On a theorem of Lehrer and Zhang

Let K be an arbitrary field of characteristic not equal to 2. Let m, n ε N and V be an m dimensional orthogonal space over K. There is a right action of the Brauer algebra B_n(m) on the n-tensor space V ^⊗n which centralizes the left action of the orthogonal group O(V ). Recently G.I. Lehrer and R.B. Zhang defined certain quasiidempotents E_i in B_n(m) (see (1.1)) and proved that the annihilator of V ^⊗n in B_n(m) is always equal to the two-sided ideal generated by E_[(m+1)/2] if char K = 0 or char K > 2(m+1). In this paper we extend this theorem to arbitrary field K with char K ≠ 2 as conjectured by Lehrer and Zhang. As a byproduct, we discover a combinatorial identity which relates to the dimensions of Specht modules over the symmetric groups of different sizes and a new integral basis for the annihilator of V ^⊗m+1 in B_m+1(m).