Quelle notation.xml Sprache: XML

<Chapter Label="Notation">
<Heading>Notation</Heading>

This chapter explains the notation
of the package &WPE;,
mainly influenced by the accompanying publication <Cite Key="WPE"/>.

<Section Label="Wreath Product">
<Heading>Wreath Products</Heading>

Let <M>G = K \wr H</M> be a wreath product of two groups,
where <M>H</M> is a permutation group of degree <M>m</M>.
The wreath product is defined as the semidirect product
of the function space <M>K^m</M> with <M>H</M>,
where <M>\pi \in H</M> acts on <M>f \in K^m</M>
by setting <M>f^{{\pi}} : \{1, \ldots, m\} \rightarrow K, i \mapsto [(i)\pi^{{-1}}]f</M>.
Note that <M>G</M> naturally embeds into the <E>parent wreath product</E>,
that is <M>P = K \wr \textrm{Sym}(m) \geq G</M>. <P/>

Formally we can write an element of <M>G</M> as a tuple <M>g = (f, \pi) \in G</M>,
where <M>f \in K^m </M> is a function <M>f : \{1, \ldots, m\} \rightarrow K </M>
and <M>\pi \in H \leq \textrm{Sym}(m)</M> is a permutation on <M>m</M> points.
We call <M>f</M> the <E>base component</E> and <M>\pi</M> the <E>top component</E> of <M>g</M>. <P/>

We can naturally identify a map <M>f \in K^m</M> with a tuple <M>(g_1, \ldots, g_m)</M>,
where each <M>g_i \in K</M> is the image of <M>i \in \{1, \ldots, m\}</M> under <M>f</M>.
This yields a second useful notation for elements in <M>G</M> by writing
<M>g = (g_1, \ldots, g_m; \pi)</M>.
Note that we use a semicolon to seperate the base component from the top component.
Further we call the element <M>g_i</M> the <E><M>i</M>-th base component</E> of <M>g</M>. <P/>

Analogously, the subgroup <M>B = K^m \times \langle 1_H \rangle \leq G</M> is called the <E>base group</E> of <M>G</M>
and the subgroup <M>T = \langle 1_K \rangle^m \times H \leq G</M> is called the <E>top group</E> of <M>G</M>. <P/>

With the above notation, the multiplication of two elements
<Display Mode="M">
g = (f, \pi) = (g_1, \ldots, g_m; \pi),\\
h = (d, \sigma) = (h_1, \ldots, h_m; \sigma)
</Display>
of <M>G = K \wr H</M>, a wreath product of finite groups, can be written as
<Display Mode="M">
g \cdot h = (f \cdot d^{(\pi^{-1})}, \pi \cdot \sigma) = (g_1 \cdot h_{1^\pi}, \ldots, g_m \cdot h_{m^\pi}; \pi \cdot \sigma)\,.
</Display>

</Section>

<Section Label="Wreath Cycle">
<Heading>Wreath Cycles</Heading>

In a permutation group we have the well-known concept of a cycle decomposition.
For wreath products we have a similar concept called <E>wreath cycle decomposition</E>
that allows us to solve certain computational tasks more efficiently. <P/>

Detailed information on <E>wreath cycle decompositions</E> can be found in Chapter 2 in <Cite Key="WPE"/>.
Chapters 3-5 in <Cite Key="WPE"/> describe how these can be exploited
for finding conjugating elements, conjugacy classes, and centralisers in wreath products,
and Chapter 6 in <Cite Key="WPE"/> contains a table of timings of sample computations done with &WPE; vs. native &GAP;. <P/>

We use the notation from Section <Ref Sect="Wreath Product"/> in order to introduce the following concepts. <P/>

<M>\bf{Definition:}</M> We define the <E>territory</E> of an element <M>g = (g_1, \ldots, g_m; \pi) \in G</M>
by <M>\textrm{terr}(g) := \textrm{supp}(\pi) \cup \{i : g_i \neq 1\}</M>,
where <M>\textrm{supp}(\pi)</M> denotes the set of moved points of <M>\pi</M>. <P/>

<M>\bf{Definition:}</M> Two elements <M>g, h \in G</M> are said to be <E>disjoint</E> if their territories are disjoint. <P/>

<M>\bf{Lemma:}</M> Disjoint elements in <M>G</M> commute. <P/>

<M>\bf{Definition:}</M> An element <M>g = (g_1, \ldots, g_m; \pi) \in G</M> is called a <E>wreath cycle</E> if
either <M>\pi</M> is a cycle in <M>\textrm{Sym}(n)</M> and <M>\textrm{terr}(g) = \textrm{supp}(\pi)</M>,
or <M>|\textrm{terr}(g)| = 1</M>.<P/>

