Related papers: Sutherland-type Trigonometric Models, Trigonometri…

I derive explicitly all polynomial relations in the character ring of $E_8$ of the form $\chi_{\wedge^k \mathfrak{e}_8} - \mathfrak{p}_{k} (\chi_{1}, \dots, \chi_{8})=0$, where $\wedge^k \mathfrak{e}_8$ is an arbitrary exterior power of the…

The octonionic flag manifold $Fl(\mathbb{O})$ is the space of all pairs in $\mathbb{O}P^2\times \mathbb{O}P^2$ (where $\mathbb{O}P^2$ denotes the octonionic projective plane) which satisfy a certain "incidence" relation. It comes equipped…

We rewrite the recently proposed differential expansion formula for HOMFLY polynomials of the knot $4_1$ in arbitrary rectangular representation $R=[r^s]$ as a sum over all Young sub-diagrams $\lambda$ of $R$ with extraordinary simple…

We present an algebraic theory of orthogonal polynomials in several variables that includes classical orthogonal polynomials as a special case. Our bottom line is a straightforward connection between apolarity of binary forms and the inner…

For every rational homology 3-sphere with 2-torsion only we construct a unified invariant (which takes values in a certain cyclotomic completion of a polynomial ring), such that the evaluation of this invariant at any odd root of unity…

Recently, L.Rozansky and E.Witten (hep-th/9612216) associated to any hyperKaehler manifold X a system of "weights" (numbers, one for each trivalent graph) and used them to construct invariants of topological 3-manifolds. We give a very…

The classification of graded non-alternating Hamiltonian Lie algebras over perfect field of characteristic 2 is obtained. It is shown that the filtered deformations of such algebras correspond to non-alternating Hamiltonian forms with…

We use a decomposition of the tensor of the fundamental representation of the quantum group $U_q(\mathfrak{sl}_N)$ and the Rosso-Jones formula to establish a peculiar ``panhandle'' shape of the HOMFLY-PT polynomial of the reverse parallel…

Recently, Gomez-Ullate et al. (1) have studied a particular N-particle quantum problem with an elliptic function potential supplemented by an external field. They have shown that the Hamiltonian operator preserves a finite dimensional space…

A new class of bivariate poly-analytic Hermite polynomials is considered. We show that they are realizable as the Fourier-Wigner transform of the univariate complex Hermite functions and form a nontrivial orthogonal basis of the classical…

Let R and S be two irreducible root systems spanning the same vector space and having the same Weyl group W, such that S (but not necessarily R) is reduced. For each such pair (R,S) we construct a family of W-invariant orthogonal…

It is shown that the 3-body trigonometric G_2 integrable system is exactly-solvable. If the configuration space is parametrized by certain symmetric functions of the coordinates then, for arbitrary values of the coupling constants, the…

The ring of projective invariants of eight ordered points on the line is a quotient of the polynomial ring on V, where V is a fourteen-dimensional representation of S_8, by an ideal I_8, so the modular fivefold (P^1)^8 // GL(2) is Proj(Sym*…

We find a two-parameter family of ordinary differential systems in dimension five with the affine Weyl group symmetry of type $D_3^{(2)}$. We show its symmetry and holomorphy conditions. This is the second example which gave higher order…

We have briefly analyzed the existence of the pseudofermionic structure of multilevel pseudo-Hermitian systems with odd time-reversal and higher order involutive symmetries. We have shown that 2N-level Hamiltonians with N-order eigenvalue…

We show that when integral polytopes are deformed while keeping the same facet normal vectors, the coefficients of weighted Ehrhart and $h^*$-polynomials are piecewise polynomial functions in the ``right hand sides'' of the linear…

We study a class of nonlinear PDEs that admit the same bi-Hamiltonian structure as WDVV equations: a Ferapontov-type first-order Hamiltonian operator and a homogeneous third-order Hamiltonian operator in a canonical Doyle--Potemin form,…

In this article we study persistence features of topological entropy and periodic orbit growth of Hamiltonian diffeomorphisms on surfaces with respect to Hofer's metric. We exhibit stability of these dynamical quantities in a rather strong…

We describe the 8-dimensional Wolf spaces as cohomogeneity one SU(3)-manifolds, and discover perturbations of the quaternion-kaehler metric on the simply-connected 8-manifold G_2/SO(4) that carry a closed fundamental 4-form but are not…

We present the supersymmetric extension of the recently constructed E$_{8(8)}$ exceptional field theory -- the manifestly U-duality covariant formulation of the untruncated ten- and eleven-dimensional supergravities. This theory is…