Quantum automorphism groups of homogeneous graphs

Details Quantum automorphism groups of homogeneous graphs Quantum automorphism groups of homogeneous graphs

2003-11-23

arxiv.org/abs/math/0311402v4

J. Funct. Anal. 224 (2005), 243-280

Mathematics

Quantum Algebra

30 pages

Scientific paper

Associated to a finite graph $X$ is its quantum automorphism group $G$. The main problem is to compute the Poincar\'e series of $G$, meaning the series $f(z)=1+c_1z+c_2z^2+...$ 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.

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