Integration over quantum permutation groups

By T. Banica, Benoît Collins. 2006

73 citations recorded by the source corpus.

Abstract We find a combinatorial formula for the Haar measure of quantum permutation groups. This leads to a dynamic formula for laws of diagonal coefficients, explaining the Poisson/free Poisson convergence result for characters.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ad4613127ee5bf566186153737c42dc1427b02b4

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In Algebra and Number Theory

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In Mathematical Physics

Citations in the corpus

Quantum groups acting on 4 points (2007, methodology): “A general matrix representation was constructed in [12], which, using the explicit description of the Haar measure [10, 11], was shown to be faithful at n = 4, giving an embedding C(Q4) ⊂ M4(C(SU(2))) [11].”
Liberation of orthogonal Lie groups (2008, methodology): “We use a standard method, going back to the papers of Weingarten [26] and of Collins and Śniady [13] in the classical case, and developed for the free quantum groups in the series of papers [7], [8], [6], [4].”
Classification results for easy quantum groups (2009, background): “The lack of symmetry in this fundamental computation was conceptually understood in [7], where it was shown that the correct invariant to look at is the law of the truncated character χt, with t ∈ (0, 1].”
Algebraic quantum permutation groups (2007, background): “See [3, 8, 6] for probabilistic and representation theoretic meanings of freeness.”
Reduced operator algebras of trace-preserving quantum automorphism groups (2012, methodology): “In addition, combinatorial “Weingarten” formulas for the Haar state on the quantum permutation groups S n were obtained in [6].”
De Finetti theorems for easy quantum groups (2009, methodology): “then Z ui1j1 ···ui kj = X π,σ∈D(k) π≤keri σ≤kerj Wkn(π,σ). Remark 3.3. The statement of the theorem above is from [7], but goes back to work of Weingarten [33] and was developed in a series of papers [2, 3, 11, 12]. Note that this reduces the problem of evaluating integrals over G to computing the e”