The Orthogonal Weingarten Formula in Compact Form

We present a compact formulation of the orthogonal Weingarten formula, with the traditional quantity I(i1, . . . , i2k : j1, . . . , j2k) = ∫ On ui1j1 . . . ui2kj2k du replaced by the more advanced quantity I(a) = ∫ On Πu aij ij du, depending on a matrix of exponents a ∈ Mn(N). Among consequences, we establish a number of basic facts regarding the integrals I(a): vanishing condition, sign, possible poles, asymptotic behavior. Introduction The computation of polynomial integrals over the orthogonal group On is a key problem in mathematical physics. These integrals are indeed known to appear in a wealth of concrete situations, coming from random matrices, lattice models, combinatorics. The standard approach to the computation of such integrals is via the Weingarten formula. This formula, originating from Weingarten’s paper [12], and worked out by Collins in [6], then by Collins and Śniady in [8], is an identity of the following type: ∫ On ui1j1 . . . ui2kj2k du = ∑ π,σ δπ(i)δσ(j)Wkn(π, σ) Here the sum is over all pairings of {1, . . . , 2k}, also called Brauer diagrams [5], and the delta symbols, describing the coupling between indices and diagrams, are 0 or 1. As for Wkn, this is the key combinatorial ingredient: the Weingarten function. The exact or asymptotic computation of Wkn is a quite subtle mathematical problem, and several results have been recently obtained on this subject. Let us mention here the work of Collins and Matsumoto [7], and Matsumoto and Novak [11], providing a deep insight into the combinatorics of Wkn. Also, the foundational part of theory has benefited from several abstract versions and generalizations, developed in [1], [9], [10]. The starting point for the considerations in the present paper is the following elementary reformulation of the Weingarten formula: ∫