Tensor Products Of Irreducible Representations Research Articles

On classical tensor categories attached to the irreducible representations of the general linear supergroups GL(nvert n)

We study the quotient of mathcal {T}_n = Rep(GL(n|n)) by the tensor ideal of negligible morphisms. If we consider the full subcategory mathcal {T}_n^+ of mathcal {T}_n of indecomposable summands in iterated tensor products of irreducible representations up to parity shifts, its quotient is a semisimple tannakian category Rep(H_n) where H_n is a pro-reductive algebraic group. We determine the H_n and the groups H_{lambda } corresponding to the tannakian subcategory in Rep(H_n) generated by an irreducible representation L(lambda ). This gives structural information about the tensor category Rep(GL(n|n)), including the decomposition law of a tensor product of irreducible representations up to summands of superdimension zero. Some results are conditional on a hypothesis on 2-torsion in pi _0(H_n).

Bases of tensor products and geometric Satake correspondence

The geometric Satake correspondence can be regarded as a geometric construction of rational representations of a complex connected reductive group $G$. In their study of this correspondence, Mirković and Vilonen introduced algebraic cycles that provide a linear basis in each irreducible representation. Generalizing this construction, Goncharov and Shen define a linear basis in each tensor product of irreducible representations. We investigate these bases and show that they share many properties with the dual canonical bases of Lusztig.

EXTREMAL RAYS IN THE HERMITIAN EIGENVALUE PROBLEM FOR ARBITRARY TYPES

The Hermitian eigenvalue problem asks for the possible eigenvalues of a sum of Hermitian matrices given the eigenvalues of the summands. This is a problem about the Lie algebra of the maximal compact subgroup of G = SL(n). There is a polyhedral cone (the \eigencone) determining the possible answers to the problem. These eigencones can be defined for arbitrary semisimple groups G, and also control the (suitably stabilized) problem of existence of non-zero invariants in tensor products of irreducible representations of G. We give a description of the extremal rays of the eigencones for arbitrary semisimple groups G by first observing that extremal rays lie on regular facets, and then classifying extremal rays on an arbitrary regular face. Explicit formulas are given for some extremal rays, which have an explicit geometric meaning as cycle classes of interesting loci, on an arbitrary regular face. The remaining extremal rays on that face are understood by a geometric process we introduce, and explicate numerically, called induction from Levi subgroups. Several numerical examples are given. The main results, and methods, of this paper generalize [B3] which handled the case of G = SL(n).

Tensor product Markov chains

We analyze families of Markov chains that arise from decomposing tensor products of irreducible representations. This illuminates the Burnside-Brauer theorem for building irreducible representations, the McKay correspondence, and Pitman's 2M−X theorem. The chains are explicitly diagonalizable, and we use the eigenvalues/eigenvectors to give sharp rates of convergence for the associated random walks. For modular representations, the chains are not reversible, and the analytical details are surprisingly intricate. In the quantum group case, the chains fail to be diagonalizable, but a novel analysis using generalized eigenvectors proves successful.

On the Saturation Conjecture for Spin(2n)

In this article we examine the saturation conjecture on decompositions of tensor products of irreducible representations for complex semisimple algebraic groups of type D (the even spin groups: Spin for an integer), extending work done by Kumar–Kapovich–Millson on Spin(8). Our main theorem asserts that the saturation conjecture holds for Spin(10) and Spin(12): for all triples of dominants weights such that is in the root lattice, and for any N > 0, if and only iffor or . Some related results for groups of other types are listed as well.

Semiclassical asymptotics of $$\mathbf {GL}_N({\mathbb {C}})$$GLN(C) tensor products and quantum random matrices

The Littlewood--Richardson process is a discrete random point process arising from the isotypic decomposition of tensor products of irreducible representations of $\operatorname{GL}_N(\mathbb{C})$. Biane--Perelomov--Popov matrices are quantum random matrices obtained as the geometric quantization of random Hermitian matrices with deterministic eigenvalues and uniformly random eigenvectors. As first observed by Biane, correlation functions of certain global observables of the LR process coincide with correlation functions of linear statistics of sums of classically independent BPP matrices, thereby enabling a random matrix approach to the statistical study of $\operatorname{GL}_N(\mathbb{C})$ tensor products. In this paper, we prove an optimal result: classically independent BPP matrices become freely independent in any semiclassical/large-dimension limit. This proves and generalizes a conjecture of Bufetov and Gorin, and leads to a Law of Large Numbers for the BPP observables of the LR process which holds in any and all semiclassical scalings.

