Let F = C(x1, . . . , x`, x`+1, . . . , xm), where x1, . . . , x` are continuous variables, and x`+1, . . . , xm are discrete variables. We show that a hyperexponential function, which is algebraic over F , is of form g(x1, . . . , xm)q(x1, . . . , x`) 1 t ω x`+1 j+1 · · ·ωxm m where g ∈ F , q ∈ C(x1, . . . , x`), t ∈ Z and ω`+1, . . . , ωm are roots of unity. Furthermore, we present an algorit...

We study sums of the form ∑ ζ R(ζ), where R is a rational function and the sum is over all nth roots of unity ζ (often with ζ = 1 excluded). We call these generalized Dedekind sums, since the most well-known sums of this form are Dedekind sums. We discuss three methods for evaluating such sums: The method of factorization applies if we have an explicit formula for ∏ ζ(1− xR(ζ)). Multisection ca...

Some Realization of gl q (n)-covariant Oscillator Algebra and gl q (n)-covariant q-Virasoro Algebra with q a root of unity. Abstract In this paper some realization of gl q (n)-covariant oscillators is obtained when q is a root of unity. And the gl q (n)-covariant q-Virasoro algebra is presented by using the gl q (n)-covariant oscillators.

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In this paper we are concerned with the problem of determining the character values and Schur indices of a finite group of Lie type over Fq. We show that (under some conditions on q) these values lie in the ring of algebraic integers generated by (1 + ñq)/2 and roots of unity of order prime to q. Furthermore, we determine the Schur indices for some of the (nonrational) unipotent characters in ...

We consider the problem of classifying irreducible Specht modules for the Iwahori–Hecke algebra of type B with parameters Q, q. We solve this problem completely in the case where q is not a root of unity, and in the case q = −1 we reduce the problem to the corresponding problem in type A.

Let p be a prime, F a field containing a primitive pth root of unity, and E/F a cyclic extension of degree p. Using the Bloch–Kato Conjecture we determine precise conditions for the cohomology group Hn(E) := H(GE ,Fp) to be free or trivial as an Fp[Gal(E/F )]-module, and we examine when these properties for Hn(E) are inherited by Hk(E), k > n. By analogy with cohomological dimension, we introdu...

Our goal in this section will be to collect some standard examples of finite groups. The main emphasis will be on realizing them as subgroups of GL(n,R) or GL(n,C). Cyclic groups: For a natural number n, let Z/nZ be the standard cyclic group of order n. We denote its elements by 0, 1, . . . , n − 1, and 1 is a generator. Another model for the cyclic group of order n is the nth roots of unity, o...

Polylogarithms appear in many diverse fields of mathematics. Herein, we investigate relations amongst Nielsen polylogarithms, both generic and at special arguments including the sixth roots of unity. The relations are explicitly exhibited in the case of low weights. All of this work relies heavily on symbolic and numeric computation. For use in applications all results are implemented and acces...