Large Lepton Mixing in a Coset-space Family Unification on E7/SU(5)×U(1)3E_7/SU(5) \times U(1)^3

We study a coset-space unification model for families based on $E_7/SU(5) \times U(1)^3$. We find that qualitative structure of quark and lepton mass matrices in this model describes very well the observation. We stress, in particular, that the large mixing angle, $\sin^22\theta_{\nu_\mu\nu_\tau} \simeq 1$, required for the atmospheric neutrino oscillation reported by the SuperKamiokande collaboration, is naturally obtained, which is a consequence of unparallel family structure in the present coset-space unification.Comment: 8 pages, Latex2

Whittaker vectors of the Virasoro algebra in terms of Jack symmetric polynomial

We give explicit formulae of Whittaker vectors for Virasoro algebra in terms of the Jack symmetric polynomials. Our fundamental tools are the Feigin-Fuchs bosonization and the split expression of the Calogero-Sutherland model given by Awata-Matsuo-Odake-Shiraishi.Comment: revised, 22 page