A new algorithm for computing branching rules and Clebsch–Gordan coefficients of unitary representations of compact groups

A numerical algorithm that computes the decomposition of any finite-dimensional unitary reducible representation of a compact Lie group is presented. The algorithm, which does not rely on an algebraic insight into the group structure, is inspired by quantum mechanical notions. After generating two adapted states (these objects will be conveniently defined in Definition II.1) and after appropriate algebraic manipulations, the algorithm returns the block matrix structure of the representation in terms of its irreducible components. It also provides an adapted orthonormal basis. The algorithm can be used to compute the Clebsch–Gordan coefficients of the tensor product of irreducible representations of a given compact Lie group. The performance of the algorithm is tested on various examples: the decomposition of the regular representation of two finite groups and the computation of Clebsch–Gordan coefficients of two examples of tensor products of representations of SU(2).

POVM construction: A simple recipe with applications to symmetric states

We propose a simple method for constructing positive operator-valued measures (POVMs) using any set of matrices which form an orthonormal basis for the space of complex matrices. Considering the orthonormal set of irreducible spherical tensors, we examine the properties of the construction on the [Formula: see text]-dimensional subspace of the [Formula: see text]-dimensional Hilbert space of [Formula: see text] qubits comprising the permutationally symmetric states. Using the notion of vectorization, the constructed POVMs are interpretable as projection operators in a higher-dimensional space. We then describe a method to physically realize the constructed POVMs for symmetric states using the Clebsch–Gordan decomposition of the tensor product of irreducible representations of the rotation group. We illustrate the proposed construction on a spin-1 system, and show that it is possible to generate entangled states from separable ones.

A presentation of the double Pieri algebra

The double Pieri algebra, constructed by Howe, Lee, and the author in [10,12], encodes information on the decomposition of Pieri type tensor products of irreducible representations for complex classical groups. In this paper we study its finite presentation in terms of generators and relations, and then prove that it admits the structure of a cluster algebra. We also study its sl2-module structure.

Two Constructions of Markov Chains on the Dual of U(n)

We provide two new constructions of Markov chains which had previously arisen from the representation theory of the infinite-dimensional unitary group. The first construction uses the combinatorial rule for the Littlewood-Richardson coefficients, which arise from tensor products of irreducible representations of the unitary group. The second arises from a quantum random walk on the group von Neumann algebra of U(n), which is then restricted to the center. Additionally, the restriction to a maximal torus can be expressed in terms of weight multiplicities, explaining the presence of tensor products.

Degenerate integrability of quantum spin Calogero–Moser systems

The main result of this note is the proof of degenerate quantum integrability of quantum spin Calogero--Moser systems and the description of the spectrum of quantum Hamiltonians in terms of the decomposition of tensor products of irreducible representations of corresponding Lie algebra.

Graphical tensor product reduction scheme for the Lie algebras [formula omitted], [formula omitted], and [formula omitted

We develop in detail a graphical tensor product reduction scheme, first described by Antoine and Speiser, for the simple rank 2 Lie algebras so(5)=sp(2), su(3), and g(2). This leads to an efficient practical method to reduce tensor products of irreducible representations into sums of such representations. For this purpose, the 2-dimensional weight diagram of a given representation is placed in a “landscape” of irreducible representations. We provide both the landscapes and the weight diagrams for a large number of representations for the three simple rank 2 Lie algebras. We also apply the algebraic “girdle” method, which is much less efficient for calculations by hand for moderately large representations. Computer code for reducing tensor products, based on the graphical method, has been developed as well and is available from the authors upon request.

