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Spectra of elements in the group ring of SU(2)

Spectra of elements in the group ring of SU(2) We present a new method for establishing the “gap” property for finitely generated subgroups of SU(2), providing an elementary solution of Ruziewicz problem on S2 as well as giving many new examples of finitely generated subgroups of SU(2) with an explicit gap. The distribution of the eigenvalues of the elements of the group ring R[SU(2)] in the N-th irreducible representation of SU(2) is also studied. Numerical experiments indicate that for a generic (in measure) element of R[SU(2)], the “unfolded” consecutive spacings distribution approaches the GOE spacing law of random matrix theory (for N even) and the GSE spacing law (for N odd) as N→∞; we establish several results in this direction. For certain special “arithmetic” (or Ramanujan) elements of R[SU(2)] the experiments indicate that the unfolded consecutive spacing distribution follows Poisson statistics; we provide a sharp estimate in that direction. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of the European Mathematical Society Springer Journals

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Publisher
Springer Journals
Copyright
Copyright © 1999 by Springer-Verlag Berlin Heidelberg & EMS
Subject
Mathematics; Mathematics, general
ISSN
1435-9855
DOI
10.1007/PL00011157
Publisher site
See Article on Publisher Site

Abstract

We present a new method for establishing the “gap” property for finitely generated subgroups of SU(2), providing an elementary solution of Ruziewicz problem on S2 as well as giving many new examples of finitely generated subgroups of SU(2) with an explicit gap. The distribution of the eigenvalues of the elements of the group ring R[SU(2)] in the N-th irreducible representation of SU(2) is also studied. Numerical experiments indicate that for a generic (in measure) element of R[SU(2)], the “unfolded” consecutive spacings distribution approaches the GOE spacing law of random matrix theory (for N even) and the GSE spacing law (for N odd) as N→∞; we establish several results in this direction. For certain special “arithmetic” (or Ramanujan) elements of R[SU(2)] the experiments indicate that the unfolded consecutive spacing distribution follows Poisson statistics; we provide a sharp estimate in that direction.

Journal

Journal of the European Mathematical SocietySpringer Journals

Published: Jan 1, 1999

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