On a theorem of Lehrer and Zhang

2011/05/26 by Jun Hu, Zhankui Xiao, Hu, Jun +1
Mathematics · #Advanced Mathematical Identities #FOS: Mathematics #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1105.5287

openalex publication_date 2011/05/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let K be an arbitrary field of characteristic not equal to 2. Let m, n∈\N and V an m dimensional orthogonal space over K. There is a right action of the Brauer algebra \bb_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 quasi-idempotents E_i in \bb_n(m) (see (\refkeydfn)) and proved that the annihilator of V^{⊗ n} in \bb_n(m) is always equal to the two-sided ideal generated by E_[(m+1)/2] if \ch K=0 or \ch K>2(m+1). In this paper we extend this theorem to arbitrary field K with \ch 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 symmetric groups of different sizes and a new integral basis for the annihilator of V^{⊗ m+1} in \bb_m+1(m).

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