Sum Of Irreducible Representations Research Articles

Skew Howe duality and limit shapes of Young diagrams

AbstractWe consider the skew Howe duality for the action of certain dual pairs of Lie groups on the exterior algebra as a probability measure on Young diagrams by the decomposition into the sum of irreducible representations. We prove a combinatorial version of this skew Howe duality for the pairs , , , and using crystal bases, which allows us to interpret the skew Howe duality as a natural consequence of lattice paths on lozenge tilings of certain partial hexagonal domains. The ‐representation multiplicity is given as a determinant formula using the Lindström–Gessel–Viennot lemma and as a product formula. These admit natural ‐analogs that we show equals the ‐dimension of a ‐representation (up to an overall factor of ), giving a refined version of the combinatorial skew Howe duality. Using these product formulas (at ), we take the infinite rank limit and prove that the diagrams converge uniformly to the limit shape.

Positive energy representations of Sobolev diffeomorphism groups of the circle

We show that any positive energy projective unitary representation of mathrm{Diff}_+(S^1) extends to a strongly continuous projective unitary representation of the fractional Sobolev diffeomorphisms mathcal {D}^s(S^1) for any real s>3, and in particular to C^k-diffeomorphisms mathrm{Diff}_+^k(S^1) with kge 4. A similar result holds for the universal covering groups provided that the representation is assumed to be a direct sum of irreducibles. As an application we show that a conformal net of von Neumann algebras on S^1 is covariant with respect to mathcal {D}^s(S^1), s > 3. Moreover every direct sum of irreducible representations of a conformal net is also mathcal {D}^s(S^1)-covariant.

Formula for the Product of Gauss Hypergeometric Functions and Applications

We deduce a formula for the product of two Γ-series in four variables, connected with the lattice B = ℤ〈(1, −1, −1, 1)〉. As a consequence, we obtain a formula for the product of Gauss hypergeometric functions F2,1, which can be interpreted as a part of the explicit decomposition of the tensor product of two representations of 𝔤𝔩3 into the direct sum of irreducible representations. Bibliography: 12 titles.

On the Monodromy of Meromorphic Cyclic Opers on the Riemann Sphere

Abstract We study the monodromy of meromorphic cyclic $\textrm{SL}(n,{\mathbb{C}})$-opers on the Riemann sphere with a single pole. We prove that the monodromy map, sending such an oper to its Stokes data, is an immersion in the case where the order of the pole is a multiple of $n$. To do this, we develop a method based on the work of Jimbo, Miwa, and Ueno from the theory of isomonodromic deformations. Specifically, we introduce a system of equations that is equivalent to the isomonodromy equations of Jimbo–Miwa–Ueno, but which is adapted to the decomposition of the Lie algebra $\mathfrak{sl}(n,\mathbb{C})$ as a direct sum of irreducible representations of $\mathfrak{sl}(2,\mathbb{C})$. Using properties of some structure constants for $\mathfrak{sl}(n,\mathbb{C})$ to analyze this system of equations, we show that deformations of certain families of cyclic $\textrm{SL}(n,\mathbb{C})$-opers on the Riemann sphere with a single pole are never infinitesimally isomonodromic.

Fourier algebras of parabolic subgroups

We study the following question: given a locally compact group when does its Fourier algebra coincide with the subalgebra of the Fourier-Stieltjes algebra consisting of functions vanishing at infinity? We provide sufficient conditions for this to be the case.As an application, we show that when $P$ is the minimal parabolic subgroup in one of the classical simple Lie groups of real rank one or the exceptional such group, then the Fourier algebra of $P$ coincides with the subalgebra of the Fourier-Stieltjes algebra of $P$ consisting of functions vanishing at infinity. In particular, the regular representation of $P$ decomposes as a direct sum of irreducible representations although $P$ is not compact.

Representations of Gelfand–Graev Type for the Unitriangular Group

We consider the analog of Gelfand–Graev representations for the unitriangular group. We obtain the decomposition into a sum of irreducible representations, prove that these representations are multiplicity free, and calculate the Hecke algebra.

Selection and identity rules for subductions of type A quantum Iwahori-Hecke algebras

This paper is concerned with the subduction problem of type A quantum Iwahori-Hecke algebras CH(Sf,q2) with a real deformation parameter q, i.e., the problem of decomposing irreducible representations of such algebras as direct sum of irreducible representations of the subalgebras CH(Sf1,q2)×CH(Sf2,q2), with f1+f2=f. After giving a suitable combinatorial description for the subduction issue, we provide a selection rule, based on the Richardson-Littlewood criterion, which allows to determine the vanishing coupling coefficients between standard basis vectors for such representations, and we also present an equivariance condition for the subduction coefficients. Such results extend those ones corresponding to the subduction problem in symmetric group algebras CSf↓CSf1×CSf2 which are obtained by q approaching the value of 1.

