From f85749775d93f9af4987a421fae133c3cb0864df Mon Sep 17 00:00:00 2001 From: Derived Cat Date: Mon, 29 Jul 2024 22:50:36 -0400 Subject: [PATCH] update group action --- category_theory.tex | 65 ++++++++++++++------ group.tex | 134 ++++++++++++++++++++++++++---------------- topological_group.tex | 21 ++----- 3 files changed, 137 insertions(+), 83 deletions(-) diff --git a/category_theory.tex b/category_theory.tex index 381a15e..c3010da 100644 --- a/category_theory.tex +++ b/category_theory.tex @@ -82,7 +82,7 @@ \section{Category} \end{example} -\begin{definition}{Opposite Category}{} +\begin{definition}{Opposite Category}{functor_op} Suppose $\mathsf{C}$ is a category. The \textbf{opposite category} $\mathsf{C}^{\mathrm{op}}$ is defined as follows \begin{itemize} \item Objects: $\mathrm{Ob}(\mathsf{C}^{\mathrm{op}})=\mathrm{Ob}(\mathsf{C})$ @@ -120,7 +120,7 @@ \section{Category} \mathsf{D} \& \& \& \mathsf{D}^{\mathrm{op}} \end{tikzcd} \] - The functor ${}^{\mathrm{op}}$ is an involution, i.e. ${}^{\mathrm{op}}\circ {}^{\mathrm{op}}=\mathrm{id}_{\mathsf{CAT}}$. Hence ${}^{\mathrm{op}}$ is an automorphism of $\mathsf{CAT}$. + The functor ${}^{\mathrm{op}}$ is an involution, i.e. ${}^{\mathrm{op}}\circ {}^{\mathrm{op}}=\mathrm{id}_{\mathsf{CAT}}$. Hence ${}^{\mathrm{op}}\in \mathrm{Aut}(\mathsf{CAT})$. \end{example} @@ -2530,26 +2530,11 @@ \section{Limit and Colimit} \end{theorem} -The next result shows that functor categories inherit limits and colimits, defined ``objectwise" in the target category: that is, given a $\mathsf{J}$-indexed diagram in $[\mathsf{A},\mathsf{C}]$ whose objects are functors $F_j : \mathsf{A} \to \mathsf{C}$, the value of the limit functor $\varprojlim_{j\in J} F_j : \mathsf{A} \to \mathsf{C}$ at an object $a \in \mathsf{A}$ is the limit of the $\mathsf{J}$-indexed diagram in $\mathsf{C}$ whose objects are the objects $F_j(a) \in \mathrm{Ob}(C)$. - -\begin{proposition}{Evaluation Functor Preserves Limits}{ev_functor_preserves_limits} - Let $\mathsf{A}$ be a small category and $\mathsf{C}$ be a category. Given a diagram $F: \mathsf{J} \to\left[\mathsf{A},\mathsf{C}\right]$ with $\mathsf{J}$ small, if for any $a\in \mathsf{A}$, the diagram - \[ - \mathrm{ev}_a \circ F:\mathsf{J}\xrightarrow{\quad F\quad}\left[\mathsf{A},\mathsf{C}\right]\xrightarrow{\quad\mathrm{ev}_a\quad}\mathsf{C} - \] - has a limit, then - \begin{enumerate}[(i)] - \item $\varprojlim F$ exists. - \item For any $a\in \mathsf{A}$, $\mathrm{ev}_a$ preserves $\varprojlim F$. - \end{enumerate} -\end{proposition} - - \begin{proposition}{}{} Suppose $\mathsf{A}$ is a small category. Denote $\mathsf{Ob}(A)=\mathsf{Disc}\left(\mathrm{Ob} \left(\mathsf{A}\right)\right)$. Then the forgetful functor $U:[\mathsf{A},\mathsf{C}] \rightarrow \left[\mathsf{Ob}(A), \mathsf{C}\right]\cong \prod\limits_{a\in \mathrm{Ob} \left(\mathsf{A}\right)}\mathsf{C}$ \[ \begin{tikzcd}[ampersand replacement=\&] - [\mathsf{A},\mathsf{C}] \&[-25pt]\&[+10pt]\&[-30pt]\prod\limits_{i\in \mathrm{Ob} \left(\mathsf{A}\right)}\mathsf{C}\&[-30pt]\&[-30pt] \\ [-15pt] + [\mathsf{A},\mathsf{C}] \&[-25pt]\&[+10pt]\&[-30pt]\prod\limits_{a\in \mathrm{Ob} \left(\mathsf{A}\right)}\mathsf{C}\&[-30pt]\&[-30pt] \\ [-15pt] F \arrow[dd, Rightarrow, "\theta"{name=L, left}] \&[-25pt] \& [+10pt] \& [-30pt]\left(F(a)\right)_{a\in \mathrm{Ob} \left(\mathsf{A}\right)}\arrow[dd, "\prod\limits_{a\in \mathrm{Ob} \left(\mathsf{A}\right)} \theta_{a}"{name=R}] \\ [-10pt] @@ -2559,6 +2544,32 @@ \section{Limit and Colimit} \] strictly creates all limits and colimits that exist in $\mathsf{C}$. These limits are defined objectwise, meaning that for each $a \in \mathrm{A}$, the evaluation functor $\mathrm{ev}_a:[\mathsf{A},\mathsf{C}] \rightarrow \mathsf{C}$ preserves all limits and colimits existing in $\mathsf{C}$. \end{proposition} +\begin{proof} + Given a diagram $F:\mathsf{J}\to [\mathsf{A},\mathsf{C}]$, suppose the limit $\varprojlim U\circ F$ exists and can be written as + \[ + \left((L(a))_{a \in \operatorname{Ob}(\mathsf{A})},\left(\ell_j=\left(\ell_{j, a}:L(a)\to F(j)(a)\right)_{a \in \operatorname{Ob}(\mathsf{A})}\right)_{j \in \operatorname{Ob}(\mathsf{J})}\right). + \] + +\end{proof} + + +The next result shows that functor categories inherit limits and colimits, defined ``objectwise" in the target category: that is, given a $\mathsf{J}$-indexed diagram in $[\mathsf{A},\mathsf{C}]$ whose objects are functors $F_j : \mathsf{A} \to \mathsf{C}$, the value of the limit functor $\varprojlim_{j\in J} F_j : \mathsf{A} \to \mathsf{C}$ at an object $a \in \mathsf{A}$ is the limit of the $\mathsf{J}$-indexed diagram in $\mathsf{C}$ whose objects are the objects $F_j(a) \in \mathrm{Ob}(\mathsf{C})$, which reads as follows: +\[ + \left(\varprojlim_{j\in J} F_j\right)(a) = \varprojlim_{j\in J} F_j(a). +\] + +\begin{proposition}{Evaluation Functor Preserves Limits}{ev_functor_preserves_limits} + Let $\mathsf{A}$ be a small category and $\mathsf{C}$ be a category. Given a diagram $F: \mathsf{J} \to\left[\mathsf{A},\mathsf{C}\right]$ with $\mathsf{J}$ small, if for any $a\in \mathsf{A}$, the diagram + \[ + \mathrm{ev}_a \circ F:\mathsf{J}\xrightarrow{\quad F\quad}\left[\mathsf{A},\mathsf{C}\right]\xrightarrow{\quad\mathrm{ev}_a\quad}\mathsf{C} + \] + has a limit, then + \begin{enumerate}[(i)] + \item $\varprojlim F$ exists. + \item For any $a\in \mathsf{A}$, $\mathrm{ev}_a$ preserves $\varprojlim F$. + \end{enumerate} +\end{proposition} + \begin{proposition}{Limits Commute with Limits}{} @@ -2674,6 +2685,22 @@ \section{Limit and Colimit} \] \end{definition} +\begin{lemma}{}{} + Let $F:\mathsf{C}\to\mathsf{D}$ and $G:\mathsf{D}\to\mathsf{E}$ be functors. If both $\varinjlim G \circ F$ and $\varinjlim G$ exist, then there exists a natural morphism between colimits + \[ + \varinjlim G \circ F \longrightarrow \varinjlim G + \] +\end{lemma} +\begin{proof} + Suppose $\mu: G\circ F \Rightarrow \diagfunctor \varinjlim_{\mathsf{C}} G\circ F $ is a colimit cone in $\mathsf{Cocone}(G\circ F,\textsf{E})$ and $\nu: G\Rightarrow \diagfunctor \varinjlim_{\mathsf{D}} G$ is a colimit cone in $\mathsf{Cocone}(G,\textsf{E})$. Then by universal property of $\varinjlim G\circ F$, we can get the following commutative diagram + \[ + \begin{tikzcd} + & \varinjlim_{\mathsf{D}} G \\ + G(F(c)) \arrow[r, "\mu_c"'] \arrow[ru, "\nu_{F(c)}"] & \varinjlim_{\mathsf{C}} G\circ F \arrow[u, dashed] + \end{tikzcd} + \] +\end{proof} + \begin{proposition}{Equivalent