Eigenvalue selectors for representations of compact connected groups

2025/02/12 by Chirvasitu, Alexandru · 2 citations
#17B10 #17B22 #20F55 #22C05 #22E46 #47A10 #54D05 #55R10 #Algebraic Topology (math.AT) #FOS: Mathematics #General Topology (math.GN) #Group Theory (math.GR) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2502.08847

Abstract

A representation ρ of a compact group \mathbbG selects eigenvalues if there is a continuous circle-valued map on \mathbbG assigning an eigenvalue of ρ(g) to every g∈ \mathbbG. For every compact connected \mathbbG, we characterize the irreducible \mathbbG-representations which select eigenvalues as precisely those annihilating the intersection Z₀(\mathbbG)∩ \mathbbG' of the connected center of \mathbbG with its derived subgroup. The result applies more generally to finite-spectrum representations isotypic on Z₀(\mathbbG), and recovers as applications (noted in prior work) the existence of a continuous eigenvalue selector for the natural representation of SU(n) and the non-existence of such a selector for U(n).

Cited by

• Spectral selections, commutativity preservation and Coxeter-Lipschitz maps
• Continuous spectrum-shrinking maps between finite-dimensional algebras

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