On a Theorem of Lehrer and Zhang
Let K be an arbitrary field of characteristic not equal to 2. Let m,n ∈ N and V be an m dimensional orthogonal space over K. There is a right action of the Brauer algebra Bn(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 quasi- idempotents Ei in Bn(m) (see (1.1)) and proved that the annihilator of V ⊗n in Bn(m) is always equal to the two-sided ideal generated by E((m+1)/2) if charK = 0 or charK > 2(m+1). In this paper we extend this theorem to arbitrary field K with charK 6 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 Bm+1(m).