Braided Line and Counting Fixed Points of GL(d,F_q)

Details Braided Line and Counting Fixed Points of GL(d,F_q) Braided Line and Counting Fixed Points of GL(d,F_q)

2001-12-22

arxiv.org/abs/math/0112258v1

Mathematics

Quantum Algebra

Scientific paper

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 $-1/(q-1)$.

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