Comparison of Gelfand-Tsetlin Bases for Alternating and Symmetric Groups

Young’s orthogonal basis is a classical basis for an irreducible representation of a symmetric group. This basis happens to be a Gelfand-Tsetlin basis for the chain of symmetric groups. It is well-known that the chain of alternating groups, just like the chain of symmetric groups, has multiplicity-free restrictions for irreducible representations. Therefore each irreducible representation of an alternating group also admits Gelfand-Tsetlin bases. Moreover, each such representation is either the restriction of, or a subrepresentation of, the restriction of an irreducible representation of a symmetric group. In this article, we describe a recursive algorithm to write down the expansion of each Gelfand-Tsetlin basis vector for an irreducible representation of an alternating group in terms of Young’s orthogonal basis of the ambient representation of the symmetric group. This algorithm is implemented with the Sage Mathematical Software.     

Continuity and measurability of multiplier and projective representations

It is shown that various kinds of measurability of a multiplier representation of a locally compact group are equivalent, and are equivalent to the continuity of an ordinary representation of a related group. A unitary representation with measurable coefficients is the sum of a strongly continuous representation and a representation all of whose coefficients are locally null functions. Similar results are obtained for projective unitary representations. This work is carried out without any separability assumptions.     

On the asymptotic distribution of the spectrum of an operator over irreducible representations of its symmetry group

We show that a representation of a finite group in the eigenfunction space of an elliptic operator defined on a Riemannian manifold and commuting with the effective action of the group is asymptotically a multiple of the regular representation of the group.     

Caractérisations de la représentation de Burau

This paper is devoted to characterizations of the (reduced) Burau representation of the Artin braid group, in terms of rigid local systems. We prove that the Burau representation is the only representation of the Hecke algebra for which some local system associated to every linear representation of the braid group is irreducible and rigid in the sense of Katz. We also use previous results to give a characterization of the corresponding Knizhnik-Zamolodchikov system.     

Separate and joint analyticity in Lie groups representations

We prove that for any Lie group there exists a basis of its Lie Algebra in which for any representation of the Lie group in a Hilbert space, a vector which is analytic for every operator representing that basis is an analytic vector for the representation.     

On Generic Extensions of Representations of a Dynkin Quiver of Type D

The notion of generic extensions of representations of a Dynkin quiver plays a big role in the study of the structure of the corresponding quantum group. In this paper, we describe the generic extensions of a simple representation by any representation and that of any representation by a simple representation of a Dynkin quiver Q of type D.     

Quasi-hopf algebra actions and smash products

A classical result of Magnus asserts that a free group F has a faithful representation in the group of units of a ring of non-commuting formal power series with integral coefficients. We generalize this result to the category of A-groups, where A is an associative ring or an Abelian group. We say that a free A-group F^A has a faithful Magnus representation if there is a ring B containing A as an additive subgroup (or a subring) such that F^A is faithfully represented (exactly as in Magnus' classical result in the case A = Z)in the group of units of the ring of formal power series in non-communting indeterminater over B.The three principal results are: (I) If A is a torsion free Abelian group and F^A is a free A-groupp of Lyndon' type, then F^A has a faithful Magnus representation; (II) If A is an ω‐residually Z ring, then F^A has a faithful Magnus representation;(III) for every nontrivial torsion-free Abelian group A, F^A has a faithful Magnus representation in B[[Y]] for a suitable ring B in and only if F^Q has a faithful Magnus representation in Q[[Y]].     

Dilations for systems of imprimitivity acting on Banach spaces

Motivated by a general dilation theory for operator-valued measures, framings and bounded linear maps on operator algebras, we consider the dilation theory of the above objects with special structures. We show that every operator-valued system of imprimitivity has a dilation to a probability spectral system of imprimitivity acting on a Banach space. This completely generalizes a well-known result which states that every frame representation of a countable group on a Hilbert space is unitarily equivalent to a subrepresentation of the left regular representation of the group. We also prove that isometric group representation induced framings on a Banach space can be dilated to unconditional bases with the same structure for a larger Banach space. This extends several known results on the dilations of frames induced by unitary group representations on Hilbert spaces.     

Groupoid extensions of mapping class representations for bordered surfaces

The mapping class group of a surface with one boundary component admits numerous interesting representations including a representation as a group of automorphisms of a free group and as a group of symplectic transformations. Insofar as the mapping class group can be identified with the fundamental group of Riemann's moduli space, it is furthermore identified with a subgroup of the fundamental path groupoid upon choosing a basepoint. A combinatorial model for this, the mapping class groupoid, arises from the invariant cell decomposition of Teichmüller space, whose fundamental path groupoid is called the Ptolemy groupoid. It is natural to try to extend representations of the mapping class group to the mapping class groupoid, i.e., to construct a homomorphism from the mapping class groupoid to the same target that extends the given representations arising from various choices of basepoint.Among others, we extend both aforementioned representations to the groupoid level in this sense, where the symplectic representation is lifted both rationally and integrally. The techniques of proof include several algorithms involving fatgraphs and chord diagrams. The former extension is given by explicit formulae depending upon six essential cases, and the kernel and image of the groupoid representation are computed. Furthermore, this provides groupoid extensions of any representation of the mapping class group that factors through its action on the fundamental group of the surface including, for instance, the Magnus representation and representations on the moduli spaces of flat connections.     

