Gelfand-Zetlin Basis Research Articles

Representations of the Lie superalgebra and parastatistics Fock spaces

The algebraic structure generated by the creation and annihilation operators of a system of m parafermions and n parabosons, satisfying the mutual parafermion relations, is known to be the Lie superalgebra . The Fock spaces of such systems are then certain lowest weight representations of . In the current paper, we investigate what happens when the number of parafermions and parabosons becomes infinite. In order to analyze the algebraic structure, and the Fock spaces, we first need to develop a new matrix form for the Lie superalgebra , and construct a new Gelfand–Zetlin basis of the Fock spaces in the finite rank case. The new structures are appropriate for the situation . The algebra generated by the infinite number of creation and annihilation operators is , a well defined infinite rank version of the orthosymplectic Lie superalgebra. The Fock spaces are lowest weight representations of , with a basis consisting of particular row-stable Gelfand–Zetlin patterns.

Spinless Basis for Spin-Singlet FQH States

We investigate an alternative description of the SU(M)-singlet FQH state by using the spinless basis. The SU(M)-singlet Halperin state is obtained via the q-deformation of the Laughlin state and its root of unity limit, by applying the Yangian Gelfand-Zetlin basis for the spin Calogero-Sutherland model. The squeezing rule for the SU(M) state is also investigated in terms of the spinless basis.

Gelfand–Zetlin basis, Whittaker Vectors and a Bosonic Formula for the ${\mathfrak{sl}}_{n+1}$ Principal Subspace

We derive a bosonic formula for the character of the principal space in the level k vacuum module for \widehat{\mathfrak{sl}}_{n+1} , starting from a known fermionic formula for it. In our previous work, the latter was written as a sum consisting of Shapovalov scalar products of the Whittaker vectors for U_{v^{\pm1}}(\mathfrak{gl}_{n+1}) . In this paper we compute these scalar products in the bosonic form, using the decomposition of the Whittaker vectors in the Gelfand–Zetlin basis. We show further that the bosonic formula obtained in this way is the quasi-classical decomposition of the fermionic formula.

Explicit representations of classical Lie superalgebras in a Gelfand-Zetlin basis

An explicit construction of all finite-dimensional irreducible representations of classical Lie algebras is a solved problem and a Gelfand-Zetlin type basis is known. However the latter lacks the orthogonality property or does not consist of weight vector

An explicit basis of lowering operators for irreducible representations of unitary groups

The representation theory of the unitary groups is of fundamental significance in many areas of physics and chemistry. In order to label states in a physical system with unitary symmetry, it is necessary to have explicit bases for the irreducible representations. One systematic way of obtaining bases is to generalize the ladder operator approach to the representations of SU(2) by using the formalism of lowering operators. Here, one identifies a basis for the algebra of all lowering operators and, for each irreducible representation, gives a prescription for choosing a subcollection of lowering operators that yields a basis upon application to the highest weight vector. Bases obtained through lowering operators are particularly convenient for computing matrix coefficients of observables as the calculations reduce to the commutation relations for the standard matrix units. The best known examples of this approach are the extremal projector construction of the Gelfand–Zetlin basis and the crystal (or canonical) bases of Kashiwara and Lusztig. In this paper, we describe another simple method of obtaining bases for the irreducible representations via lowering operators. These bases do not have the algebraic canonicity of the Gelfand–Zetlin and crystal bases, but the combinatorics involved are much more straightforward, making the bases particularly suited for physical applications. Keywords: unitary group, special unitary group, irreducible representations, lowering operators, spin-free quantum chemistry, many-body problem

Derivation of Green's function of a spin Calogero–Sutherland model by Uglov's method

The hole propagator of a spin 1/2 Calogero–Sutherland model is derived using Uglov's method, which maps the exact eigenfunctions of the model, called the Yangian Gelfand-Zetlin basis, to a limit of Macdonald polynomials (gl2-Jack polynomials). To apply this mapping method to the calculation of 1-particle Green's function, we confirm that the sum of the field annihilation operator ψ↑ + ψ↓ on a Yangian Gelfand-Zetlin basis is transformed to the field annihilation operator ψ on gl2-Jack polynomials by the mapping. The resultant expression for the hole propagator for a finite-size system is written in terms of renormalized momenta and spin of quasi-holes, and the expression in the thermodynamic limit coincides with the earlier result derived by another method. We also discuss the singularity of the spectral function for a specific coupling parameter where the hole propagator of the spin Calogero–Sutherland model becomes equivalent to the dynamical colour correlation function of an SU(3) Haldane–Shastry model.

