How to recover a permutation group amidst errors

Abstract

We consider the problem of recovering a permutation group $G \leq S_n$ from an error-prone sampling process $X$. We model $X$ as an $S_n$-valued random variable, defined as a mixture of the uniform distributions on $G$ and $S_n$ . Our suite of tools recovers properties of $G$ from $X$ and bolsters our main method for recovering $G$ itself. Our algorithms are motivated by the numerical computation of monodromy groups, a setting where such error-prone sampling procedures occur organically.

Figures (14)

• Figure 1: A flowchart of our main algorithm, \ref{['alg:main_algorithm']}.
• Figure 2: Young diagrams of twelve of the $22$ cycle types of elements of $S_8$ categorized by being giant, Jordan, or primitive, and decorated with the number of permutations of that cycle type. Other cycle types are imprimitive and non-Jordan.
• Figure 3: Markov chain for $X_{\mathcal{P}}$.
• Figure 4: (Left) The probabilities $\gamma(W(E_6),p,k)$ for various $p$ and $k$. (Right) Bounds on $p$ from \ref{['cor:prob_bounds_for_half_prob']} for which $\gamma(W(E_6),p,k)>1/2$.
• Figure 5: Results of \ref{['alg:naivealgorithm']} (left) and \ref{['alg:errordetectinggrouprecovery']} (right) on $10,000$ runs with $G=W(E_6)$, $p=0.75$, and $k=3$ and $\mathcal{Q}\colon$ non-giant.

Theorems & Definitions (67)

• Proposition 2.1: Burnside's Lemma
• Corollary 2.2
• Proposition 2.3
• Example 2.4
• Remark 2.5: Computing Jordan, Giant, and Primitive proportions
• Lemma 2.6
• proof
• Lemma 2.7
• proof
• Proposition 2.8: Luczak Pyber LuczakPyber