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The spectral gap of a Markov chain can be bounded by the spectral gaps ofconstituent "restriction" chains and a "projection" chain, and the strength ofsuch 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. Wefurther define a notion of orthogonality between the restriction chains and"complementary" restriction chains. This leads to a new ComplementaryDecomposition 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 overallspectral gap. As an application, we provide a $1/n$-orthogonal decomposition ofthe nearest neighbor Markov chain over $k$-class biased monotone permutationson [$n$], as long as the number of particles in each class is at least $C\logn$. This allows us to apply the Complementary Decomposition theorem iteratively$n$ times to prove the first polynomial bound on the spectral gap when $k$ isas large as $\Theta(n/\log n)$. The previous best known bound assumed $k$ wasat most a constant.

Iterated Decomposition of Biased Permutations Via New Bounds on the Spectral Gap of Markov Chains