Skew Young diagram method in spectral decomposition of integrable lattice models

Abstract

The spectral decomposition of the path space of the vertex model associated to the vector representation of the quantized affine algebra $U_q(\hat{sl}_n)$ is studied. We give a one-to-one correspondence between the spin configurations and the semi-standard tableaux of skew Young diagrams. As a result we obtain a formula of the characters for the degeneracy of the spectrum in terms of skew Schur functions. We conjecture that our result describes the $sl_n$-module contents of the Yangian $Y(sl_n)$-module structures of the level 1 integrable modules of the affine Lie algebra $\hat{sl}_n$. An analogous result is obtained also for a vertex model associated to the quantized twisted affine algebra $U_q(A^{(2)}_{2n})$, where $Y(B_n)$ characters appear for the degeneracy of the spectrum. The relation to the spectrum of the Haldane-Shastry and the Polychronakos models are also discussed.