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In a Russian text, on the topic of Permutation Groups, the author introduces the concept of the decrement of a permutation defined as follows: (Note that this is my translation of the text, so it might be inaccurate):

Definition: The permutation decrement is the difference between the number of moving elements(*) and the number of independent cycles of length $\geq 2$ in the canonical decomposition of the permutation or, equivalently, the difference between the number of all elements of the set and the number of independent cycles (including 1-cycles) in the canonical decomposition of the permutations.

If the decrement is even, then the permutation is called even. Otherwise, the permutation is called odd.

(*) Moving elements are those that are not mapped to themselves by the permutation. I.e. $a$ is a moving element if $a \in \{ a \space | \space \pi (a_i) \neq a_i , \ \space i \in \{1, ... , n\}\}$. I'm not sure of the formal way of referring to such elements.

Coming to the question, I ran a number of searches on the decrement of a permutation but I can't find the term "decrement" used in any of the textbooks I checked (J. Gallian, M. Artin, Vinberg, Allan Clark and others) thus, I'm guessing it is a non-standard translation. Additionally, most texts seem to use other methods of determining the parity of a permutation (counting the number of transpositions). So, does anyone know of the right terminology for this definition and a source of notes on this concept of a "decrement"?

E.Nole
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1 Answers1

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A probably more widespread synonym for this notion of "decrement" is "reflection length", because the "decrement" of a permutation $w \in S_n$ is the minimum number of transpositions needed if you want to write $w$ as a product of transpositions (and from the viewpoint of Coxeter groups, transpositions are reflections). See, e.g., arXiv:1202.4765v3 for an example of the use of this concept.

To prove that the "decrement" of $w$ really is this minimum number, you just need two observations:

(1) When you compose a permutation $\sigma$ with a transposition $t_{i, j}$, the resulting permutation $\sigma \circ t_{i, j}$ will either have one fewer cycle than $\sigma$ or one more cycle than $\sigma$. (More precisely: It will have one fewer cycle if $i$ and $j$ belong to different cycles of $\sigma$, and otherwise it will have one more cycle.)

(2) Every $k$-cycle can be written as a composition of $k-1$ transpositions.

Both should be well-known and easy to prove.

As for the word "decrement", I have seen it in a few places, but I am not sure of its provenance and would not be surprised if it arose from confusion. I prefer using "decrement" as a verb for making something smaller (preferrably by $1$).

American texts tend to define the sign of a permutation using its inversions rather than its cycles (and I agree: inversions are cleaner to work with if you don't want to wave hands). And when they do count cycles, they consider the number of cycles rather than $n$ minus this number. This is probably why you aren't seeing the notion of "decrement" too often.

darij grinberg
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