  Given a certain kind of linear representation of a reductive group, referred
to as a quasi-symmetric representation in recent work of \v{S}penko and Van den
Bergh, we construct equivalences between the derived categories of coherent
sheaves of its various geometric invariant theory (GIT) quotients for suitably
generic stability parameters. These variations of GIT quotient are examples of
more complicated wall crossings than the balanced wall crossings studied in
recent work on derived categories and variation of GIT quotients.
  Our construction is algorithmic and quite explicit, allowing us to: 1)
describe a tilting vector bundle which generates the derived category of such a
GIT quotient, 2) provide a combinatorial basis for the K-theory of the GIT
quotient in terms of the representation theory of G, and 3) show that our
derived equivalences satisfy certain relations, leading to a representation of
the fundamental groupoid of a "K\"ahler moduli space" on the derived category
of such a GIT quotient. Finally, we use graded categories of singularities to
construct derived equivalences between all Deligne-Mumford hyperk\"ahler
quotients of a symplectic linear representation of a reductive group (at the
zero fiber of the algebraic moment map and subject to a certain genericity
hypothesis on the representation), and we likewise construct actions of the
fundamental groupoid of the corresponding K\"ahler moduli space.
