Anatol N. Kirillov

Exact solution of the integrable XXZ Heisenberg model with arbitrary spin. I. The ground state and the excitation spectrum

An integrable generalisation of the XXZ Heisenberg model with arbitrary spin and with light plane type anisotropy is studied. Integral equations describing the thermodynamics of the model are found. The antiferromagnetic ground state, the excitation spectrum, the quantum numbers and scattering amplitudes of the excitations are determined.

The Bethe Ansatz and the combinatorics of Young tableaux

The investigation of combinatorial aspects of the method of the inverse problem is continued in this paper.

Representations of Yangians and multiplicities of occurrence of the irreducible components of the tensor product of representations of simple Lie algebras

New combinatorial formulas are obtained for the multiplicities in the decomposition of the tensor product of the representations of simple Lie algebras into irreducible components.

The Yang-Baxter equation, symmetric functions, and Schubert polynomials

Abstract We present an approach to the theory of Schubert polynomials, corresponding symmetric functions, and their generalizations that is based on exponential solutions of the Yang-Baxter equation. In the case of the solution related to the nilCoxeter algebra of the symmetric group, we recover the Schubert polynomials of Lascoux and Schutzenberger, and provide simplified proofs of their basic properties, along with various generalizations thereof. Our techniques make use of an explicit combinatorial interpretation of these polynomials in terms of configurations of labelled pseudo-lines.

Quadratic Algebras, Dunkl Elements, and Schubert Calculus

We suggest a new combinatorial construction for the cohomology ring of the ag manifold. The degree 2 commutation relations satissed by the divided diierence operators corresponding to positive roots deene a quadratic associative algebra. In this algebra, the formal analogues of Dunkl operators generate a commuta-tive subring, which is shown to be canonically isomorphic to the cohomology of the ag manifold. This leads to yet another combinatorial version of the corresponding Schubert calculus. The paper contains numerous conjectures and open problems. We also discuss a generalization of the main construction to quantum cohomology. Contents

Combinatorics, Bethe Ansatz, and representations of the symmetric group

Techniques developed in the realms of the quantum method of the inverse problem are used to analyze combinatorial problems (Young diagrams and rigged configurations).

Affine Hecke algebras and raising operators for Macdonald polynomials

We introduce certain raising and lowering operators for Macdonald polynomials (of type

A representation of the current algebra connected with the SU (2)-invariant thirring model

A_{n-1}

COMBINATORIAL BN-ANALOGUES OF SCHUBERT POLYNOMIALS

) by means of Dunkl operators. The raising operators we discuss are a natural

The Yangians, Bethe Ansatz and combinatorics

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