Howe pairs, supersymmetry, and ratios of random characteristic polynomials for the unitary groups U(N)

Abstract

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.

arXiv comment76 pages, references added, minor corrections, chapter 2 shortened

References in corpus (3)

• Products and Ratios of Characteristic Polynomials of Random Hermitian Matrices
• Supersymmetry for Systems with Unitary Disorder: Circular Ensembles
• On the averages of characteristic polynomials from classical groups

Cited by in corpus (22)

• PSU(2,2|4) Character of Quasiclassical AdS/CFT
• A Symplectic Test of the L-Functions Ratios Conjecture
• Nuclei, Primes and the Random Matrix Connection
• Wronskian solutions of the T, Q and Y-systems related to infinite dimensional unitarizable modules of the general linear superalgebra gl(M|N)
• A few remarks on Colour-Flavour Transformations,truncations of random unitary matrices, Berezin reproducing kernels and Selberg type integrals
• Gaussian free fields at the integer quantum Hall plateau transition
• Mixed moments of characteristic polynomials of random unitary matrices
• Applications of the L-functions ratios conjectures
• Haar expectations of ratios of random characteristic polynomials
• Chaotic maps and flows: Exact Riemann-Siegel lookalike for spectral fluctuations
• Autocorrelation of ratios of L-functions
• Averages of ratios of characteristic polynomials in circular beta-ensembles and super-Jack polynomials
• Maxima of log-correlated fields: some recent developments
• A weighted one-level density of families of $L$-functions
• Selected results on Lie supergroups and their radial operators
• Generalised unitary group integrals of Ingham-Siegel and Fisher-Hartwig type
• Almost complex structures on real Lie supergroups
• On permanental polynomials of certain random matrices
• Correlations of eigenvalues and Riemann zeros
• Color-Flavor Transformation Revisited
• Lower Order Terms for the One-Level Density of Elliptic Curve L-Functions
• Bi-orthogonal systems on the unit circle, Regular Semi-Classical Weights and Integrable Systems - II