Howe pairs, supersymmetry, and ratios of random characteristic polynomials for the unitary groups U(N)
| dc.creator | Conrey, J. B. | |
| dc.creator | Farmer, D. W. | |
| dc.creator | Zirnbauer, M. R. | |
| dc.date | 2005-11-06 | |
| dc.date | 2007-09-09 | |
| dc.date.accessioned | 2026-07-07T08:28:02Z | |
| dc.date.available | 2026-07-07T08:28:02Z | |
| dc.description | For the classical compact Lie groups K = U(N) the autocorrelation functions of ratios of random characteristic polynomials are studied. Basic to our treatment is a property shared by the spinor representation of the spin group with the Shale-Weil representation of the metaplectic group: in both cases the character is the analytic square root of a determinant or the reciprocal thereof. By combining this fact with Howe's theory of supersymmetric dual pairs (g,K), we express the K-Haar average product of p ratios of characteristic polynomials and q conjugate ratios as a character which is associated with an irreducible representation of the Lie superalgebra g = gl(n|n) for n = p+q. This primitive character is shown to extend to an analytic radial section of a real supermanifold related to gl(n|n), and is computed by invoking Berezin's description of the radial parts of Laplace-Casimir operators for gl(n|n). The final result for the character looks like a natural transcription of the Weyl character formula to the context of highest-weight representations of Lie supergroups. While several other works have recently reproduced our results in the stable range where N is no less than max(p,q), the present approach covers the full range of matrix dimensions N. To make this paper accessible to the non-expert reader, we have included a chapter containing the required background material from superanalysis. | |
| dc.description | 76 pages, references added, minor corrections, chapter 2 shortened | |
| dc.identifier | https://arxiv.org/abs/math-ph/0511024 | |
| dc.identifier | http://arxiv.org/abs/math-ph/0511024 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/137477 | |
| dc.subject | Mathematical Physics | |
| dc.title | Howe pairs, supersymmetry, and ratios of random characteristic polynomials for the unitary groups U(N) | |
| dc.type | text |