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Register a new userFermionic formulas for eigenfunctions of the difference Toda Hamiltonian
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We use the Whittaker vectors and the Drinfeld Casimir element to show that eigenfunctions of the difference Toda Hamiltonian can be expressed via fermionic formulas. Motivated by the combinatorics of the fermionic formulas we use the representation theory of the quantum groups to prove a number of identities for the coefficients of the eigenfunctions.
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We study a class of representations of the Lie algebra of Laurent polynomials with values in the nilpotent subalgebra of sl(3). We derive Weyl-type (bosonic) character formulas for these representations. We establish a connection between the bosonic formulas and the Whittaker vector in the Verma module for the quantum group $U_v sl(3)$. We also obtain a fermionic formula for an eigenfunction of the sl(3) quantum Toda Hamiltonian.
Explicit determinant formulas are presented for the $tau$ functions of the generalized Painleve equations of type $A$. This result allows an interpretation of the $tau$-functions as the Plucker coordinates of the universal Grassmann manifold.
For $mathfrak g$ a Kac-Moody algebra of affine type, we show that there is an $text{Aut}, mathcal O$-equivariant identification between $text{Fun},text{Op}_{mathfrak g}(D)$, the algebra of functions on the space of ${mathfrak g}$-opers on the disc, and $Wsubset pi_0$, the intersection of kernels of screenings inside a vacuum Fock module $pi_0$. This kernel $W$ is generated by two states: a conformal vector, and a state $delta_{-1}left|0right>$. We show that the latter endows $pi_0$ with a canonical notion of translation $T^{text{(aff)}}$, and use it to define the densities in $pi_0$ of integrals of motion of classical Conformal Affine Toda field theory. The $text{Aut},mathcal O$-action defines a bundle $Pi$ over $mathbb P^1$ with fibre $pi_0$. We show that the product bundles $Pi otimes Omega^j$, where $Omega^j$ are tensor powers of the canonical bundle, come endowed with a one-parameter family of holomorphic connections, $ abla^{text{(aff)}} - alpha T^{text{(aff)}}$, $alphain mathbb C$. The integrals of motion of Conformal Affine Toda define global sections $[mathbf v_j dt^{j+1} ] in H^1(mathbb P^1, Piotimes Omega^j, abla^{text{(aff)}})$ of the de Rham cohomology of $ abla^{mathrm{(aff)}}$. Any choice of ${mathfrak g}$-Miura oper $chi$ gives a connection $ abla^{mathrm{(aff)}}_chi$ on $Omega^j$. Using coinvariants, we define a map $mathsf F_chi$ from sections of $Pi otimes Omega^j$ to sections of $Omega^j$. We show that $mathsf F_chi abla^{text{(aff)}} = abla^{text{(aff)}}_chi mathsf F_chi$, so that $mathsf F_chi$ descends to a well-defined map of cohomologies. Under this map, the classes $[mathbf v_j dt^{j+1} ]$ are sent to the classes in $H^1(mathbb P^1, Omega^j, abla^{text{(aff)}}_chi)$ defined by the ${mathfrak g}$-oper underlying $chi$.
A unified approach is given to kernel functions which intertwine Ruijsenaars difference operators of type A and of type BC. As an application of the trigonometric cases, new explicit formulas for Koornwinder polynomials attached to single columns and single rows are derived.
We study solutions of the Bethe ansatz equations of the non-homogeneous periodic XXX model associated to super Yangian $mathrm Y(mathfrak{gl}_{m|n})$. To a solution we associate a rational difference operator $mathcal D$ and a superspace of rational functions $W$. We show that the set of complete factorizations of $mathcal D$ is in canonical bijection with the variety of superflags in $W$ and that each generic superflag defines a solution of the Bethe ansatz equation. We also give the analogous statements for the quasi-periodic supersymmetric spin chains.
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