  Let $G$ be a finite group. Let $H, K$ be subgroups of $G$ and $H \backslash G
/ K$ the double coset space. Let $Q$ be a probability on $G$ which is constant
on conjugacy classes ($Q(s^{-1} t s) = Q(t)$). The random walk driven by $Q$ on
$G$ projects to a Markov chain on $H \backslash G /K$. This allows analysis of
the lumped chain using the representation theory of $G$. Examples include
coagulation-fragmentation processes and natural Markov chains on contingency
tables. Our main example projects the random transvections walk on $GL_n(q)$
onto a Markov chain on $S_n$ via the Bruhat decomposition. The chain on $S_n$
has a Mallows stationary distribution and interesting mixing time behavior. The
projection illuminates the combinatorics of Gaussian elimination. Along the
way, we give a representation of the sum of transvections in the Hecke algebra
of double cosets. Some extensions and examples of double coset Markov chains
with $G$ a compact group are discussed.
