Quantum automorphism groups of homogeneous graphs

By T. Banica. 2003

157 citations recorded by the source corpus.

Associated to a finite graph X is its quantum automorphism group G. The main problem is to compute the Poincare series of G, meaning the series f(z)=1+c1z+c2z2+⋯ whose coefficients are multiplicities of 1 into tensor powers of the fundamental representation. In this paper we find a duality between certain quantum groups and planar algebras, which leads to a planar algebra formulation of the problem. Together with some other results, this gives f for all homogeneous graphs having 8 vertices or less.

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

View Paper: https://www.semanticscholar.org/paper/a12a76dbb5d3945e75a9934ee23025e8a5899140

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In Algebra and Number Theory

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In Mathematical Physics

Citations in the corpus

Quantum automorphism groups of vertex-transitive graphs of order ≤ 11 (2006, background): “Taking now the cube of the above relation gives and once again since v preserves eigenspaces, we get: As already mentioned, we don’t know if these computations are particular cases of some general result.”
Integration over quantum permutation groups (2006, background): “The following result is known since Wang’s paper [14]; for the magic unitary formulation, see [1], [2].”
Tannaka–Kreĭn duality for compact quantum homogeneous spaces. I. General theory (2012, background): “We also mention that a different kind of Tannaka-Krĕın duality for classical homogeneous spaces was developed in [17], and for actions on finite quantum spaces in [5, 4] within the framework of planar algebras.”
Ergodic Coactions with Large Multiplicity and Monoidal Equivalence of Quantum Groups (2005, background): “In the 1980’s, Woronowicz introduced the notion of a compact quantum group and generalized the classical Peter-Weyl representation theory.”