On a theorem of Lehrer and Zhang

Abstract

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 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 quasiidempotents 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 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 Bm+1(m).