scholarly journals Degree of reductivity of a modular representation

2017 ◽  
Vol 19 (03) ◽  
pp. 1650023 ◽  
Author(s):  
Martin Kohls ◽  
Müfi̇t Sezer
Keyword(s):  
Fixed Point ◽  
Lower Bound ◽  
Cyclic Subgroup ◽  
Representation Formula ◽  
Modular Representation ◽  
Dimensional Representation ◽  
Finite Dimensional Representation ◽  
Finite Dimensional ◽  
Modular Groups ◽  
Sharp Lower Bound

For a finite-dimensional representation [Formula: see text] of a group [Formula: see text] over a field [Formula: see text], the degree of reductivity [Formula: see text] is the smallest degree [Formula: see text] such that every nonzero fixed point [Formula: see text] can be separated from zero by a homogeneous invariant of degree at most [Formula: see text]. We compute [Formula: see text] explicitly for several classes of modular groups and representations. We also demonstrate that the maximal size of a cyclic subgroup is a sharp lower bound for this number in the case of modular abelian [Formula: see text]-groups.

scholarly journals Holomorphic Cohomology of Complex Analytic Linear Groups

1966 ◽  
Vol 27 (2) ◽  
pp. 531-542 ◽  
Author(s):  
G. Hochschild ◽  
G. D. Mostow
Keyword(s):  
Linear Group ◽  
Structure Theory ◽  
Linear Groups ◽  
Dimensional Representation ◽  
Lie Algebra Cohomology ◽  
Analytic Group ◽  
Finite Dimensional Representation ◽  
Semidirect Products ◽  
Finite Dimensional ◽  
Rational Cohomology

Let G be a complex analytic group, and let A be the representation space of a finite-dimensional complex analytic representation of G. We consider the cohomology for G in A, such as would be obtained in the usual way from the complex of holomorphic cochains for G in A. Actually, we shall use a more conceptual categorical definition, which is equivalent to the explicit one by cochains. In the context of finite-dimensional representation theory, nothing substantial is lost by assuming that G is a linear group. Under this assumption, it is the main purpose of this paper to relate the holomorphic cohomology of G to Lie algebra cohomology, and to the rational cohomology, in the sense of [1], of algebraic hulls of G. This is accomplished by using the known structure theory for complex analytic linear groups in combination with certain easily established results concerning the cohomology of semidirect products. The main results are Theorem 4.1 (whose hypothesis is always satisfied by a complex analytic linear group) and Theorems 5.1 and 5.2. These last two theorems show that the usual abundantly used connections between complex analytic representations of complex analytic groups and rational representations of algebraic groups extend fully to the superstructure of cohomology.

scholarly journals Random walks on projective spaces

2014 ◽  
Vol 150 (9) ◽  
pp. 1579-1606 ◽  
Author(s):  
Yves Benoist ◽  
Jean-François Quint
Keyword(s):  
Local Fields ◽  
Probability Measures ◽  
Stationary Probability ◽  
Semisimple Lie Group ◽  
Dimensional Representation ◽  
Dense Subgroup ◽  
Finite Dimensional Representation ◽  
Empirical Measures ◽  
Finite Dimensional ◽  
Strongly Irreducible

AbstractLet $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}G$ be a connected real semisimple Lie group, $V$ be a finite-dimensional representation of $G$ and $\mu $ be a probability measure on $G$ whose support spans a Zariski-dense subgroup. We prove that the set of ergodic $\mu $-stationary probability measures on the projective space $\mathbb{P}(V)$ is in one-to-one correspondence with the set of compact $G$-orbits in $\mathbb{P}(V)$. When $V$ is strongly irreducible, we prove the existence of limits for the empirical measures. We prove related results over local fields as the finiteness of the set of ergodic $\mu $-stationary measures on the flag variety of $G$.

GENERALIZED HECKE ALGEBRAS AND C*-COMPLETIONS

2009 ◽  
Vol 20 (01) ◽  
pp. 45-76
Author(s):  
MAGNUS B. LANDSTAD ◽  
NADIA S. LARSEN
Keyword(s):  
Heisenberg Group ◽  
Hecke Algebra ◽  
Continuous Extension ◽  
Finite Range ◽  
Structure And Properties ◽  
Projection Formula ◽  
Dimensional Representation ◽  
Finite Dimensional Representation ◽  
Finite Dimensional ◽  
The Given

