Related papers: Polynomial Toda maps are transfer matrices
Let L be a bounded distributive lattice. We give several characterizations of those L^n --> L mappings that are polynomial functions, i.e., functions which can be obtained from projections and constant functions using binary joins and…
Intertwiners between representations of Lie groups can be used to obtain relations for matrix elements. We apply this technique to obtain different identities for the wave functions of the open Toda chain, in particular raising operators…
We extend the previous construction of loop transfer matrix to the case of nonzero self-intersection coupling constant $\kappa$. The loop generalization of Fourier transformation allows to diagonalize transfer matrices depending on…
This paper establishes three relations between the Toda field theory associated to a simple Lie algebra and the integral curves of the standard differential system on the corresponding complete flag variety. The motivation comes from the…
Given a finite subset $A$ of a distributive lattice, its total orderization $to(A)$ is a natural transformation of $A$ into a totally ordered set. Recently, the author showed that multivariate maps on distributive lattices which remain…
The algebraic and Hamiltonian structures of the multicomponent dispersionless Benney and Toda hierarchies are studied. This is achieved by using a modified set of variables for which there is a symmetry between the basic fields. This…
This is a short review of some recent results obtained by the author. These results are related the problem of obtaining polynomial identities (computational formulas) for some matrix functions by means of the known polarization theorem,…
An affine Toda-Sutherland system is a quasi-exactly solvable multi-particle dynamics based on an affine simple root system. It is a `cross' between two well-known integrable multi-particle dynamics, an affine Toda molecule and a Sutherland…
We follow the example of Tutte in his construction of the dichromate of a graph (that is, the Tutte polynomial) as a unification of the chromatic polynomial and the flow polynomial in order to construct a new polynomial invariant of maps…
For any classical Lie algebra $g$, we construct a family of integrable generalizations of Toda mechanics labeled a pair of ordered integers $(m,n)$. The universal form of the Lax pair, equations of motion, Hamiltonian as well as Poisson…
The spectral decomposition for an explicit second-order differential operator $T$ is determined. The spectrum consists of a continuous part with multiplicity two, a continuous part with multiplicity one, and a finite discrete part with…
We consider the relation of the multi-component 2D Toda hierarchy with matrix orthogonal and biorthogonal polynomials. The multi-graded Hankel reduction of this hierarchy is considered and the corresponding generalized matrix orthogonal…
There is now a renewed interest to the Hurwitz tau-function, counting the isomorphism classes of Belyi pairs, arising in the study of equilateral triangulations and Grothiendicks's dessins d'enfant. It is distinguished by belonging to a…
In a recent paper we constructed an integrable generalization of the Toda law on the square lattice. In this paper we construct other examples of integrable dynamics of Toda-type on the square lattice, as well as on the triangular lattice,…
By exhibiting the corresponding Lax pair representations we propose a wide class of integrable two-dimensional (2D) fermionic Toda lattice (TL) hierarchies which includes the 2D N=(2|2) and N=(0|2) supersymmetric TL hierarchies as…
In this paper, modified Toda (mToda) equation is generalized to form an integrable hierarchy in the framework of Sato theory, which is therefore called mToda hierarchy. Inspired by the fact that Toda hierarchy is 2-component generalization…
If L is a complete ortholattice, f any partial function from L^n to L, then there is a complete ortholattice L* containing L as a subortholattice, and an ortholattice polynomial with coefficients in L* which represents f on L^n. Iterating…
For the root system of each complex semi-simple Lie algebra of rank two, and for the associated 2D Toda chain $E=\{u_{xy}=\exp(K u)\}$, we calculate the two first integrals of the characteristic equation $D_y(w)=0$ on $E$. Using the…
A family of polynomial \tau-functions for the NLS-Toda hierarchy is constructed. The hierarchy is associated with the homogeneous vertex operator representation of the affine algebra \g of type A_1^{(1)}. These \tau-functions are given…
This work is divided into three parts. The first part concerns polynomials in one variable with all real roots. We consider linear transformations that preserve real rootedness, as well as matrices that preserve interlacing. The second part…