Knot invariants and higher representation theory

We construct knot invariants categorifying the quantum knot variants for all representations of quantum groups. We show that these invariants coincide with previous invariants defined by Khovanov for sl_2 and sl_3 and by Mazorchuk-Stroppel and Sussan for sl_n. Our technique is to study 2-representations of 2-quantum groups (in the sense of Rouquier and Khovanov-Lauda) categorifying tensor products of irreducible representations. These are the representation categories of certain finite dimensional algebras with an explicit diagrammatic presentation, generalizing the cyclotomic quotient of the KLR algebra. When the Lie algebra under consideration is $\mathfrak{sl}_n$, we show that these categories agree with certain subcategories of parabolic category O for gl_k. We also investigate the finer structure of these categories: they are standardly stratified and satisfy a double centralizer property with respect to their self-dual modules. The standard modules of the stratification play an important role as test objects for functors, as Vermas do in more classical representation theory. The existence of these representations has consequences for the structure of previously studied categorifications. It allows us to prove the non-degeneracy of Khovanov and Lauda's 2-category (that its Hom spaces have the expected dimension) in all symmetrizable types, and that the cyclotomic quiver Hecke algebras are symmetric Frobenius. In work of Reshetikhin and Turaev, the braiding and (co)evaluation maps between representations of quantum groups are used to define polynomial knot invariants. We show that the categorifications of tensor products are related by functors categorifying these maps, which allow the construction of bigraded knot homologies whose graded Euler characteristics are the original polynomial knot invariants.

Unitary representations of the universal cover of SU(1,1) and tensor products

In this paper we initiate a study of the relation between weight modules for simple Lie algebras and unitary representations of the corresponding simply-connected Lie groups. In particular we consider in detail from this point of view the universal covering group of SU(1,1), including new results on the discrete part of tensor products of irreducible representations. As a consequence of these results, we show that the set of smooth vectors of the tensor product intersects trivially some of the representations in the discrete spectrum.

Deformations of polyhedra and polygons by the unitary group

We introduce the set of framed (convex) polyhedra with N faces as the symplectic quotient \documentclass[12pt]{minimal}\begin{document}${\mathbb {C}}^{2N}//\mathrm{SU}(2)$\end{document}C2N// SU (2). A framed polyhedron is then parametrized by N spinors living in \documentclass[12pt]{minimal}\begin{document}${\mathbb {C}}^{2}$\end{document}C2 satisfying suitable closure constraints and defines a usual convex polyhedron plus extra U(1) phases attached to each face. We show that there is a natural action of the unitary group U(N) on this phase space, which changes the shape of faces and allows to map any (framed) polyhedron onto any other with the same total (boundary) area. This identifies the space of framed polyhedra to the Grassmannian space U(N)/ (SU(2)×U(N−2)). We show how to write averages of geometrical observables (polynomials in the faces' area and the angles between them) over the ensemble of polyhedra (distributed uniformly with respect to the Haar measure on U(N)) as polynomial integrals over the unitary group and we provide a few methods to compute these integrals systematically. We also use the Itzykson-Zuber formula from matrix models as the generating function for these averages and correlations. In the quantum case, a canonical quantization of the framed polyhedron phase space leads to the Hilbert space of SU(2) intertwiners (or, in other words, SU(2)-invariant states in tensor products of irreducible representations). The total boundary area as well as the individual face areas are quantized as half-integers (spins), and the Hilbert spaces for fixed total area form irreducible representations of U(N). We define semi-classical coherent intertwiner states peaked on classical framed polyhedra and transforming consistently under U(N) transformations. And we show how the U(N) character formula for unitary transformations is to be considered as an extension of the Itzykson-Zuber to the quantum level and generates the traces of all polynomial observables over the Hilbert space of intertwiners. We finally apply the same formalism to two dimensions and show that classical (convex) polygons can be described in a similar fashion trading the unitary group for the orthogonal group. We conclude with a discussion of the possible (deformation) dynamics that one can define on the space of polygons or polyhedra. This work is a priori useful in the context of discrete geometry but it should hopefully also be relevant to (loop) quantum gravity in 2+1 and 3+1 dimensions when the quantum geometry is defined in terms of gluing of (quantized) polygons and polyhedra.

Classification of double flag varieties of complexity 0 and 1

We obtain a classification of double flag varieties of complexity 0 and 1 and consider an application to the problem of decomposing tensor products of irreducible representations of semisimple Lie groups.

Arithmetic harmonic analysis on character and quiver varieties

We present a conjecture generalizing the Cauchy formula for Macdonald polynomials. This conjecture encodes the mixed Hodge polynomials of the character varieties of representations of the fundamental group of a Riemann surface of genus g to GL_n(C) with fixed generic semi-simple conjugacy classes at k punctures. Using the character table of GL_n(F_q) we calculate the E-polynomial of these character varieties and confirm that it is as predicted by our main conjecture. Then, using the character table of gl_n(F_q), we calculate the E-polynomial of certain associated comet-shaped quiver varieties, the additive analogues of our character variety, and find that it is the pure part of our conjectured mixed Hodge polynomial. Finally, we observe that the pure part of our conjectured mixed Hodge polynomial also equals certain multiplicities in the tensor product of irreducible representations of GL_n(F_q). This implies a curious connection between the representation theory of GL_n(F_q) and Kac-Moody algebras associated with comet-shaped, typically wild, quivers.

