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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Complex equiangular Parseval frames and Seidel matrices containing $ p$th roots of unity


Authors: Bernhard G. Bodmann and Helen J. Elwood
Journal: Proc. Amer. Math. Soc. 138 (2010), 4387-4404
MSC (2010): Primary 42C15, 52C17; Secondary 05B20
Published electronically: May 27, 2010
MathSciNet review: 2680063
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Abstract: We derive necessary conditions for the existence of complex Seidel matrices containing $ p$th roots of unity and having exactly two eigenvalues, under the assumption that $ p$ is prime. The existence of such matrices is equivalent to the existence of equiangular Parseval frames with Gram matrices whose off-diagonal entries are a common multiple of the $ p$th roots of unity. Explicitly examining the necessary conditions for $ p=5$ and $ p=7$ rules out the existence of many such frames with a number of vectors less than 50, similar to previous results in the cube roots case. On the other hand, we confirm the existence of $ p^2\times p^2$ Seidel matrices containing $ p$th roots of unity, and thus the existence of the associated complex equiangular Parseval frames, for any $ p\ge 2$. The construction of these Seidel matrices also yields a family of previously unknown Butson-type complex Hadamard matrices.


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Additional Information

Bernhard G. Bodmann
Affiliation: Department of Mathematics, University of Houston, Houston, Texas 77204-3008
Email: bgb@math.uh.edu

Helen J. Elwood
Affiliation: Department of Mathematics, University of Houston, Houston, Texas 77204-3008
Email: helwood@math.uh.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10435-5
PII: S 0002-9939(2010)10435-5
Received by editor(s): September 21, 2009
Received by editor(s) in revised form: February 9, 2010
Published electronically: May 27, 2010
Additional Notes: This research was partially supported by NSF Grant DMS-0807399 and by NSF Grant DMS-0914021
Communicated by: Michael T. Lacey
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.