Schur Orthogonality Relations Research Articles

Asymptotic Schur orthogonality in hyperbolic groups with application to monotony

We prove a generalization of Schur orthogonality relations for certain classes of representations of Gromov hyperbolic groups. We apply the obtained results to show that representations of non-abelian free groups associated to the Patterson-Sullivan measures corresponding to a wide class of invariant metrics on the group are monotonous in the sense introduced by Kuhn and Steger. This in particular includes representations associated to harmonic measures of a wide class of random walks.

Matrix-valued orthogonal polynomials related to the quantum analogue of (mathrm{SU}(2) times mathrm{SU}(2), mathrm{diag})

Matrix-valued spherical functions related to the quantum symmetric pair for the quantum analogue of (mathrm{SU}(2) times mathrm{SU}(2), mathrm{diag}) are introduced and studied in detail. The quantum symmetric pair is given in terms of a quantised universal enveloping algebra with a coideal subalgebra. The matrix-valued spherical functions give rise to matrix-valued orthogonal polynomials, which are matrix-valued analogues of a subfamily of Askey–Wilson polynomials. For these matrix-valued orthogonal polynomials, a number of properties are derived using this quantum group interpretation: the orthogonality relations from the Schur orthogonality relations, the three-term recurrence relation and the structure of the weight matrix in terms of Chebyshev polynomials from tensor product decompositions, and the matrix-valued Askey–Wilson type q-difference operators from the action of the Casimir elements. A more analytic study of the weight gives an explicit LDU-decomposition in terms of continuous q-ultraspherical polynomials. The LDU-decomposition gives the possibility to find explicit expressions of the matrix entries of the matrix-valued orthogonal polynomials in terms of continuous q-ultraspherical polynomials and q-Racah polynomials.

Matrix-Valued Orthogonal Polynomials Related to (SU(2)×SU(2), diag)

The matrix-valued spherical functions for the pair (K x K, K), K=SU(2), are studied. By restriction to the subgroup A the matrix-valued spherical functions are diagonal. For suitable set of representations we take these diagonals into a matrix-valued function, which are the full spherical functions. Their orthogonality is a consequence of the Schur orthogonality relations. From the full spherical functions we obtain matrix-valued orthogonal polynomials of arbitrary size, and they satisfy a three-term recurrence relation which follows by considering tensor product decompositions. An explicit expression for the weight and the complete block-diagonalization of the matrix-valued orthogonal polynomials is obtained. From the explicit expression we obtain right-hand sided differential operators of first and second order for which the matrix-valued orthogonal polynomials are eigenfunctions. We study the low-dimensional cases explicitly, and for these cases additional results, such as the Rodrigues formula and being eigenfunctions to first order differential-difference and second order differential operators, are obtained.

Infinite dimensional multiplicity free spaces III: matrix coefficients and regular functions

In earlier papers we studied direct limits \({(G,\,K) = \varinjlim\, (G_n,K_n)}\) of two types of Gelfand pairs. The first type was that in which the G n /K n are compact Riemannian symmetric spaces. The second type was that in which \({G_n = N_n\rtimes K_n}\) with N n nilpotent, in other words pairs (G n , K n ) for which G n /K n is a commutative nilmanifold. In each we worked out a method inspired by the Frobenius–Schur Orthogonality Relations to define isometric injections \({\zeta_{m,n}: L^2(G_n/K_n) \hookrightarrow L^2(G_m/K_m)}\) for m ≧ n and prove that the left regular representation of G on the Hilbert space direct limit \({L^2(G/K) := \varinjlim L^2(G_n/K_n)}\) is multiplicity-free. This left open questions concerning the nature of the elements of L 2(G/K). Here we define spaces \({\mathcal{A}(G_n/K_n)}\) of regular functions on G n /K n and injections \({\nu_{m,n} : \mathcal{A}(G_n/K_n) \to \mathcal{A}(G_m/K_m)}\) for m ≧ n related to restriction by \({\nu_{m,n}(f)|_{G_n/K_n} = f}\). Thus the direct limit \({\mathcal{A}(G/K) := \varinjlim \{\mathcal{A}(G_n/K_n), \nu_{m,n}\}}\) sits as a particular G-submodule of the much larger inverse limit \({\varprojlim \{\mathcal{A}(G_n/K_n), {\rm restriction}\}}\). Further, we define a pre Hilbert space structure on \({\mathcal{A}(G/K)}\) derived from that of L 2(G/K). This allows an interpretation of L 2(G/K) as the Hilbert space completion of the concretely defined function space \({\mathcal{A}(G/K)}\), and also defines a G-invariant inner product on \({\mathcal{A}(G/K)}\) for which the left regular representation of G is multiplicity-free.

