  The spectral gap of a Markov chain can be bounded by the spectral gaps of
constituent "restriction" chains and a "projection" chain, and the strength of
such a bound is the content of various decomposition theorems. In this paper,
we introduce a new parameter that allows us to improve upon these bounds. We
further define a notion of orthogonality between the restriction chains and
"complementary" restriction chains. This leads to a new Complementary
Decomposition theorem, which does not require analyzing the projection chain.
For $\epsilon$-orthogonal chains, this theorem may be iterated $O(1/\epsilon)$
times while only giving away a constant multiplicative factor on the overall
spectral gap. As an application, we provide a $1/n$-orthogonal decomposition of
the nearest neighbor Markov chain over $k$-class biased monotone permutations
on [$n$], as long as the number of particles in each class is at least $C\log
n$. This allows us to apply the Complementary Decomposition theorem iteratively
$n$ times to prove the first polynomial bound on the spectral gap when $k$ is
as large as $\Theta(n/\log n)$. The previous best known bound assumed $k$ was
at most a constant.
