Classically, a permutation on a set $X$ can be thought of in one of two different ways: either as a bijection from $X$ to itself, or as a pair of linear orderings on $X$. In the case of a set that is already equipped with a natural ordering (such as a finite ordinal $X = \{1,\dots,n\}$), these two ways of defining permutations are interchangeable, but they correspond to different abstract structures that are more salient in different contexts, and to different natural ways of comparing permutations. Typically, the definition of permutations-as-automorphisms is important in group theory, while the definition of permutations-as-linear-orders is more common in combinatorics. Both views of permutations are relevant to the theory of symmetric operads.
As automorphisms $\sigma : X \to X$ in Set, the permutations of $X$ naturally form a group under composition, called the symmetric group (or permutation group) on $X$. This group may be denoted $S_X$, $\Sigma_X$, or $X!$. When $X$ is the finite set $(n) = \{1,\dots,n\}$, then its symmetric group is a finite group of cardinality $n!$ = “$n$ factorial”, and one typically writes $S_n$ or $\Sigma_n$.
Two permutations $\sigma : X \to X$ and $\tau : Y \to Y$ are said to be conjugate (written $\sigma \cong \tau$) just in case there is a bijection $f : X \to Y$ such that $\sigma = f^{-1} \tau f$, or equivalently, when there is a commuting square of bijections:
Observe that conjugacy is an equivalence relation, and that conjugacy classes of permutations of $X$ are in one-to-one correspondence with partitions of $X$ (see below).
If $X$ is a set equipped with a linear ordering $\lt$, then a permutation of $X$ is the same thing as a second (unrelated) linear ordering $\lt_\sigma$ on $X$. Indeed, suppose we label the elements of $X$ according to the $\lt$ order as $x_1 \lt x_2 \lt \dots$, and according to the $\lt_\sigma$ order as $\sigma_1 \lt_\sigma \sigma_2 \lt_\sigma \dots$. Then we can define a bijection $\sigma : X \to X$ as the function sending $x_1 \mapsto \sigma_1$, $x_2 \mapsto \sigma_2$, and so on. Conversely, any bijection $\sigma : X \to X$ induces a linear ordering $\lt_\sigma$ on $X$ by defining $x \lt_\sigma x'$ iff $\sigma(x) \lt \sigma(x')$.
This way of viewing permutations naturally gives rise to “array notation” (or “one-line notation”), where a permutation $\sigma : X \to X$ is represented as the list of elements $\sigma = (\sigma_1,\sigma_2,\dots)$, and it may be contrasted with “cycle notation” (see below). Whereas cycle notation makes it easy to compare permutations for conjugacy, array notation leads to a different natural way of comparing permutations known as pattern containment.
Given two linearly ordered sets $(X,\lt_X)$ and $(Y,\lt_Y)$, a permutation (or “pattern”) $\sigma : X \to X$ is said to be contained in a permutation $\tau : Y \to Y$ (written $\sigma \preceq \tau$) just in case there exists a monotone injective function $f : X \to Y$ such that
for all $i,j \in X$, or in other words, if $\tau = (\tau_1,\tau_2,\dots)$ contains a subsequence of elements whose relative ordering (in $Y$) is the same as the relative ordering (in $X$) of the sequence $\sigma = (\sigma_1,\sigma_2,\dots)$. Note that this definition of $\sigma \preceq \tau$ is equivalent to asking for the existence of a commuting square:
where $f$ and $g$ are monotone injections (each uniquely determined by the other).
Whereas conjugacy defines an equivalence relation on the permutations of a particular set $X$, pattern containment defines a partial order on the permutations of arbitrary linearly ordered sets (which restricts to the discrete order on the permutations of $X$).
In combinatorics, one sometimes also wants a slight generalisation of the notion of permutation-as-linear-order: for any natural number $r$, an $r$-permutation of $X$ is an injective function $\sigma : (r) \to X$. An $r$-permutation corresponds to a list of $r$ distinct elements of $X$, so that for a finite set $X$ of cardinality $n = |X|$, an $n$-permutation is the same thing as an ordinary permutation of $X$ (it is surjective and therefore bijective, since Set is a balanced category). More generally, the number of $r$-permutations of a set of cardinality $n$ is counted by the falling factorial $n^{\underline{r}} = n(n-1)\dots(n-r+1)$.