Group-theoretical structure of the atomic f shell: connection with the non-Euclidean heptakisoctahedral (didodecahedral) group

This paper discusses the use of the non-Euclidean heptakisoctahedral group, 7 O, of order 168, to study the atomic f shell of interest in lanthanide and actinide chemistry. In this group only the representations of the spherical harmonics of dimensions 0, 1, or 6 (mod 7) can be reduced to sums of irreducible representations (irreps). These include the S, F, I, K, and N spherical harmonics with ℓ values of 0, 3, 6, 7, and 10, respectively. The other spherical harmonics must be combined into pairs with total dimension a multiple of 7 in order to have representations reducible into sums of irreps of 7 O. Such pairs of interest in the atomic f shell include the PH and DG pairs of total dimension 14, the GM and HL pairs of total dimension 28, and the LQ and MO pairs of total dimension 42. If the subgroup chain SO(7) ⊃ G 2 ⊃ 7 O is used to study the atomic f shell, terms corresponding to these paired spherical harmonics, such as PH and DG pairs found in the highest spin f n configurations, appear together under a given irrep of SO(7) or G 2.

Jahn–Teller distortions considered as steady state bifurcations

The steady state bifurcation model can be used to determine the Jahn–Teller distortion modes of a molecule having symmetry point group G to lower symmetry structures having symmetry point groups H j ⊂ G. In order to find such distortion modes, the trace formula arising from bifurcation theory is applied to all subgroups H j of G to give a sum of irreducible representations Γ(G → H j ) corresponding to the possible distortion modes of G to H j . These subgroups H j correspond to epikernel subgroups of G for any irreducible representation of G found in Γ(G → H j ). This method has been applied to Jahn–Teller distortions of square (D 4 h ), octahedral (O h ), and icosahedral (I h ) molecules leading to lower symmetry structures.

Bond valence and finite graph symmetries

Abstract The graph theory prediction of bond valence distributions in a crystal network is reformulated in a form more familiar to chemists, that of symmetry-adapted functions. The graph theory terms relevant to this exposition are explained, including graph matrices, graph vector spaces and graph automorphism groups. It was found that the vertex and edge sets of a chemical graph form representations of the graph automorphism group. These representations may be expressed as a sum of irreducible representations using the group character table in the usual way. Alternatively the classes of the individual orbits spanned may be used to derive these irreducible representations. The orthogonal cut and cycle subspaces of the edge space can be defined so as to also transform as a sum of irreducible representations. Hence appropriate symmetry-adapted orthogonal basis vectors may be chosen for the vertex and edge spaces. Removing the usual Equal Valence Rule constraint creates a solution space, a range of possible solutions for a particular prob lem. This solution space may be found by combining a set of contributions from the cut space vectors fixed by the atom valences, with a convex polytope in the cycle space which is defined by the non-negativity of the individual bond valences. The complete procedure is illustrated for the β-Ga2O3 structure.

Poincaré representations of the massless class

The (proper) Poincaré group has four classes of representations. The (discrete-spin, finite-dimensional) representations with zero-mass differ from the others in significant, and interesting ways since the representations of the little group, SE(2), are nondecomposable, so nonunitary; matrix elements are thus arbitrary functions. As the matrices (which are found for physically relevant representations) are not decomposable, the bases are connections, not tensors, giving gauge invariance — a partial statement of Poincaré invariance for zero mass, only. Helicity m and algebra representation dimension l are related by m = ( 1 2 )(l − 1) ; the helecity is integral or half-integral. The photon has three (not four) components. Product representations need not be decomposable into sums of irreducible representations.

