Group Elements Whose Character Values are Roots of Unity

Closed paths whose steps are roots of unity

We give explicit formulas for the number Un(N) of closed polygonal paths of length N (starting from the origin) whose steps are n roots of unity, as well as asymptotic expressions for these numbers when N → ∞. We also prove that the sequences (Un(N))N≥0 are P -recursive for each fixed n ≥ 1 and leave open the problem of determining the values of N for which the dual sequences (Un(N))n≥1 are P -...

Multiple polylogarithm values at roots of unity

For any positive integer N let μN be the group of the N th roots of unity. In this note we shall study the Q-linear relations among the values of multiple polylogarithms evaluated at μN . We show that the standard relations considered by Racinet do not provide all the possible relations in the following cases: (i) level N = 4, weight w = 3 or 4, and (ii) w = 2, 7 <N < 50, and N is a power of 2 ...

On group Elements Having Square Roots

Character Values at Regular Elements

There has been a good deal of work in the last decade or so on computing upper bounds of irreducible character values for finite simple groups [G1, G2, GM, LS2, LP, MS-P, R]. Some of this work concerns permutation groups, some groups of Lie type. Some is aimed at worst-case behavior, some at the character values of typical or at least fairly well-behaved elements. Various applications have been...

Standard Relations of Multiple Polylogarithm Values at Roots of Unity

Let N be a positive integer. In this paper we shall study the special values of multiple polylogarithms atNth roots of unity, called multiple polylogarithm values (MPVs) of levelN . These objects are generalizations of multiple zeta values and alternating Euler sums, which was studied by Euler, and more recently, many mathematicians and theoretical physicists.. Our primary goal in this paper is...