-form expression of HOMFLY-PT invariants of double twist knots colored by rectangular Young diagrams where the twist is encoded in interpolation Macdonald polynomials. We also put forth a conjecture of cyclotomic expansions of HOMFLY-PT polynomials colored by rectangular Young diagrams for any knot.

Integrability approach to Fehér-Némethi-Rimányi-Guo-Sun type identities for factorial Grothendieck polynomials

Recently, Guo and Sun derived an identity for factorial Grothendieck polynomials which is a generalization of the one for Schur polynomials by Fehér, Némethi and Rimányi. We analyze the identity from the point of view of quantum integrability, based on the correspondence between the wavefunctions of a five-vertex model and the Grothendieck polynomials. We give another proof using the quantum inverse scattering method. We also apply the same idea and technique to derive an identity for factorial Grothendieck polynomials for rectangular Young diagrams. Combining with the Guo-Sun identity, we get a duality formula. We also discuss a q-deformation of the Guo-Sun identity.

Fluctuations of Rectangular Young Diagrams of Interlacing Wigner Eigenvalues

We prove a new CLT for the difference of linear eigenvalue statistics of a Wigner random matrix $H$ and its minor $\hat H$ and find that the fluctuation is much smaller than the fluctuations of the individual linear statistics, as a consequence of the strong correlation between the eigenvalues of $H$ and $\hat H$. In particular our theorem identifies the fluctuation of Kerov's rectangular Young diagrams, defined by the interlacing eigenvalues of $H$ and $\hat H$, around their asymptotic shape, the Vershik-Kerov-Logan-Shepp curve. This result demonstrates yet another aspect of the close connection between random matrix theory and Young diagrams equipped with the Plancherel measure known from representation theory. For the latter a CLT has been obtained in [18] which is structurally similar to our result but the variance is different, indicating that the analogy between the two models has its limitations. Moreover, our theorem shows that Borodin's result [7] on the convergence of the spectral distribution of Wigner matrices to a Gaussian free field also holds in derivative sense.

On the gauge symmetries of Maxwell-like higher-spin Lagrangians

In their simplest form, metric-like Lagrangians for higher-spin massless fields are usually assumed to display constrained gauge symmetries, unless auxiliary fields are introduced or locality is foregone. Specifically, in its standard incarnation, gauge invariance of Maxwell-like Lagrangians relies on parameters with vanishing divergence. We find an alternative form of the corresponding local symmetry involving unconstrained gauge parameters of mixed-symmetry type, described by rectangular two-row Young diagrams and entering high-derivative gauge transformations. The resulting gauge algebra appears to be reducible and we display the full pattern of gauge-for-gauge parameters, testing its correctness via the corresponding counting of degrees of freedom. The algebraic techniques applied in this work also allow us to elucidate some general properties of linear gauge systems. In particular, we establish the general fact that any linear local field theory always admits unconstrained, local, and finitely reducible parametrization of the gauge symmetry. Incidentally, this shows that massless higher spins admit a local unconstrained formulation with no need for auxiliary fields.

Baxter operators and Hamiltonians for “nearly all” integrable closed [formula omitted] spin chains

We continue our systematic construction of Baxter Q-operators for spin chains, which is based on certain degenerate solutions of the Yang–Baxter equation. Here we generalize our approach from the fundamental representation of gl(n) to generic finite-dimensional representations in quantum space. The results equally apply to non-compact representations of highest or lowest weight type. We furthermore fill an apparent gap in the literature, and provide the nearest-neighbor Hamiltonians of the spin chains in question for all cases where the gl(n) representations are described by rectangular Young diagrams, as well as for their infinite-dimensional generalizations. They take the form of digamma functions depending on operator-valued shifted weights.

Bases in coset conformal field theory from AGT correspondence and Macdonald polynomials at the roots of unity

We continue our study of the AGT correspondence between instanton counting on C^2/Z_p and Conformal field theories with the symmetry algebra A(r,p). In the cases r=1, p=2 and r=2, p=2 this algebra specialized to: A(1,2)=H+sl(2)_1 and A(2,2)=H+sl(2)_2+NSR. As the main tool we use a new construction of the algebra A(r,2) as the limit of the toroidal gl(1) algebra for q,t tend to -1. We claim that the basis of the representation of the algebra A(r,2) (or equivalently, of the space of the local fields of the corresponding CFT) can be expressed through Macdonald polynomials with the parameters q,t go to -1. The vertex operator which naturally arises in this construction has factorized matrix elements in this basis. We also argue that the singular vectors of the $\mathcal{N}=1$ Super Virasoro algebra can be realized in terms of Macdonald polynomials for a rectangular Young diagram and parameters q,t tend to -1.

Solution to the Ward identities for superamplitudes

Supersymmetry and R-symmetry Ward identities relate on-shell amplitudes in a supersymmetric field theory. We solve these Ward identities for (Next-to)^K MHV amplitudes of the maximally supersymmetric N=4 and N=8 theories. The resulting superamplitude is written in a new, manifestly supersymmetric and R-invariant form: it is expressed as a sum of very simple SUSY and SU(N)_R-invariant Grassmann polynomials, each multiplied by a "basis amplitude". For (Next-to)^K MHV n-point superamplitudes the number of basis amplitudes is equal to the dimension of the irreducible representation of SU(n-4) corresponding to the rectangular Young diagram with N columns and K rows. The linearly independent amplitudes in this algebraic basis may still be functionally related by permutation of momenta. We show how cyclic and reflection symmetries can be used to obtain a smaller functional basis of color-ordered single-trace amplitudes in N=4 gauge theory. We also analyze the more significant reduction that occurs in N=8 supergravity because gravity amplitudes are not ordered. All results are valid at both tree and loop level.

