Multiplicity Of Irreducible Representation Research Articles

Efficient Multi Port-Based Teleportation Schemes

In this manuscript we analyse generalised port-based teleportation (PBT) schemes, allowing for transmitting more than one unknown quantum state (or a composite quantum state) in one go, where the state ends up in several ports at Bob’s side. We investigate the efficiency of our scheme discussing both deterministic and probabilistic case, where parties share maximally entangled states. It turns out that the new scheme gives better performance than various variants of the optimal PBT protocol used for the same task. All the results are presented in group-theoretic manner depending on such quantities like dimensions and multiplicities of irreducible representations in the Schur-Weyl duality. The presented analysis was possible by considering the algebra of permutation operators acting on <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> systems distorted by the action of partial transposition acting on more than one subsystem. Considering its action on the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n-$ </tex-math></inline-formula> fold tensor product of the Hilbert space with finite dimension, we present construction of the respective irreducible matrix representations, which are in fact matrix irreducible representations of the Walled Brauer Algebra. I turns out that the introduced formalism, and symmetries beneath it, appears in many aspects of theoretical physics and mathematics - theory of anti ferromagnetism, aspects of gravity theory or in the problem of designing quantum circuits for special task like for example inverting an unknown unitary.

Convex Bodies Associated to Actions of Reductive Groups

We associate convex bodies to a wide class of graded G- algebras where G is a connected reductive group. These convex bodies give information about the Hilbert function as well as multiplicities of irreducible representations appearing in the graded algebra. We extend the notion of Duistermaat-Heckman measure to graded G-algebras and prove a Fujita type approximation theorem and a Brunn-Minkowski in- equality for this measure. This in particular applies to arbitrary G-line bundles giving an equivariant version of the theory of volumes of line bundles. We generalize the Brion-Kazarnowskii formula for the degree of a spherical variety to arbitrary G-varieties. Our approach follows some of the previous works of A. Okounkov. We use the asymptotic theory of semigroups of integral points and Newton-Okounkov bodies developed in the authors' paper 'Okounkov bodies, semigroups of inte- gral points, graded algebras and intersection theory'. 2000 Math. Subj. Class. Primary: 14L30, 53D20; Secondary: 52A39.

Witten deformation and the equivariant index

Let M be a compact Riemannian manifold endowed with an isometric action of a compact, connected Lie group. The method of the Witten deformation is used to compute the virtual representation-valued equivariant index of a transversally elliptic, first order differential operator on M. The multiplicities of irreducible representations in the index are expressed in terms of local quantities associated to the isolated singular points of an equivariant bundle map that is locally Clifford multiplication by a Killing vector field near these points.

Geometric proof of a conjecture of Fulton

We give a geometric proof of a conjecture of Fulton on the multiplicities of irreducible representations in a tensor product of irreducible representations for GL ( r ) .

Tomographic representation of spin and quark states

We present a short review of the general principles of constructing tomograms of quantum states. We derive a general tomographic reconstruction formula for the quantum density operator of a system with a dynamical Lie group. In the reconstruction formula, the multiplicity of irreducible representation in Clebsch–Gordan decomposition is taken into account. Various approaches to spin tomography are discussed. An integral representation for the tomographic probability is found and a contraction of the spin tomogram to the photon-number tomography distribution is considered. The case of SU(3) tomography is discussed with the examples of quark states (related to the simplest triplet representations) and octet states.

A graded representation model for Macdonald's polynomials.

We define doubly graded Sn modules Rmu for which we conjecture that the multiplicities of irreducible representations in various bi-degrees are given by the Macdonald coefficients Klambdamu. Assuming one fundamental conjecture, the modules Rmu can be given several equivalent definitions, which we discuss. We prove the conjectures in various special cases.

Representations of Sn*U(N) in repeated tensor products of the unitary groups

The n-fold tensor product space generated by a given irreducible representation of the unitary group U(N) is a representation space for the symmetric group Sn as well as for U(N). Using ideas from the theory of dual pairs, such tensor product spaces are decomposed into irreducible representations of U(N) times reducible representations of Sn. Computationally effective formulae for the multiplicity of irreducible representations of Sn are given. Generating sets of invariant polynomials from the enveloping algebra of U(N) that commute with the Sn action are exhibited; it is shown that the eigenvalues of such operators can be used to break the multiplicity occurring in the dual U(N)xSn action.

Upper bounds for multiplicities in a tensor product of representations and in a restriction of group representations to a subgroup

Upper bounds for multiplicities of irreducible representations of a semisimple compact Lie group in the tensor product of representations and multiplicities of irreducible representations of a subgroup in an irreducible representation of a group are derived. Pairs of groups and subgroups appear from the representation theory of non-compact groups.

Relation Between ``Outer'' and ``Inner'' Multiplicity for SU(2) and SU(3)

The Clebsch-Gordan series for SU(3) is given in terms of irreducible representations of SU(3) such that the ``outer'' multiplicity of the Clebsch-Gordan series is related to the ``inner'' multiplicity of irreducible representations.