We obtain two formulae for the higher Frobenius-Schur indicators: one for a spherical fusion category in terms of the twist of its center and the other one for a modular tensor category in terms of its twist. The first one is a categorical generalization of an analogous result by Kashina, Sommerhäuser, and Zhu for Hopf algebras, and the second one extends Bantay's 2nd indicator formula for a conformal field theory to higher degree. These formulae imply the sequence of higher indicators of an object in these categories is periodic. We define the notion of Frobenius-Schur (FS-)exponent of a pivotal category to be the global period of all these sequences of higher indicators, and we prove that the FS-exponent of a spherical fusion category is equal to the order of the twist of its center. Consequently, the FS-exponent of a spherical fusion category is a multiple of its exponent, in the sense of Etingof, by a factor not greater than 2. As applications of these results, we prove that the exponent and the dimension of a semisimple quasi-Hopf algebra H have the same prime divisors, which answers two questions of Etingof and Gelaki affirmatively for quasi-Hopf algebras. Moreover, we prove that the FS-exponent of H divides dim(H) 4 . In addition, if H is a group-theoretic quasi-Hopf algebra, the FS-exponent of H divides dim(H) 2 , and this upper bound is shown to be tight.

In this paper, we obtain a canonical central element ν H for each semi-simple quasi-Hopf algebra H over any field k and prove that ν H is invariant under gauge transformations. We show that if k is algebraically closed of characteristic zero then for any irreducible representation of H which affords the character χ, χ(ν H ) takes only the values 0, 1 or -1, moreover if H is a Hopf algebra or a twisted quantum double of a finite group then χ(ν H ) is the corresponding Frobenius-Schur Indicator. We also prove an analog of a Theorem of Larson-Radford for split semi-simple quasi-Hopf algebra over any field k. Using this result, we establish the relationship between the antipode S, the values of χ(ν H ), and certain associated bilinear forms when the underlying field k is algebraically closed of characteristic zero.

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The definitions of the n th Gauss sum and the associated n th central charge are introduced for premodular categories C and n ∈ Z. We first derive an expression of the n th Gauss sum of a modular category C, for any integer n coprime to the order of the Tmatrix of C, in terms of the first Gauss sum, the global dimension, the twist and their Galois conjugates. As a consequence, we show for these n, the higher Gauss sums are d-numbers and the associated central charges are roots of unity. In particular, if C is the Drinfeld center of a spherical fusion category, then these higher central charges are 1. We obtain another expression of higher Gauss sums for de-equivariantization and local module constructions of appropriate premodular and modular categories. These expressions are then applied to prove the Witt invariance of higher central charges for pseudounitary modular categories.