Relative dimension, towers of projections and commuting squares of subfactors (Q913202)

If \(N\subset M\) is a subfactor of the type \(II_ 1\) factor M the index of N in M, [M:N], introduced by V. Jones, can only take the values \(\{4 \cos^ 2\pi /(n+2):\) \(n\geq 0\}\cup [4,\infty]\). It can also be defined as the dimension of the smallest nonzero projection in M which expected on N is a scalar multiple of the identity. This paper deals with the set \(\Lambda\) (M,N) of nonzero scalars appearing this way. The main result is the following: Let \(N\subset M\) be a subfactor of finite index. (i) If \([M:N]=4 \cos^ 2\pi /(n+2)\) then \(\Lambda (M,N)=\{0\}\cup \{P_{k-1}([M:N]^{-1})/[M:N]P_ k([M:N]^{- 1\quad}):\quad 0\leq k\leq n-1\},\) (ii) If \([M:N]\geq 4\) and \(t\leq 1/2\) is so that \(t(1-t)=[M:N]^{-1}\) then \(\Lambda (M,N)\cap (0,t)=\{P_{k- 1}([M:N]^{-1})/[M:N]P_ k([M:N]^{-1\quad}):\quad k\geq 0\}.\) Where \(P_ k(x)\) is the Jones' polynomial defined by \(P_{-1}\equiv 1\), \(P_ 0\equiv 1\), \(P_{n+1}(x)=P_ n(x)-xP_{n-1}(x)\).