MO640 - Exercises - Algebraic theory, Feijao and Meidanis 2013

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4. Find a sorting rearrangement with weight 1 or less taking π to σ of the previous exercise, according to the algebraic concept of distance.

Answer:

Algebraic representation of the genomes:

π_chr = (a+ b+ d- f+ c- e+ e- c+ f- d+ b- a-)
σ_chr = (d+ f+ f- d-)(a+ c- b- e+ e- b+ c+ a-)(z+ z-)

We need to find a divisor of σ_chr π_chr^-1, then invert and conjugate it by π_adj and see if they are disjoint. So π_chr^-1 and σ_chr π_chr^-1 must be calculated:

π_chr^-1 = (a- b- d+ f- c+ e- e+ c- f+ d- b+ a+)
σ_chr π_chr^-1 = (a- e+ b- f+ d+ d- c+ b+ c- f-)(z+ z-)

We choose μ = (a- d-), which is equal to μ^-1. Its conjugate is:

π_adj μ π_adj^-1 = (b- b+)

Since μ and π_adj μ^-1 π_adj^-1 are disjoint, their product is a sorting operation:

sorting operation = (b- b+)(a- d-).

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