Recursive structure of MOS scales: Difference between revisions
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Without loss of generality assume r ≥ 1 (otherwise flip the roles of L and s). Let W'(λ, σ) be the reduced word with step sizes λ (corresponding to the chunk of L's of size r+1) and σ (corresponding the chunk of size r), and assume that W' is not a mos. Then for some k, W' must have k-steps of the following sizes: | Without loss of generality assume r ≥ 1 (otherwise flip the roles of L and s). Let W'(λ, σ) be the reduced word with step sizes λ (corresponding to the chunk of L's of size r+1) and σ (corresponding the chunk of size r), and assume that W' is not a mos. Then for some k, W' must have k-steps of the following sizes: | ||
# p₁ λ's and q₁ σ's, represented by subword W₁(λ, σ) in W'. | # p₁ λ's and q₁ σ's, represented by subword W₁(λ, σ) in W'. | ||
# p₂ λ's and q₂ σ's, represented by subword W₂(λ, σ) in W'. | # p₂ λ's and q₂ σ's, represented by subword W₂(λ, σ) in W'. We can assume W₂ begins in λ; otherwise we can slink W₂ to the right until it begins in λ, which is guaranteed never decrease the number of λ's | ||
Here, pᵢ + qᵢ = k and we assume p₂ - p₁ ≥ 2. | Here, pᵢ + qᵢ = k and we assume p₂ - p₁ ≥ 2. | ||