Limiting behaviour of Gelfand-Zetlin basis of O(n) as n→infinity

We show that the matrix elements of the generators of O(n) with respect to the canonical Gelfand-Zetlin basis states of that group become, in the limit n→infinity, identical to the matrix elements of a combination of two generators of U(n) with respect to canonical Gelfand-Zetlin states of U(n) carrying the same labels as those of the O(n) states. This property holds when the state labels msub(r,2r) r = 1,2,..., (n/2) are equal to zero, and the index r or the label msub(re) which is modified by +-1 under the action of the generators, remains finite. A connection of these results with the limit form of basis states of O(n) realized in terms of traceless bosons is discussed. In the particular case when all the non-vanishing entries of the Gelfand pattern are confined to an upper left triangle with rows (j<<n), the limit process is shown to give matrix elements of generators of U(j) with respect to Gelfand-Zetlin states of U(j). (author)