Jones-Ocneanu trace
introduction
- linear functional on the Hecke algebra of type \(A_n\)
- Jones related a trace found by Ocneanu with HOMFLY polynomial
construction
- let \(W=S_n\), the symmetric group on \(n\) letters and \(S\) the set of transpositions \(s_i: = (i,i+1)\)
- Coxeter matrix \[m_{ii}=1\] and \(m_{i,i+1}=3\), 2 otherwise
Let \(A\) be a commutative ring with 1 and fix two invertible elements \(u, v \in A\).
For \(n\geq 1\), consider \(H_A(S_n)\) associated with \(S_n\) over the ring \(A\) and with parameters \(a_{s_i} = u\), \(b_{s_i} = v\) for \(1\leq i \leq n - 1\).
For simplicity, let \(H_n: = H_A(S_n)\).
Regard \(H_n\) as a subalgebra of \(H_{n+1}\).
- thm (Jones, Ocneanu)
There is a unique family \(\{\tau_n\}_{n\geq1}\) of \(A\)-linear maps \(\tau_n : H_n \to A\) s.t. the following conditions hold : \[ \begin{array}{ll} (M1) & \tau_1(T_e)=1 \\ (M2) & \tau_{n+1}(hT_{s_n}^{\pm})=\tau_{n}(h) \quad& \text{for \]n\geq1\( and \)h\in H_n\(} \\ (M3) & \tau_n(hh')=\tau_{n}(h'h) & \text{for \)n\geq1\( and \)h,h'\in H_n\(} \end{array} \) Moreover, \(\tau_{n+1}(h)=v^{-1}(1-u)\tau_{n}(h)\) for all \(n\geq 1\) and \(h\in H_n\).
- proof
Let us define \(\tau_n\) recursively as follows.
For \(n=1\), set \(\tau_1(T_e)=1\).
Let \(n\geq 1\) and assume that \(\tau_n\) has been defined. Then we set \begin{equation}\label{star} \tau_{n+1}(a+b T_{s_n}c):=\frac{1-u}{v}\tau_n(a)+\tau_n(bc), \, a,b,c\in H_n \end{equation}
need to check that \(\tau_{n+1}\) is well-defined and satisfies M2, M3.
we need the isomorphism of \(A\)-modules for \(n\geq 2\):
\[ \psi_n: H_n\oplus (H_n\otimes_{H_{n-1}} H_n) \to H_{n+1},\, a\oplus (b\otimes c)\mapsto a+bT_{s_n}c \] ■
- \(v^{-1}\) is used when we define \(\tau_n\)
- note that to have \(T_{s_n}^{-1}\), we need \(u^{-1}\).