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Determining a semisimple group from its representation degrees. (English) Zbl 1073.22009
The author investigates to what extent a compact semisimple Lie group \(G\) is determined by the multiset of degrees of its representations. The zeta function \(\zeta _{G}(s)\) associated to the group \(G\) is defined and it is proven that it determines the Lie algebra of \(G\) up to isomorphism. The Weyl dimension formula, which expresses the dimension of each irreducible representation of \(G\) as a product over the set of roots of \(G\), is used as a basic tool. The geometry of the root system of \(G\) is recovered from the factorizations of the representation degrees. The construction of pairs of nonisomorphic compact semisimple Lie groups with the same zeta function is given. The efficiency of root systems is defined and some examples of calculus are given for all Lie groups appearing in the Dynkin scheme of the classification.

22E46 Semisimple Lie groups and their representations
20F40 Associated Lie structures for groups
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