Schur Orthogonality Relations for the Finitary Infinite Symmetric Group

Let S f be the finitary infinite symmetric group. For a certain class of irreducible unitary representations of S f , a version of Schur orthogonality relations is proved. That is, we construct an invariant inner product on the matrix coefficient space of each representation and show that matrix coefficients for distinct representations are orthogonal with respect to these norms.

Schur orthogonality relations and invariant sesquilinear forms

Important connections between the representation theory of a compact group G G and L 2 ( G ) L^{2}(G) are summarized by the Schur orthogonality relations. The first part of this work is to generalize these relations to all finite-dimensional representations of a connected semisimple Lie group G . G. The second part establishes a general framework in the case of unitary representations ( π , V ) (\pi , V) of a separable locally compact group. The key step is to identify the matrix coefficient space with a dense subset of the Hilbert-Schmidt endomorphisms on V V .

An inducing construction of quantum codes from classical error correcting codes

The aim of the present paper is to construct error correcting quantum codes from classical error correcting group codes by using the Schur orthogonality relations for characters of a finite abelian group and the Knill–Laflamme criterion for a quantum code to correct a pre-assigned family of errors. This is an abstraction and generalization of Shor's example of a nine qubit single error correcting quantum code.

An inducing construction of quantum codes from classical error correcting codes

The aim of the present paper is to construct error correcting quantum codes from classical error correcting group codes by using the Schur orthogonality relations for characters of a finite abelian group and the Knill–Laflamme criterion for a quantum code to correct a pre-assigned family of errors. This is an abstraction and generalization of Shor's example of a nine qubit single error correcting quantum code.

Spectral analysis and the Haar functional on the quantumSU (2) group

The Haar functional on the quantumSU(2) group is the analogue of invariant integration on the groupSU(2). If restricted to a subalgebra generated by a self-adjoint element the Haar functional can be expressed as an integral with a continuous measure or with a discrete measure or by a combination of both. These results by Woronowicz and Koornwinder have been proved by using the corepresentation theory of the quantumSU(2) group and Schur's orthogonality relations for matrix elements of irreducible unitary corepresentations. These results are proved here by using a spectral analysis of the generator of the subalgebra. The spectral measures can be described in terms of the orthogonality measures of orthogonal polynomials by using the theory of Jacobi matrices.

Les noyaux de Bergman et Szegö pour des domaines strictement pseudo-convexes qui généralisent la boule

Let G be a complex semi-simple group with a compact maximal group K and an irreducible holomorphic representation ? on a finite dimensional space V. There exists on V a K-invariant Hermitian scalar product. Let O be the intersection of the unit ball of V with the G-orbit of a dominant vector. O is a generalization of the unit ball (case obtained for G = SL(n,C) and ? the natural representation on Cn). We prove that for such manifolds, the Bergman and Szego kernels as for the ball are rational fractions of the scalar products and these fractions can be computed explicitely, using invariants of ?. To compute this kernels, one uses a good orthonormal basis related to ?, and then proves that one has a rational fraction, using Schur's orthogonality relations and Weyl's dimensional formula for V.