Theory of neutrino masses and mixing

We motivate the usage of finite groups as symmetries of the Lagrangian. After a presentation of basic group-theoretical concepts, we introduce the notion of characters and character tables in the context of irreducible representations and discuss their applications. We exemplify these theoretical concepts with the groups S_4 and A_4. Finally, we discuss the relation between tensor products of irreducible representations and Yukawa couplings and describe a model for tri-bimaximal lepton mixing based on A_4.

CleGo: A package for automated computation of Clebsch–Gordan coefficients in tensor product representations for Lie algebras [formula omitted

We present a program that allows for the computation of tensor products of irreducible representations of Lie algebras A–G based on the explicit construction of weight states. This straightforward approach (which is slower and more memory-consumptive than the standard methods to just calculate dimensions of the tensor product decomposition) produces Clebsch–Gordan coefficients that are of interest for instance in discussing symmetry breaking in model building for grand unified theories. For that purpose, multiple tensor products have been implemented as well as means for analyzing the resulting effective operators in particle physics. Program summaryProgram title: CleGoCatalogue identifier: AEIQ_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEIQ_v1_0.htmlProgram obtainable from: CPC Program Library, Queenʼs University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 3641No. of bytes in distributed program, including test data, etc.: 34 536Distribution format: tar.gzProgramming language: OʼCamlComputer: i386–i686, x86_64Operating system: Cross-platform, for definiteness though we assume some UNIX system.RAM: ⩾4 GB commendable, though in general memory requirements depend on the size of the Lie algebras and the representations involved.Classification: 4.2, 11.1Nature of problem: Clebsch–Gordan coefficients are widely used in physics. This program has been written as a means to analyze symmetry breaking in the context of grand unified theories in particle physics. As an example, we computed the singlets appearing in higher-dimensional operators 27⊗27⊗27⊗78 and 27⊗27⊗27⊗650 in an E6-symmetric GUT.Solution method: In contrast to very efficient algorithms that also produce tensor product decompositions (as far as outer multiplicities/Littlewood–Richardson coefficients are concerned) we proceed straightforwardly by constructing all the weight states, i.e. the Clebsch–Gordan coefficients. This obviously comes at the expense of high memory and CPU-time demands. Applying Dynkin arithmetic in weight space, the algorithm is an extension of the one for the addition of angular momenta in su(2)≈A1, for reference see [1]. Note that, in general, Clebsch–Gordan coefficients are basis-dependent and therefore need to be understood with respect to the chosen basis. However, singlets appearing in (multiple) tensor products are less basis-dependent.Restrictions: Generically, only tensor products of non-degenerate or adjoint representations can be computed. However, the irreps appearing therein can subsequently be used as new input irreps for further tensor product decomposition so in principle there is no restriction on the irreps the tensor product is taken of. In practice, though, it is by the very nature of the explicit algorithm that input is restricted by memory and CPU runtime requirements.Unusual features: Analytic computation instead of float numerics.Additional comments: The program can be used in “notebook style” using a suitable OʼCaml toplevel. Alternatively, an OʼCaml input file can be compiled which results in processing that is approximately a factor of five faster. The latter mode is commendable when large irreps need to be constructed.Running time: Varies depending on the input from parts of seconds to weeks for very large representations (because of memory exhaustion).Reference:[1]J. Fuchs, C. Schweigert, Symmetries, Lie Algebras and Representations, Cambridge Univ. Press, Cambridge, UK, 1997, 438 p.

A combinatorial model for crystals of Kac-Moody algebras

We present a simple combinatorial model for the characters of the irreducible integrable highest weight modules for complex symmetrizable Kac-Moody algebras. This model can be viewed as a discrete counterpart to the Littelmann path model. We describe crystal graphs and give a Littlewood-Richardson rule for decomposing tensor products of irreducible representations. The new model is based on the notion of a $\lambda$-chain, which is a chain of positive roots defined by certain interlacing conditions.