Two-dimensional l-adic representations of the Galois group of a global field of characteristic P and automorphic forms on Gl(2)

It is well known (Ref. Zh. Mat., 1978, 1A405) that to each parabolic representation of the group GL (2) over adeles of a global field k of characteristic p there corresponds an irreducible two-dimensionall-adic representation of the Galois group of this field. In this paper, it is proved that, conversely also, to each irreducible two-dimensionall-adic representation of the Galois group there corresponds a parabolic representation of the group GL(2) over adeles. The proof of Langland's hypotheses for GL(2,k) is thereby completed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Maternaticheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 134, pp. 138–156, 1984.     

基于李群方法的贝塞尔函数数学实现

基于李群的表示理论,首先讨论了欧拉群的表示及其性质;然后,从该群的表示理论出发,分别导出了第一类贝塞尔函数的积分形式和幂级数形式.该研究表明了群方法可以求解对称边界问题的解析波函数,并为用群方法求解电磁场问题创造了条件.     

Asymptotic linearity of the mapping class group and a homological version of the Nielsen–Thurston classification

We study the action of the mapping class group on the integral homology of finite covers of a topological surface. We use the homological representation of the mapping class to construct a faithful infinite-dimensional representation of the mapping class group. We show that this representation detects the Nielsen–Thurston classification of each mapping class. We then discuss some examples that occur in the theory of braid groups and develop an analogous theory for automorphisms of free groups. We close with some open problems.     

Hyperbolic cone-manifold structures with prescribed holonomy I: punctured tori

We consider the relationship between hyperbolic cone-manifold structures on surfaces, and algebraic representations of the fundamental group into a group of isometries. A hyperbolic cone-manifold structure on a surface, with all interior cone angles being integer multiples of 2π, determines a holonomy representation of the fundamental group. We ask, conversely, when a representation of the fundamental group is the holonomy of a hyperbolic cone-manifold structure. In this paper we prove results for the punctured torus; in the sequel, for higher genus surfaces. We show that a representation of the fundamental group of a punctured torus is a holonomy representation of a hyperbolic cone-manifold structure with no interior cone points and a single corner point if and only if it is not virtually abelian. We construct a pentagonal fundamental domain for hyperbolic structures, from the geometry of a representation. Our techniques involve the universal covering group [(PSL₂\mathbb R)\tilde]{\widetilde{{\it PSL}_2{\mathbb R}}} of the group of orientation-preserving isometries of \mathbb H²{{\mathbb H}^2} and Markoff moves arising from the action of the mapping class group on the character variety.     

Induced Representations of Abelian Groups of Finite Rank

We prove that any irreducible faithful representation of an almost torsion-free Abelian group G of finite rank over a finitely generated field of characteristic zero is induced from an irreducible representation of a finitely generated subgroup of the group G.     

Spectra of elements in the range of strongly continuous unitary representations of Lie groups

The main result proved in this work is that, for a strongly continuous unitary representation of a Lie group, we have the following dichotomy: The representation is norm continuous or on an open neighborhood of the unit (which is the whole group for a weakly exponential Lie group), there is a residual set of elements whose range admits the entire torus as spectrum.     

Graphical Calculus for the Double Affine Q-Dependent Braid Group

In this paper, we present a straightforward pictorial representation of the double affine Hecke algebra (DAHA) which enables us to translate the abstract algebraic structure of a DAHA into an intuitive graphical calculus suitable for a physics audience. Initially, we define the larger double affine Q-dependent braid group. This group is constructed by appending to the braid group a set of operators Q _i , before extending it to an affine Q-dependent braid group. We show specifically that the elliptic braid group and the DAHA can be obtained as quotient groups. Complementing this, we present a pictorial representation of the double affine Q-dependent braid group based on ribbons living in a toroid. We show that in this pictorial representation, we can fully describe any DAHA. Specifically, we graphically describe the parameter q upon which this algebra is dependent and show that in this particular representation q corresponds to a twist in the ribbon.     

Homotopy equivalent group representations and Picard groups op the burnside ring and the character ring

We are concerned with the homotopy theory of group representations and its relation to character theory and the theory of the Burnside ring. We combine the methods of tom Dieck — Petrie [4] and torn Dieck [3] to show that the canonical map from the J-group jO(G), a subquotient of the representation ring RO(G), into the Picard group of the rational representation ring is injective for p-groups G. Moreover we compute the order of the cokernel of this map. We show that the Picard group of the rational representation ring is a direct summand in the Picard group of the Burnside ring. Finally we compute the Picard groups if G is abelian and indicate a computation for general G.     

On decomposition theory for unitary representations of locally compact groups

The main purpose of this note is to give a simple example of a unitary representation of a (discrete) group, which cannot be decomposed as a topological direct integral of factor representations. But it is shown that the regular representation of any discrete group is a direct integral (on a compact space) of factor representations.     

On Primitive Representations of Finite Alternating and Symmetric Groups with a 2-Transitive Subconstituent

In this paper we analyse primitive permutation representations of finite alternating and symmetric groups which have a 2-transitive subconstituent. We show that either the representation belongs to an explicit list of known examples, or the point stabiliser is a known almost-simple 2-transitive group and acts primitively in the natural representation of the associated alternating or symmetric group.