Braided Line and Counting Fixed Points of GL(d,F_q)
Abstract
We interpret a recent formula for counting orbits of $GL(d,F_q)$ in terms of counting fixed points as addition in the affine braided line. The theory of such braided groups (or Hopf algebras in braided categories) allows us to obtain the inverse relationship, which turns out to be the same formula but with $q$ and $q^{1}$ interchanged (a perfect duality between counting orbits and counting fixed points). In particular, the probability that an element of $GL(d,F_q)$ has no fixed points is found to be the order$d$ truncated $q$exponential of $1/(q1)$.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 December 2001
 arXiv:
 arXiv:math/0112258
 Bibcode:
 2001math.....12258C
 Keywords:

 Quantum Algebra;
 Number Theory