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We consider some remarkable central elements of the universal enveloping algebraU(gl(n)) which we call quantum immanants.
      
The least upper bound for the degrees of elements in a system of generators turns out to be independent of the number of vector variables.
      
An important class consists of those that we calln-coset groups; they arise as orbit spaces of groupsG modulo a group of automorphisms withn elements.
      
We give a characterization of those elements ofW whose reduced expressions avoid substrings of the formsts wheres andt are noncommuting generators.
      
A basis is calledmonomial if each of its elements is the result of applying to a (fixed) highest weight vector a monomial in the Chevalley basis elementsYα, α a simple root, in the opposite Borel subalgebra.
      
Let ζ-1/2 be the inverse on the set of regular elements ofu of a square root of the discriminant of.
      
We prove that the group of invertible elements of an irreducible algebraic monoid is an algebraic group, open in the monoid.
      
We also describe the multiplication rules of the above basos elements ofY.
      
We introduce a notion of transversality for pairs of elements inS, and then study the action ofG on the set of triples of mutually transversal points inS.
      
We prove that the elements A\leqslant defined by Lusztig in a completion of the periodic module actually live in the periodic module (in the type A case).
      
In order to prove this, we compare, using the Schur duality, these elements with the Kashiwara canonical basis of an integrable module.
      
In case p >amp;gt; 0, assume G is defined and split over the finite field of p elements Fp.
      
Here we show that this statement remains true for extensions of finite complex reflection groups by elements in their normalizer.
      
For a finite-dimensional representation $\rho: G \rightarrow \mathrm{GL}(M)$ of a group G, the diagonal action of G on $M^p,$ p-tuples of elements of M, is usually poorly understood.
      
From elements of the invariant algebra C[V]G we obtain, by polarization, elements of C[kV]G, where k ≥ 1 and kV denotes the direct sum of k copies of V.
      
Implementing this point of view, Poisson Summation Formulas are proved in several spaces including integrable functions of bounded variation (where the result is known) and elements of mixed norm spaces.
      
The basic technique uses factorization of group elements and Gel'fand-Tsetlin bases to simplify the computations, and may be extended to treat the computation of Fourier transforms of finitely supported distributions on the group.
      
A frame in a Hilbert space allows every element in to be written as a linear combination of the frame elements, with coefficients called frame coefficients.
      
The matrix elements of the irreducible unitary representations are calculated and the Fourier transform of functions on the motion group is defined.
      
A continuous frame is a family of vectors in a Hilbert space which allows reproductions of arbitrary elements by continuous superpositions.
      
 

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