For a Hecke pair (G, H) and a finite-dimensional representation σ of H on Vσ with finite range, we consider a generalized Hecke algebra [Formula: see text], which we study by embedding the given Hecke pair in a Schlichting completion (Gσ, Hσ) that comes equipped with a continuous extension σ of Hσ. There is a (non-full) projection [Formula: see text] such that [Formula: see text] is isomorphic to [Formula: see text]. We study the structure and properties of C*-completions of the generalized Hecke algebra arising from this corner realisation, and via Morita–Fell–Rieffel equivalence, we identify, in some cases explicitly, the resulting proper ideals of [Formula: see text]. By letting σ vary, we can compare these ideals. The main focus is on the case with dim σ = 1 and applications include ax + b-groups and the Heisenberg group.

scholarly journals SOLVABILITY OF THE F4 INTEGRABLE SYSTEM

2001 ◽  
Vol 16 (29) ◽  
pp. 4769-4801 ◽  
Author(s):  
KONSTANTIN G. BORESKOV ◽  
JUAN CARLOS LOPEZ VIEYRA ◽  
ALEXANDER V. TURBINER
Keyword(s):  
Weyl Group ◽  
Differential Operators ◽  
Coupling Constants ◽  
Dimensional Representation ◽  
Algebraic Form ◽  
Finite Dimensional Representation ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Exactly Solvable ◽  
New Variables

It is shown that the F4 rational and trigonometric integrable systems are exactly-solvable for arbitrary values of the coupling constants. Their spectra are found explicitly while eigenfunctions are by pure algebraic means. For both systems new variables are introduced in which the Hamiltonian has an algebraic form being also (block)-triangular. These variables are invariant with respect to the Weyl group of F4 root system and can be obtained by averaging over an orbit of the Weyl group. An alternative way of finding these variables exploiting a property of duality of the F4 model is presented. It is demonstrated that in these variables the Hamiltonian of each model can be expressed as a quadratic polynomial in the generators of some infinite-dimensional Lie algebra of differential operators in a finite-dimensional representation. Both Hamiltonians preserve the same flag of spaces of polynomials and each subspace of the flag coincides with the finite-dimensional representation space of this algebra. Quasi-exactly-solvable generalization of the rational F4 model depending on two continuous and one discrete parameters is found.

scholarly journals A QUADRATIC DEFORMATION OF THE HEISENBERG–WEYL AND QUANTUM OSCILLATOR ENVELOPING ALGEBRAS

1993 ◽  
Vol 08 (20) ◽  
pp. 3479-3493 ◽  
Author(s):  
JENS U. H. PETERSEN
Keyword(s):  
Virasoro Algebra ◽  
Heisenberg Algebra ◽  
Roots Of Unity ◽  
Quantum Oscillator ◽  
Dimensional Representation ◽  
Finite Dimensional Representation ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Quadratic Deformation ◽  
Two Parameter

A new two-parameter quadratic deformation of the quantum oscillator algebra and its one-parameter deformed Heisenberg subalgebra are considered. An infinite-dimensional Fock module representation is presented, which at roots of unity contains singular vectors and so is reducible to a finite-dimensional representation. The semicyclic, nilpotent and unitary representations are discussed. Witten's deformation of sl 2 and some deformed infinite-dimensional algebras are constructed from the 1d Heisenberg algebra generators. The deformation of the centerless Virasoro algebra at roots of unity is mentioned. Finally the SL q(2) symmetry of the deformed Heisenberg algebra is explicitly constructed.

scholarly journals The solution of the functional equation of D'Alembert's type for commutative groups

1982 ◽  
Vol 5 (2) ◽  
pp. 315-335 ◽  
Author(s):  
A. L. Rukhin
Keyword(s):  
Functional Equation ◽  
Commutative Group ◽  
Matrix Elements ◽  
Dimensional Representation ◽  
Finite Dimensional Representation ◽  
Finite Dimensional

A functional equation of the formϕ1(x+y)+ϕ2(x−y)=∑inαi(x)βi(y), where functionsϕ1,ϕ2,αi,βi,i=1,…,nare defined on a commutative group, is solved. We also obtain conditions for the solutions of this equation to be matrix elements of a finite dimensional representation of the group.

scholarly journals Diagonal torsion matrices associated with modular data

2021 ◽  
Vol 32 (1) ◽  
pp. 127-137
Author(s):  
G. Singh ◽  
Keyword(s):  
Modular Group ◽  
Dimensional Representation ◽  
Finite Dimensional Representation ◽  
Finite Dimensional ◽  
Modular Data ◽  
Mathematics And Physics

Modular data are commonly studied in mathematics and physics. A modular datum defines a finite-dimensional representation of the modular group SL2(Z). Cuntz (2007) defined isomorphic integral modular data. Here we discuss isomorphic integral and non-integral modular data as well as non-isomorphic but closely related modular data. In this paper, we give some insights into diagonal torsion matrices associated to modular data.

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