Polynomial representations of the symplectic groups

The irreducible finite dimensional representations of the symplectic groups are realized as polynomials on the irreducible representation spaces of the corresponding general linear groups. It is shown that the number of times an irreducible representation of a maximal symplectic subgroup occurs in a given representation of a symplectic group, is related to the betweenness conditions of representations of the corresponding general linear groups. Using this relation, it is shown how to construct polynomial bases for the irreducible representation spaces of the symplectic groups in which the basis labels come from the representations of the symplectic subgroup chain, and the multiplicity labels come from representations of the odd dimensional general linear groups, as well as from subgroups. The irreducible representations of Sp(4) are worked out completely, and several examples from Sp(6) are given.     

Representations of cyclic groups acting on complete simplicial fans

Let σ be a complete simplicial fan in finite dimensional real Euclidean space V, and let G be a cyclic subgroup of GL(V) which acts properly on σ. We show that the representation of G carried by the cohomology of Xσ, the toric variety associated to σ, is a permutation representation.     

Lattice universality of free burnside groups

We generalize Whitman's theorem on the representation of lattices by partition lattices or, which is the same, by subgroup lattices of a suitable group. A sufficient condition is stated for a group variety to be lattice-universal (i.e., every lattice has a presentation by the subgroup lattice of a group in this variety). As a consequence, we infer that every couniable lattice is representable by the subgroup lattice of a finitely generated free Burnside group of a large enough odd exponent. Translated fromAlgebra i Logika, Vol. 35, No. 5, pp. 587–611, September–October, 1996.     

Parseval frames for ICC groups

We analyze Parseval frames generated by the action of an ICC group on a Hilbert space. We parametrize the set of all such Parseval frames by operators in the commutant of the corresponding representation. We characterize when two such frames are strongly disjoint. We prove an undersampling result showing that if the representation has a Parseval frame of equal norm vectors of norm , the Hilbert space is spanned by an orthonormal basis generated by a subgroup. As applications we obtain some sufficient conditions under which a unitary representation admits a Parseval frame which is spanned by a Riesz sequences generated by a subgroup. In particular, every subrepresentation of the left-regular representation of a free group has this property.     

On the first cohomology of cocompact arithmetic groups

Results of Matsushima and Raghunathan imply that the first cohomology of a cocompact irreducible lattice in a semisimple Lie groupG, with coefficients in an irreducible finite dimensional representation ofG, vanishes unless the Lie group isSO(n, 1) orSU(n, 1) and the highest weight of the representation is an integral multiple of that of the standard representation. We show here that every cocompact arithmetic lattice inSO(n, 1) contains a subgroup of finite index whose first cohomology is non-zero when the representation is one of the exceptional types mentioned above.     

Regular representations of finite groups as automorphism groups of k‐tournaments

It was shown by Babai and Imrich [2] that every finite group of odd order except and admits a regular representation as the automorphism group of a tournament. Here, we show that for k ≥ 3, every finite group whose order is relatively prime to and strictly larger than k admits a regular representation as the automorphism group of a k‐tournament. Our constructions are elementary, suggesting that the problem is significantly simpler for k‐tournaments than for binary tournaments. © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 238–248, 2002     

Deformations of permutation representations of Coxeter groups

The permutation representation afforded by a Coxeter group W acting on the cosets of a standard parabolic subgroup inherits many nice properties from W such as a shellable Bruhat order and a flat deformation over ?[q] to a representation of the corresponding Hecke algebra. In this paper we define a larger class of “quasiparabolic” subgroups (more generally, quasiparabolic W-sets), and show that they also inherit these properties. Our motivating example is the action of the symmetric group on fixed-point-free involutions by conjugation.     

Pappus’ theorem and a construction of complex Kleinian groups with rich dynamics

Pappus’ theorem is used to produce and study discrete subgroups of PSL(3, ?) with very rich dynamics. We get an example of a subgroup of PSL(3, ?) which is not conjugate to any subgroup of PU(2, 1) nor to any subgroup of Aff(?²), its Kulkarni region of discontinuity is non-empty and its complement, the Kulkarni limit set, contains infinitely many complex projective lines in general position. This construction is based on a representation of the group SL(2, ?) in the projective group PSL(3, ?).     

On circle diffeomorphisms with discontinuous derivatives and quasi-invariance subgroups of Malliavin-Shavgulidze measures

This note is devoted to construction of the domain of the regular representation of Diff₊(S¹). We consider a subgroup of diffeomorphism group such that the Malliavin-Shavgulidze measure is quasi-invariant with respect to the left action of the subgroup. The measure appears to be quasi-invariant with respect to the left action of diffeomorphisms with discontinuous second derivative. We derive an expression for quasi-invariance density and obtain an additional PSL(2,R) symmetry breaking term.     

