an:03856051
Zbl 0538.58016
Kirillov, A. N.
Completeness of states of the generalized Heisenberg magnet
RU
Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 134, 169-189 (1984).
00221105
1984
j
53D50 82D20 22E70
Bethe model; character; irreducible representations
The author proves the completeness of the multiplet systems based on Bethe vectors for the general Heisenberg magnet model. The method is based on the connection between the algebraic model of Bethe and the representation theory of the Lie group \(\text{SU}(p+1)\). Firstly, the basic results on the general Heisenberg magnet are given. The algebraic system of equations describing a Heisenberg magnet is studied for the case \(N\to \infty\), where \(N\) is the number of all possible states of the magnet. The solutions of this system are parametrized by Bethe vectors. The number of these vectors is calculated and the hypothesis of completeness is rigorously formulated. Secondly, a fundamental combinatorial formula is proven, based on the following relation between the characters \(\omega^ m_ k\) of \(\text{SU}(p+1):\omega^ m_ k\otimes \omega^ m_ k=\omega_ k^{m-1}\otimes \omega_ k^{m+1}+\omega^ m_{k-1}\otimes \omega^ m_{k+1}.\) This formula is used then to prove the completeness for the general Heisenberg magnet. The completeness hypothesis results from the equality of the number \(Z(N,s| \mu)\) of Bethe vectors and the multiplicity \(e_{\mu}\) of the irreducible representation \(V_{\mu}\) of \(\text{SU}(p+1)\), which signature \(\mu\), in the tensorial product \(\otimes_{n}W_ n=\Sigma e_{\mu}V_{\mu}\) of representations. Finally, a method for obtaining the irreducible representations \(V_{\eta}\) in the tensorial product \(V_{\lambda}\otimes V_{\mu}\) of irreducible representations is given.
G. Zet