<M>\bf{Example:}</M> For example, if we consider the wreath product <M> \textrm{Sym}(4) \wr \textrm{Sym}(5) </M>, the element
<Display Mode="M">
{\bf(} \;\; (),\, (1,2,3),\, (),\, (1,2),\, ();\, (1,2,4) \;\; {\bf)}
</Display>
is a wreath cycle as described in the first case and the element
<Display Mode="M">
{\bf(} \; (),\, (),\, (1,3),\, (),\, ();\, () \;\; {\bf)}
</Display>
is a wreath cycle as described in the second case. Moreover, these elements are disjoint and thus commute.<P/>

<M>\bf{Theorem:}</M> Every element of <M>G</M> can be written as a finite product
of disjoint wreath cycles in <M>P</M>.
This decomposition is unique up to ordering of the factors.
We call such a decomposition a <E>wreath cycle decomposition</E>. <P/>

</Section>

<Section Label="Sparse Wreath Cycle">
<Heading>Sparse Wreath Cycles</Heading>

We use the notation from Section <Ref Sect="Wreath Product"/> in order to introduce the following concepts. <P/>

The main motivation for introducing the concept of <E>sparse wreath cycles</E>
is the efficient computation of centralisers of wreath product elements.
Simply put, we compute the centraliser <M>C_G(g)</M> of an arbitrary element <M>g \in P</M> in <M>G</M>
by conjugating it in <M>P</M> to a restricted representative <M>h = g^c \in P</M>,
computing the centraliser of <M>h</M> in <M>G</M> and then conjugating it back.
The wreath cycle decomposition of the representative <M>h</M>
consists only of sparse wreath cycles.<P/>

More information on <E>sparse wreath cycles</E> and centralisers of wreath product elements
can be found in Chapter 5 in <Cite Key="WPE"/>.<P/>

<M>\bf{Definition:}</M> We say that a wreath cycle <M>g = (g_1, \ldots, g_m; \pi) \in G</M> is a <E>sparse wreath cycle</E>,
if there exists an <M>i_0</M> such that <M>g_i = 1</M> for all <M>i \neq i_0</M>.<P/>

<M>\bf{Example:}</M> For example, if we consider the wreath product <M> \textrm{Sym}(4) \wr \textrm{Sym}(5) </M>, the element
<Display Mode="M">
{\bf(} \;\; (),\, (1,2,3),\, (),\, (),\, ();\, (1,2,4) \;\; {\bf)}
</Display>
is a sparse wreath cycle, as well as the element
<Display Mode="M">
{\bf(} \;\; (),\, (),\, (1,3),\, (),\, ();\, () \;\; {\bf)} \;.
</Display><P/>

A very important invariant under conjugation is the <E>yade</E> of a wreath cycle. <P/>

<M>\bf{Definition:}</M> For a wreath cycle <M>g = (f, \pi) \in G</M> and a point <M>i \in \textrm{terr}(g)</M> we define the <E>yade</E> of <M>g</M> in <M>i</M> as
<Display Mode="M">
[(i)\pi^{0}]f \cdot [(i)\pi^{1}]f \cdots [(i)\pi^{|\pi| - 1}]f \;.
</Display>

<M>\bf{Example:}</M> Consider the wreath product <M> \textrm{Sym}(4) \wr \textrm{Sym}(5) </M>, and the wreath cycle
<Display Mode="M">
g = (f, \pi) = {\bf(} \;\; (),\, (1,2,3),\, (),\, (1,2),\, ();\, (1,2,4) \;\; {\bf)}.
</Display>
The yade evaluated at <M>i = 1</M> is given by
<Display Mode="M">
[(1)\pi^{0}]f \cdot [(1)\pi^{1}]f \cdot [(1)\pi^{2}]f = [1]f \cdot [2]f \cdot [4]f = () \cdot (1,2,3) \cdot (1,2) = (2,3)
</Display>
and the yade evaluated at <M>j = 4</M> is given by
<Display Mode="M">
[(4)\pi^{0}]f \cdot [(4)\pi^{1}]f \cdot [(4)\pi^{2}]f = [4]f \cdot [1]f \cdot [2]f = (1,2) \cdot () \cdot (1,2,3) = (1,3) \;.
</Display><P/>

Up to conjugacy, the yade is independent under the chosen evaluation point <M>i</M>.
Moreover, wreath cycles are conjugate over <M>G</M> if and only if the top components are conjugate over <M>H</M> and the yades are conjugate over <M>K</M>. More specific, we can conjugate a wreath cycle <M>g</M> to a sparse wreath cycle <M>h</M> such that the <M>i</M>-th base component of <M>h</M> contains the yade of <M>g</M> in <M>i</M>. This leads to the following result. <P/>

<M>\bf{Theorem:}</M> Every element <M>g \in P</M> can be conjugated by some <M>c \in K^m \times \langle 1_H \rangle \leq P</M> to an element <M>h = g^c \in P</M>
such that the wreath cycle decomposition of <M>h</M>
consists only of sparse wreath cycles.

</Section>



</Chapter>

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