We show that the algebra of commuting Hamiltonians of the homogeneous XXX Heisenberg model has simple spectrum on the subspace of singular vectors of the tensor product of two-dimensional $gl_2$ -modules. As a byproduct we show that there exist exactly $\binom {n}{l}-\binom{n}{l-1}$ two-dimensional vector subspaces $V \subset \C[u]$ with a basis $f,g\in V$ such that $°f = l, °g = n-l+1$ and $f(u)g(u-1) - f(u-1)g(u) = (u+1)^n$ .We show that the algebra of commuting Hamiltonians of the homogeneous XXX Heisenberg model has simple spectrum on the subspace of singular vectors of the tensor product of two-dimensional $gl_2$-modules. As a byproduct we show that there exist exactly $\binom {n}{l}-\binom{n}{l-1}$ two-dimensional vector subspaces $V \subset \C[u]$ with a basis $f,g\in V$ such that $°f = l, °g = n-l+1$ and $f(u)g(u-1) - f(u-1)g(u) = (u+1)^n$.

arXiv:0706.0688Jun 5, 20077 citations

Three-state Heisenberg model on the Bethe lattice

S. V. Semkin, V. P. Smagin

Bethe Algebra of Homogeneous XXX Heisenberg Model has Simple Spectrum

Alexander N. Varchenko, Evgeny Mukhin, V. Tarasov

Jul 1, 20214 citations

We show that the algebra of commuting Hamiltonians of the homogeneous XXX Heisenberg model has simple spectrum on the subspace of singular vectors of the tensor product of two-dimensional

gl_2

-modules. As a byproduct we show that there exist exactly

\binom {n}{l}-\binom{n}{l-1}

two-dimensional vector subspaces $V \subset \C[u]$ with a basis

f,g\in V

such that

°f = l, °g = n-l+1

and

f(u)g(u-1) - f(u-1)g(u) = (u+1)^n