Algorithms for Computing U(N) Clebsch Gordan Coefficients

If V (m) is an irreducible representation space for the unitary group U(N), then the r-fold tensor product space, $V^{(m)_1}\otimes...\otimes V^{(m)_r}$ is in general reducible. Such a reducible representation can be reduced to a direct sum of irreducible representation spaces, albeit with multiplicity. Clebsch–Gordan coefficients are the overlap coefficients between an orthonormal basis in the tensor product space and an orthonormal basis in the direct sum space. Since such coefficients are basis dependent, bases in the U(N) irrep spaces must be chosen. In this paper we use the Gelfand–Zetlin basis, built out of the chain of subgroups U(N)⊃...⊃U(1). Our first result is to derive algorithms for generating Gelfand–Zetlin bases from Gelfand–Zetlin tableaux. Given these concrete basis realizations we develop algorithms for computing the Clebsch–Gordan coefficients themselves.

Representations of \(\mathcal{U}_q (sl(3))\) at roots of 1 (In the restricted specialisation)

Irreducible representations of \(\mathcal{U}_q (sl(3))\) at roots of unity in the restricted specialisation are described with the Gelfand-Zetlin basis. This basis is redefined to allow the Casimir operator of the quantum subalgebra \(\mathcal{U}_q (sl(2)) \subset \mathcal{U}_q (sl(3))\) not to be completely diagonalised. Some irreducible representations of \(\mathcal{U}_q (sl(3))\) indeed contain indecomposable \(\mathcal{U}_q (sl(2))\)-modules. The set of redefined (mixed) states is described as a teepee inside the pyramid made with the whole representation.

A calculus for SU(3) leading to an algebraic formula for the Clebsch–Gordan coefficients

We develop a simple computational tool for SU(3) analogous to Bargmann’s calculus for SU(2). Crucial new inputs are (i) explicit representation of the Gelfand–Zetlin basis in terms of polynomials in four variables and positive or negative integral powers of a fifth variable, (ii) an auxiliary Gaussian measure with respect to which the Gelfand–Zetlin states are orthogonal but not normalized, (iii) simple generating functions for generating all basis states and also all invariants. As an illustration of our techniques, an algebraic formula for the Clebsch–Gordan coefficients is obtained for the first time. This involves only Gaussian integrations. Thus SU(3) is made as accessible for computations as SU(2) is.

ON PERIODIC REPRESENTATIONS OF QUANTUM GROUPS

We present some results on representations of quantum groups at the root of unity. In the case of SL(2)q, the classification of the finite dimensional irreducible representations is given. For [Formula: see text] with [Formula: see text] a semi-simple or affine Lie algebra and q an mth root of unity (m odd), we classify the representations of dimension [Formula: see text] on which the actions of the Chevalley generators are injective. Finally, we adapt the Gelfand-Zetlin basis to the case of SL(N)q and IU(N)q.

Periodic and partially periodic representations ofSU(N) q

The Gelfand-Zetlin basis is adapted toSU(N)q forq a root of unit. Extra parameters are incorporated in the matrix elements of the generators to obtain all the invariants corresponding to the augmented center. A crucial identity is derived and proved, which guarantees the periodicity of the action of the generators. Full periodicity is relaxed by stages, some raising and lowering operators remaining injective while others become nilpotent with corresponding changes in the dimension of the representation. In the extreme case of highest weight representations. all the raising and lowering operators are nilpotent. As an alternative approach an auxiliary algebra giving all the periodic representations is presented. An explicit solution of this system forN=3, while fully equivalent to the G.-Z. basis, turns out to be much simpler.

Q-analogs of IU(n) and U(n,1)

The starting point is the nonsemisimple, inhomogeneous Lie algebra Un×I2n [denoted also as IU(n)], where I2n represents an Abelian subalgebra in semidirect product with the homogeneous part U(n). This is realized by explicitly giving the matrix elements of the generators on a modified Gelfand–Zetlin basis that allows representations of infinite dimensions. The enveloping algebra is q lifted by introducing q brackets in the matrix elements giving Uq(IU(n)). The deformation of the Abelian structure of I2n is studied for q≠1. Some implications are pointed out. The important invariants are constructed for arbitrary n. The results are compared to the corresponding ones for Jimbo’s construction of Uq (U(n+1)) on a Gelfeld–Zetlin basis. Finally, the related construction of Uq(U(n,1)) is presented and discussed. Here, Uq(SU(1,1)), the q-analog of relativistic motion in a plane, is analyzed in the context of this formalism.

Gelfand-Zetlin basis for Uq(gl(N+1)) modules

The Gelfand-Zetlin basis of Uq(gl(N+1)) modules is constructed via the lowering operator method.

Spectrum-generating SU(4) in particle physics:SU(n)and particle assignments

The conventions and representations of SU(n) as used in the spectrum-generating-group approach to particle physics are described. The particle assignments are given in terms of Gelfand-Zetlin basis vectors, and the most useful Clebsch-Gordan coefficients are listed. The electric charge operator and the electromagnetic current operator are defined and the matrix elements of the electromagnetic current operator are discussed.