Mean eigenvalues for simple, simply connected, compact Lie groups

Author： Kaiser N.  

Publisher： IOP Publishing

ISSN： 0305-4470

Source： Journal of Physics A: Mathematical and General, Vol.39, Iss.49, 2006-12, pp. : 15287-15298

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Abstract

We determine for each of the simple, simply connected and compact Lie groups SU(n), Spin(4n + 2) and E6 with a complex fundamental representation that particular region inside the unit disc in the complex plane which is filled by their mean eigenvalues. We give analytical parameterizations for the boundary curves of these so-called trace figures. The area enclosed by a trace figure turns out to be a rational multiple of π in each case. We calculate also the length of the boundary curve and determine the radius of the largest circle that fits into a trace figure. The discrete centre of the corresponding compact Lie group shows up prominently in the form of cusp points of the trace figure placed symmetrically on the unit circle. For the exceptional Lie groups G2, F4 and E8 with trivial centre we determine the (negative) lower bounds on their mean eigenvalues lying within the real interval [- 1, 1]. We find the rational boundary values -2/7, - 3/13 and -1/31 for G2, F4 and E8, respectively.

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