4 Mutations
4.1 Definition of mutations
A pair \((X, Y)\) of chromosomes forms a mutation if:
\(X \ne Y\),
\(\mathrm{sig}(X) = \mathrm{sig}(Y)\),
\(Y \le X\) (dominance).
4.2 Primitive mutations on Pi
Given \(\varepsilon \), \(k \le m\): \(X_1 = (m, \varepsilon ) + (k, -\varepsilon )\) and \(Y_1 = (m+1, -\varepsilon ) + (k-1, \varepsilon )\) (when \(k \ge 1\)).
Given \(\varepsilon \), \(k \le m\): \(X_2 = (m, \varepsilon ) + (k, \varepsilon )\) and \(Y_2 = (m+1, \varepsilon ) + (k-1, \varepsilon )\) (when \(k \ge 1\)).
Given \(\varepsilon \), \(k \le m\): \(X_3 = (m, \varepsilon ) + (k, -\varepsilon )\) and \(Y_3 = (m+2, -\varepsilon ) + (k-2, \varepsilon )\) (when \(k \ge 2\)).
A primitive mutation on \(\mathrm{Pi}\) is one of the three types above (up to adding a common chromosome \(Z\)).
A step on \(\mathrm{Pi}\) is a primitive mutation plus an arbitrary summand: \(\mathrm{Step}(X+Z, Y+Z)\) whenever \(\mathrm{Primitive}(X, Y)\).
4.3 Properties of mutations on Pi
Every primitive mutation satisfies the mutation conditions.
Type 1 mutations preserve the signature at all levels: \(\mathrm{sig}(X_1^{(k)}) = \mathrm{sig}(Y_1^{(k)})\) for all \(k\).
Type 1 mutations satisfy \(Y_1 \le X_1\).
Type 2 mutations preserve the signature at all levels.
Type 2 mutations satisfy \(Y_2 \le X_2\).
Type 3 mutations preserve the signature at all levels.
Type 3 mutations satisfy \(Y_3 \le X_3\).
4.4 Main mutation theorem
For any \(X, Y \in \mathrm{Pi}\) with \(\mathrm{rank}(X) = \mathrm{rank}(Y) = n\) and \(Y \le X\), either \(X = Y\) or there exists a step mutation from \(X\) towards \(Y\): there exist \(X' \in \mathrm{Pi}\) with \(\mathrm{Step}(X, X')\) and \(Y \le X' {\lt} X\).