Let W W be a Weyl group and let W ′ W’ be a parabolic subgroup of W W . Define A A as follows: \[ A = R ⊗ Q [ u ] A ( W ) A = R{ \otimes _{{\mathbf {Q}}[u]}}\mathcal {A}(W) \] where A ( W ) \mathcal {A}(W) is the generic algebra of type A n {A_n} over Q [ u ] {\mathbf {Q}}[u] an indeterminate, associated with the group W W , and R R is a Q [ u ] {\mathbf {Q}}[u] -algebra, possibly of infinite rank, in which u u is invertible. Similarly, we define A ′ A’ associated with W ′ W’ . Let M M be an A − A A - A bimodule, and let b ∈ M b \in M . Define the relative norm [14] \[ N W , W ′ ( b ) = ∑ t ∈ T u − l ( t ) a t − 1 b a t {N_{W,W’}}(b) = \sum \limits _{t \in T} {{u^{ - l(t)}}{a_{{t^{ - 1}}}}b{a_t}} \] where T T is the set of distinguished right coset representives for W ′ W’ in W W . We show that if b ∈ Z M ( A ′ ) = { m ∈ M | m a ′ = a ′ m ∀ a ′ ∈ A ′ } b \in {Z_M}(A’) = \{ m \in M|ma’ = a’m\quad \forall a’ \in A’\} , then N W , W ′ ( b ) ∈ Z M ( A ) {N_{W,W’}}(b) \in {Z_M}(A) . In addition, other properties of the relative norm are given and used to develop a theory of induced modules for generic Hecke algebras including a Markey decomposition. This section of the paper is previously unpublished work of P. Hoefsmit and L. L. Scott. Let α = ( k 1 , k 2 , … , k z ) \alpha = ({k_1},{k_2}, \ldots ,{k_z}) be a partition of n n and let S α = Π i = 1 Z S k i {S_\alpha } = \Pi _{i = 1}^Z{S_{{k_i}}} be a "left-justified" parabolic subgroup of S n {S_n} of shape α \alpha . Define \[ b α = N S n , S α ( N α ) {b_\alpha } = {N_{{S_n},{S_\alpha }}}({\mathcal {N}_\alpha }) \] , where \[ N α = ∏ i = 1 z N S k i − 1 , S 1 ( a w i ) {\mathcal {N}_\alpha } = \prod \limits _{i = 1}^z {{N_{{S_{{k_i} - 1}},{S_1}}}({a_{{w_i}}})} \] with w i {w_i} a k i {k_i} -cycle of length k i − 1 {k_i} - 1 in S k i {S_{{k_i}}} . Then the main result of this paper is Theorem. The set { b α | α ⊢ n } \{ {b_\alpha }|\alpha \vdash n\} is a basis for Z A ( S n ) ( A ( S n ) ) {Z_{A({S_n})}}(A({S_n})) over Q [ u , u − 1 ] {\mathbf {Q}}[u,{u^{ - 1}}] . Remark. The norms b α {b_\alpha } in Z A ( S n ) ( A ( S n ) ) {Z_{A({S_n})}}(A({S_n})) are analogs of conjugacy class sums in the center of Q S n {\mathbf {Q}}{S_n} and, in fact, specialization of these norms at u = 1 u = 1 gives the standard conjugacy class sum basis of the center of Q S n {\mathbf {Q}}{S_n} up to coefficients from Q {\mathbf {Q}} .