| Let V be the three-dimensional irreducible (?)l2(C)-module. For any n∈N, let (?)n(3) be the Brauer algebra [1] over the complex field C with parameter 3. It has been proved in [8] that there is an algebraic epimorphismη:(?)n(3)→End(?)l2(C)(V(?)n).In order to describe the kernel ofη, Lehrer and Zhang constructed the two-sided ideal (?)n(3)Φ(?)n(3) for n≥4, whereΦis an element in (?) (3). They conjectured that kerη= (?)n(3)Φ(?)n(3). This conjecture has been verified for n = 4,5,6, in [8].We will prove that (?)n(3)Φ(?)n(3) is a C(?)2n-module and will give a conjecture to describe the decomposition of (?)n(3)Φ(?)n(3) into irreducible C(?)2n-module. Then it will be possible for us to use the representation theorem of the symmetric group to describe kerη. We will verify our conjecture for n = 4.
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