Primary ideals associated to the linear strands of Lascoux's resolution and syzygies of the corresponding irreducible representations of the Lie superalgebra [formula omitted

We investigate equivariant Koszul duality between primary ideals I a × b of S = S ( M ( m × n ) ∗ ) associated to rectangular Young diagrams a × b and the corresponding atypical irreducible mixed supertensor representations X a × b of gl ( m | n ) in characteristic zero. We show I a × b to be H 0 ( S ⊗ X a × b ) of the Koszul dual S ⊗ X a × b and we compute all the higher cohomology of S ⊗ X a × b to be direct sums of I ( a + r ) × ( b + r ) with multiplicities being given by coefficients of Gauss polynomials. Utilizing this we are able to describe the equivariant syzygies of X a × b over S ! = Λ ( M ( m × n ) ) and determine the homological dimension of I a × b over S.

Singular structure of Toda lattices and cohomology of certain compact Lie groups

We study the singularities (blow-ups) of the Toda lattice associated with a real split semisimple Lie algebra g . It turns out that the total number of blow-up points along trajectories of the Toda lattice is given by the number of points of a Chevalley group K ( F q ) related to the maximal compact subgroup K of the group G ˇ with g ˇ = Lie ( G ˇ ) over the finite field F q . Here g ˇ is the Langlands dual of g . The blow-ups of the Toda lattice are given by the zero set of the τ -functions. For example, the blow-ups of the Toda lattice of A -type are determined by the zeros of the Schur polynomials associated with rectangular Young diagrams. Those Schur polynomials are the τ -functions for the nilpotent Toda lattices. Then we conjecture that the number of blow-ups is also given by the number of real roots of those Schur polynomials for a specific variable. We also discuss the case of periodic Toda lattice in connection with the real cohomology of the flag manifold associated to an affine Kac–Moody algebra.

Explicit formula for singular vectors of the Virasoro algebra with central charge less than 1

We calculate explicitly the singular vectors of the Virasoro algebra with the central charge $c\leq 1$. As a result, we have an infinite sequence of the singular vectors for each Fock space with given central charge and highest weight, and all its elements can be written in terms of the Jack symmetric functions with rectangular Young diagram.

Rational solutions of the Garnier system in terms of Schur polynomials

We present a family of rational solutions of the Garnier system HN and show that the corresponding τ-functions form special polynomials Rm,n(m,n ∈ ℤ≥0) under a certain normalization. We give an explicit formula for the special polynomials Rm,n in terms of the Schur polynomial attached to a rectangular Young diagram.

Rational solutions to the Pfaff lattice and Jack polynomials

The finite Pfaff lattice is given by a commuting Lax pair involving a finite matrix L (zero above the first subdiagonal) and a projection onto sp(N). The lattice admits solutions such that the entries of the matrix L are rational in the time parameters t_1,t_2,\dotsc, after conjugation by a diagonal matrix. The sequence of polynomial \tau-functions, solving the problem, belongs to an intriguing chain of subspaces of Schur polynomials, associated to Young diagrams, dual with respect to a finite chain of rectangles. Also, this sequence of \tau-functions is given inductively by the action of a fixed vertex operator.As an example, one such sequence is given by Jack polynomials for rectangular Young diagrams, while another chain starts with any two-column Jack polynomial.

Linearly Independent Products of Rectangularly Complementary Schur Functions

Fix a rectangular Young diagram $R$, and consider all the products of Schur functions $s_\lambda s_\lambda^{c}$, where $\lambda$ and $\lambda^{c}$ run over all (unordered) pairs of partitions which are complementary with respect to $R$. Theorem: The self-complementary products, $s_\lambda^2$ where $\lambda=\lambda^{c}$, are linearly independent of all other $s_\lambda s_\lambda^{c}$. Conjecture: The products $s_\lambda s_\lambda^{c}$ are all linearly independent.

Polynomial τ-Functions of the NLS-Toda Hierarchy and the Virasoro Singular Vectors

A family of polynomial τ-functions for the NLS-Toda hierarchy is constructed. The hierarchy is associated with the homogeneous vertex operator representation of the affine algebra \(\mathfrak{g}\) of type A1(1). These τ-functions are given explicitly in terms of Schur functions that correspond to rectangular Young diagrams. It is shown that an arbitrary polynomial τ-function which is an eigenvector of d, the degree operator of \(\mathfrak{g}\), is contained in the family. By the construction, any τ-function in the family becomes a Virasoro singular vector. This consideration gives rise to a simple proof of known results on the Fock representation of the Virasoro algebra with c = 1.

Fusion Rules for Quantum Transfer Matrices as a Dynamical System on Grassmann Manifolds

We show that the set of transfer matrices of an arbitrary fusion type for an integrable quantum model obeys these bilinear functional relations, which are identified with an integrable dynamical system on a Grassmann manifold (higher Hirota equation). The bilinear relations were previously known for a particular class of transfer matrices corresponding to rectangular Young diagrams. We extend this result for general Young diagrams. A general solution of the bilinear equations is presented.

Singular vectors of the Virasoro algebra in terms of Jack symmetric polynomials

We present an explicit formula of the Virasoro singular vectors in terms of Jack symmetric polynomials. The parametert in the Virasoro central chargec=13-6(t+1/t) is just identified with the deformation parameter α of Jack symmetric polynomialsJγ(α). As a by-product, we obtain an integral representation of Jack symmetric polynomials indexed by the rectangular Young diagrams.