As an element of the symmetric group $S_X$, every permutation $\sigma : X \to X$ generates a cyclic subgroup $\langle \sigma \rangle$, and hence inherits a group action on $X$. The orbits of this action partition the set $X$ into a disjoint union of cycles, called the cyclic decomposition of the permutation $\sigma$.
For example, let $\sigma$ be the permutation on $(6)$ defined by
The domain of the permutation is partitioned into three $\langle\sigma\rangle$-orbits
corresponding to the three cycles
We can express this more compactly by writing $\sigma$ in “cycle notation”, as the composition $\sigma = (1)(2\,4\,6)(3\,5)$, or $\sigma = (2\,4\,6)(3\,5)$ leaving implicit the action of the identity (1).
Let $n \in \mathbb{N}$ and $\Sigma(n)$ the symmetric group on $n$ elements. Then the conjugacy classes of elements of $\Sigma(n)$, hence of permutations of $n$ elements, correspond to the cycle structures: two elements are conjugate to each other precisely if they have the same number of distinct cycles of the same length, or in other words if they define the same underlying partition of $n$.
For the symmetric group on three elements there are three such classes:
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One may regard the symmetric group $S_n$ as the general linear group in dimension $n$ on the field with one element $GL(n,\mathbb{F}_1)$.
The classifying space $B \Sigma(n)$ of the symmetric group on $n$ elements may be presented by $Emb(\{1,\cdots, n\}, \mathbb{R}^\infty)/\Sigma(n)$, the Fadell's configuration space on $n$ unordered points in $\mathbb{R}^\infty$.
Write $\tau_n$ for the rank $n$ vector bundle over this which exhibits the canonical action of $\Sigma(n)$ on $\mathbb{R}^n$, by permutation of coordinates.
The Thom space $B \Sigma(n)^{\tau_n}$ of this bundle appears as the cofficients of the spectral symmetric algebra of the “absolute spectral superpoint” $Sym_{\mathbb{S}} \Sigma \mathbb{S}$ (see Rezk 10, slide 4).
See also (Hopkins-Mahowald-Sadofsky 94, around def. 2.8)
The symmetric groups and alternating groups are the first stages in a restriction of the Whitehead tower of the orthogonal group to “finite discrete ∞-groups” in the sense of homotopy type with finite homotopy groups. The homotopy fibers of the stages of the “finite Whitehead tower” are the stable homotopy groups of spheres (Epa-Ganter 16). (See also at super algebra – Abstract idea and at Platonic 2-group.)
on the right: the delooped smooth ∞-group Whitehead tower of the orthogonal group (fivebrane 6-group $\to$ string 2-group $\to$ spin group $\to$ special orthogonal group $\to$ orthogonal group);
in the middle, its restriction to deloopings of finite groups and their universal ∞-group extensions ($\cdots \to$ covering of alternating group $\to$ alternating group $\to$ symmetric group)
on the left the homotopy fibers of each stage.
Notice that the squares on the right are not homotopy pullback squares. (The homotopy pullback of the string 2-group along $\tilde A \hookrightarrow Spin(n)$ is a $\mathbf{B}U(1)$-extension of $\tilde A$, but here we get the universal finite 2-group extension, by $\mathbb{Z}/24$ instead.
Narthana Epa, Nora Ganter, Platonic and alternating 2-groups, Higher Structures 1(1):122-146, 2017 (arXiv:1605.09192)
Michael Hopkins, Mark Mahowald, Hal Sadofsky, Constructions of elements in Picard groups, Contemporary mathematics, Volume 158, 1994 in Eric Friedlander, Mark Mahowald (eds.), Topology and Representation theory (doi:10.1090/conm/158/01454)
For the operadic structure of permutations, see Volume 1, Chapter 1 of:
For more on permutation patterns, see:
Wikipedia, Permutation pattern.
Étienne Ghys, A Singular Mathematical Promenade, August 2017. arXiv:1612.06373 author pdf
Last revised on September 13, 2018 at 04:28:47. See the history of this page for a list of all contributions to it.