Semisimple types for p-adic classical groups

We construct, for any symplectic, unitary or special orthogonal group over a locally compact nonarchimedean local field of odd residual characteristic, a type for each Bernstein component of the category of smooth representations, using Bushnell–Kutzko’s theory of covers. Moreover, for a component corresponding to a cuspidal representation of a maximal Levi subgroup, we prove that the Hecke algebra is either abelian, or a generic Hecke algebra on an infinite dihedral group, with parameters which are, at least in principle, computable via results of Lusztig. In an appendix, we make a correction to the proof of a result of the second author: that every irreducible cuspidal representation of a classical group as considered here is irreducibly compactly-induced from a type.     

Fine Control over the Projective Characters of the Centre of a Group

In this article we will study the restriction of the irreducible projective characters of a finite group to a central subgroup. We will also consider under what conditions on the restriction of such irreducible projective characters to an abelian normal subgroup that we can deduce that the subgroup is central. Finally, we will investigate the relationship between the inertia subgroup and the absolute centralizer of a central subgroup relative to a fixed 2-cocycle of the group.     

Transitive groups with irreducible representations of bounded degree

A well-known theorem of Jordan states that there exists a function J(d) of a positive integer d for which the following holds: if G is a finite group having a faithful linear representation over ℂ of degree d, then G has a normal Abelian subgroup A with [G:A]≤J(d). We show that if G is a transitive permutation group and d is the maximal degree of irreducible representations of G entering its permutation representation, then there exists a normal solvable subgroup A of G such that [G:A]≤J(d) ^log ₂ ^d. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 223, 1995, pp. 108–119. Translated by S. A. Evdokimov.     

Braid groups are linear

The braid group can be defined as the mapping class group of the -punctured disk. A group is said to be linear if it admits a faithful representation into a group of matrices over . Recently Daan Krammer has shown that a certain representation of the braid groups is faithful for the case . In this paper, we show that it is faithful for all .

     

Nonstandard representations of locally compact groups

In the note, it is proved that, under natural conditions, any infinite-dimensional unitary representation T of a direct product of groups G = K × N, where K is a compact group and N is a locally compact Abelian group, is imaged by a representation of the nonstandard analog \(\tilde G\) of the group G in the group of nonstandard matrices of a fixed nonstandard size.     

On representations of affine Kac-Moody groups and related loop groups

We demonstrate a one to one correspondence between the irreducible projective representations of an affine Kac-Moody group and those of the related loop group, which leads to the results that every non-trivial representation of an affine Kac-Moody group must have its degree greater than or equal to the rank of the group and that the equivalence appears if and only if the group is of type for some . Moreover the characteristics of the base fields for the non-trivial representations are found being always zero.

     

On the symmetry of L1 algebras of locally compact motion groups,and the Wiener Tauberian theorem

Let G be a locally compact motion group, i.e., it is a semidirect product of a compact subgroup with a closed abelian normal subgroup, the action of the compact subgroup on the other one being by conjugation. The main result of this paper is that the group algebra of such a group is symmetric. This result is then used to prove that a generalization of the Wiener-Tauberian theorem holds for such groups. Precisely, it is shown that every proper closed two-sided ideal in L₁(G) is annihilated by an irreducible unitary representation of G, lifted to L₁(G).     

Separation of unitary representations of connected Lie groups by their moment sets

We show that every unitary representation π of a connected Lie group G is characterized up to quasi-equivalence by its complete moment set.Moreover, irreducible unitary representations π of G are characterized by their moment sets.     

Constructing representations of higher degrees of finite simple groups and covers

Let be a finite group and an irreducible character of . A simple method for constructing a representation affording can be used whenever has a subgroup such that has a linear constituent with multiplicity 1. In this paper we show that (with a few exceptions) if is a simple group or a covering group of a simple group and is an irreducible character of of degree between 32 and 100, then such a subgroup exists.

     

Geometric aspects of frame representations of abelian groups

We consider frames arising from the action of a unitary representation of a discrete countable abelian group. We show that the range of the analysis operator can be determined by computing which characters appear in the representation. This allows one to compare the ranges of two such frames, which is useful for determining similarity and also for multiplexing schemes. Our results then partially extend to Bessel sequences arising from the action of the group. We apply the results to sampling on bandlimited functions and to wavelet and Weyl-Heisenberg frames. This yields a sufficient condition for two sampling transforms to have orthogonal ranges, and two analysis operators for wavelet and Weyl-Heisenberg frames to have orthogonal ranges. The sufficient condition is easy to compute in terms of the periodization of the Fourier transform of the frame generators.