Unitary spectrum and the fundamental group for actions of semisimple Lie groups

Suppose a locally compact group G acts on a separable metrizable space M, preserving a finite measure #. Then there is an associated unitary representation of G on LZ(M,/~)o, the square integrable functions orthogonal to the constants. This representation will in general be far from irreducible, and in fact, when G is not compact need not have any irreducible subrepresentations. Thus, in the decomposition of LZ(G)o into irreducible constituents, one not only requires a direct integral rather than a direct sum, but the associated spectral measure on the unitary dual G (say for type I groups) will, except in special circumstances, have no atoms. The point of this paper is to give natural and general conditions, mostly involving just rq(M), under which there will be an infinite atomic spectrum. (I.e., the spectral measure has infinitely many atoms.) The basic example of actions in which the spectrum is purely atomic, i.e., when the representation is a direct sum of irreducible representations, is the case of M = G/F, where F c G is a cocompact discrete subgroup. In this case, assuming 9 G to be simply connected, we have 7tl(M ) = F. The first results of this paper show that when G is a simple Lie group with R-rank(G)=> 2, we can deduce that any action of G on a compact manifold M with fundamental group isomorphic to such a F, and satisfying a natural hypothesis of either a geometric or ergodic theoretic nature, must also have an infinite atomic spectrum. In fact, under certain natural hypotheses the same is true as long as there is a surjection of hi(M) onto F. To formulate this more precisely, let F c Ad(G) be a cocompact lattice, and let Pr be the unitary representation of Ad(G) on L2(Ad(G)/F)o . We recall that an action of a group on a manifold (or more generally on a standard space in the sense of [4]) is called engaging if there is no loss of ergodicity in passing to finite covers. (See I'4] for discussion.)

A new approach to the molecular rotational hamiltonian

The present paper is devoted to a new approach of the rotational hamiltonian of a non degenerate vibronic state of the semi-rigid molecules. It is based upon two points. Firstly the hermitian operators on a finiten dimensional vector space belong to an2 dimensional euclidian vector space. Secondly, the vector space of the rotational states is a direct sum of irreducible representations of the rotation group. Accordingly in each one of those representations the rotational hamiltonian can be represented by its set of real components on a orthonormal basis of hermitian operators. The components of the reduced hamiltonian of Watson limited to its quartic terms are determined.

Gradient property of bifurcation equation for systems with rotational symmetry

We show that any polynomial equation covariant with respect to any representation of SO(2) is a gradient equation. The same holds for the fundamental representation of SO(3). For the other representations of SO(3) we give a simple necessary and sufficient condition an equation has to satisfy in order to be a gradient one. As a side result, we obtain a formula for the decomposition of the symmetrized power of an irreducible representation of SO(3) in a sum of irreducible representations. We apply our results to symmetric bifurcation theory.

Eisenstein Series for Reductive Groups over Global Function Fields I.

Let G be the Lie group SL(2, R) and Γ a discrete subgroup of arithmetic type. The homogeneous space Γ\G can be equipped with an invariant measure so that there is a Hilbert space of square integrable functions, denoted L2(Γ\G), on which G acts by right translations. If Γ\G is compact then this Hilbert space breaks up into a countable direct sum of irreducible representations of G, each occurring with finite multiplicity. Quite often however Γ\G is not compact, but of finite volume; in this case L2(Γ\G) splits into a discrete spectrum Ld2 which behaves as if Γ\G were compact, and a continuous spectrum Lc2 which is described by the so called theory of Eisenstein series. These are generalized eigenfunctions of the Casimir operator of G, which are parametrized by a right half plane in C, and as such are analytic functions on this half-plane; in the course of describing the continuous spectrum Lc2 however, one analytically continues them to meromorphic functions over all of C, and shows them to satisfy functional equations.

The Behavior of General Theta Functions Under the Modular Group and the Characters of Binary Modular Congruence Groups. I

The general formulae for the transformation of the Jacobian theta functions under arbitrary modular substitutions were given in 1858 by Hermite.1 Since then several authors have applied Hermito's method to the study of the behaviour of other and more general functions under arbitrary modular substitutions of the variable!2 The same method will be applied in this paper to multiple theta series, formed with an arbitrary positive definite quadratic form in any number of variables, and in which the variables of summation are subject to certain congruence conditions. By a suitable choice of these congruence conditions a set of theta series is obtained with the property that if a modular transformation is applied to one of the series of the set, the result can be represented as a linear combination (with constant coefficients) of the series of the set. The formulae obtained in this way define a matrix representation of the infinite modular group. A matrix representation of modular congruence groups is then obtained by dividing the series of the set by a suitably chosen function. The remaining part of the paper is devoted to the problem of the reduction of this matric representation as a sum of irreducible representations. The modular congruence groups to modulus pX, where p is a prime number, merit special attention. For X = 1 (the binary modular group mod p) the irreducible characters have been determined by Frobenius.3 The irreducible characters of the binary modular group mod p2 have been determined almost at the same time by H. Rohrbach4 and by H. W. Praetorius.5 For X _ 3 the irreducible characters are not yet known. However, the reduction of the above mentioned matric representation by the method of this paper does not presuppose the knowledge of the irreducible characters. On the contrary this reduction can be used to determine the irreducible characters contained in this representation. In the second part of this paper the special case of the binary