Character expansions for the orthogonal and symplectic groups

We prove that general correlation functions of both ratios and products of characteristic polynomials of Hermitian random matrices are governed by integrable kernels of three different types: a) those constructed from orthogonal polynomials; b) constructed from Cauchy transforms of the same orthogonal polynomials and finally c) those constructed from both orthogonal polynomials and their Cauchy transforms. These kernels are related with the Riemann-Hilbert problem for orthogonal polynomials. For the correlation functions we obtain exact expressions in the form of determinants of these kernels. Derived representations enable us to study asymptotics of correlation functions of characteristic polynomials via Deift-Zhou steepest-descent/stationary phase method for Riemann-Hilbert problems, and in particular to find negative moments of characteristic polynomials. This reveals the universal parts of the correlation functions and moments of characteristic polynomials for arbitrary invariant ensemble of β = 2 symmetry class.

show abstract

“…. ,σ(m) ( and byπ (1) A nice proof of the formula (A.6) can be found, for example, in the paper by Balantekin and Cassak [2]). …”

Section: Summary and Discussionmentioning

confidence: 83%

Universal Results for Correlations of Characteristic Polynomials: Riemann-Hilbert Approach

Strahov

Fyodorov

2003

Commun. Math. Phys.

144

View full text Add to dashboard Cite

show abstract

“…The Schur function expansion is a special case of character expansions (see [8,9,59,55,43]). Let Γ(a) = ∞ 0 e −y y a−1 dy for a > 0 (the Gamma function) and note Γ(a + 1) = a!…”

Section: Schur Function Expansionmentioning

confidence: 99%

Noncolliding Brownian Motion and Determinantal Processes

2007

View full text Add to dashboard Cite

A system of one-dimensional Brownian motions (BMs) conditioned never to collide with each other is realized as (i) Dyson's BM model, which is a process of eigenvalues of hermitian matrixvalued diffusion process in the Gaussian unitary ensemble (GUE), and as (ii) the h-transform of absorbing BM in a Weyl chamber, where the harmonic function h is the product of differences of variables (the Vandermonde determinant). The Karlin-McGregor formula gives determinantal expression to the transition probability density of absorbing BM. We show from the Karlin-McGregor formula, if the initial state is in the eigenvalue distribution of GUE, the noncolliding BM is a determinantal process, in the sense that any multitime correlation function is given by a determinant specified by a matrix-kernel. By taking appropriate scaling limits, spatially homogeneous and inhomogeneous infinite determinantal processes are derived. We note that the determinantal processes related with noncolliding particle systems have a feature in common such that the matrix-kernels are expressed using spectral projections of appropriate effective Hamiltonians. On the common structure of matrix-kernels, continuity of processes in time is proved and general property of the determinantal processes is discussed.

show abstract

“…The character computation of a particular representation for SO(2N ) needs special attention as it possesses the notion of simple and double characters. First, we compute two character functions [35,37,38]…”

Section: Charactersmentioning

confidence: 99%

Characters and group invariant polynomials of (super)fields: road to “Lagrangian”

et al. 2020

View full text Add to dashboard Cite

The dynamics of the subatomic fundamental particles, represented by quantum fields, and their interactions are determined uniquely by the assigned transformation properties, i.e., the quantum numbers associated with the underlying symmetry of the model under consideration. These fields constitute a finite number of group invariant operators which are assembled to build a polynomial, known as the Lagrangian of that particular model. The order of the polynomial is determined by the mass dimension. In this paper, we have introduced an automated $${\texttt {Mathematica}}^{\tiny \textregistered }$$ Mathematica ® package, GrIP, that computes the complete set of operators that form a basis at each such order for a model containing any number of fields transforming under connected compact groups. The spacetime symmetry is restricted to the Lorentz group. The first part of the paper is dedicated to formulating the algorithm of GrIP. In this context, the detailed and explicit construction of the characters of different representations corresponding to connected compact groups and respective Haar measures have been discussed in terms of the coordinates of their respective maximal torus. In the second part, we have documented the user manual of GrIP that captures the generic features of the main program and guides to prepare the input file. We have attached a sub-program CHaar to compute characters and Haar measures for $$SU(N), SO(2N), SO(2N+1), Sp(2N)$$ S U ( N ) , S O ( 2 N ) , S O ( 2 N + 1 ) , S p ( 2 N ) . This program works very efficiently to find out the higher mass (non-supersymmetric) and canonical (supersymmetric) dimensional operators relevant to the effective field theory (EFT). We have demonstrated the working principles with two examples: the standard model (SM) and the minimal supersymmetric standard model (MSSM). We have further highlighted important features of GrIP, e.g., identification of effective operators leading to specific rare processes linked with the violation of baryon and lepton numbers, using several beyond standard model (BSM) scenarios. We have also tabulated a complete set of dimension-6 operators for each such model. Some of the operators possess rich flavour structures which are discussed in detail. This work paves the way towards BSM-EFT.

show abstract

Character expansions for the orthogonal and symplectic groups

Cited by 19 publications

References 54 publications

Universal Results for Correlations of Characteristic Polynomials: Riemann-Hilbert Approach

Universal Results for Correlations of Characteristic Polynomials: Riemann-Hilbert Approach

Noncolliding Brownian Motion and Determinantal Processes

Characters and group invariant polynomials of (super)fields: road to “Lagrangian”

Contact Info

Product

Resources

About