Characterization of Final Functor}{} Let $F:\mathsf{C}\to\mathsf{D}$ be a functor. The following are equivalent: \begin{enumerate}[(i)] @@ -2692,6 +2719,8 @@ \section{Limit and Colimit} A functor $F:\mathsf{C}\to\mathsf{D}$ is \textbf{initial} if the opposite functor $F^{\text{op}}:\mathsf{C}^{\text{op}}\to\mathsf{D}^{\text{op}}$ is final. \end{definition} + + \subsection{Product and Coproduct} \begin{definition}{Binary Product}{} \begin{center} diff --git a/group.tex b/group.tex index 583f297..456dd3e 100644 --- a/group.tex +++ b/group.tex @@ -18,6 +18,11 @@ \section{Basic Concepts} \] \end{definition} +\begin{proposition}{Group is Isomorphic to Its Opposite group}{} + $G$ is isomorphic to $G^{\mathrm{op}}$ through the isomorphism $x\mapsto x^{-1}$. This is same as saying that $\mathsf{B}G$ is isomorphic to $(\mathsf{B}G)^{\mathrm{op}}$ through the functor ${}^{\mathrm{op}}$ defined in \Cref{th:functor_op}. +\end{proposition} + + \begin{definition}{Subgroup}{} Let $G$ be a group. A subset $H$ of $G$ is called a \textbf{subgroup} of $G$ if $H$ is a group with respect to the binary operation of $G$. In this case, we write $H\le G$. \end{definition} @@ -344,7 +349,7 @@ \subsection{Definitions} Let $X$ and $Y$ be objects in a category $\mathsf{C}$. Then $\mathrm{Aut}_\mathsf{C}(X)$ acts on $\mathrm{Hom}_\mathsf{C}(X,Y)$ by the composition of functors \[ \begin{tikzcd}[ampersand replacement=\&, column sep=5em, row sep=3em] - \mathsf{B}\mathrm{Aut}_\mathsf{C}(X)\arrow[r] \&[-3em] \mathsf{B}\mathrm{Aut}_\mathsf{C}(X)^{\mathrm{op}} \arrow[r, hook] \&[-2em] \mathsf{C}^{\mathrm{op}} \arrow[r, "{\mathrm{Hom}_{\mathsf{C}}(-,Y)}"] \& \mathsf{Set}\\[-2.3em] + \mathsf{B}\mathrm{Aut}_\mathsf{C}(X)\arrow[r,"{(-)}^{-1}"] \&[-3em] \mathsf{B}\mathrm{Aut}_\mathsf{C}(X)^{\mathrm{op}} \arrow[r, hook] \&[-2em] \mathsf{C}^{\mathrm{op}} \arrow[r, "{\mathrm{Hom}_{\mathsf{C}}(-,Y)}"] \& \mathsf{Set}\\[-2.3em] \bullet \arrow[r, maps to] \arrow[d, "g"'] \& \bullet \arrow[r, maps to] \arrow[d, "g^{-1}"] \& X \arrow[d, "g^{-1}"] \arrow[r, maps to] \& {\mathrm{Hom}(X,Y)} \arrow[d, "\left(g^{-1}\right)^*"] \\ \bullet \arrow[r, maps to] \& \bullet \arrow[r, maps to] \& X \arrow[r, maps to] \& {\mathrm{Hom}(X,Y)} \end{tikzcd} @@ -369,17 +374,22 @@ \subsection{Definitions} \end{align*} \end{example} -\begin{example}{Actions on $X$ Induce Actions on $Y^X$}{acting_on_functions} - If $G$ acts on $X$ through a functor $\sigma(-):\mathsf{B}G\to\mathsf{Set}$, then it also acts on $Y^X$ for any set $Y$ by the composition of functors +\begin{example}{Actions on $X$ Induce Actions on $\mathrm{Hom}_{\mathsf{Set}}(X,Y)$}{acting_on_functions} + If $G$ acts on $X$ through a functor $\sigma(-):\mathsf{B}G\to\mathsf{Set}$, then it also acts on $\mathrm{Hom}_{\mathsf{Set}}(X,Y)$ for any set $Y$ by the composition of functors \[ \begin{tikzcd}[ampersand replacement=\&, column sep=5em, row sep=3em] - \mathsf{B}G \arrow[r] \&[-2em] \mathsf{B}G^{\mathrm{op}} \arrow[r, "\sigma(-)^{\mathrm{op}}"] \&[-2.2em] \mathsf{Set}^{\mathrm{op}} \arrow[r, "{\mathrm{Hom}_{\mathsf{Set}}(-,Y)}"] \& \mathsf{Set} + \mathsf{B}G \arrow[r,"{(-)}^{-1}"] \&[-2em] \mathsf{B}G^{\mathrm{op}} \arrow[r, "\sigma(-)^{\mathrm{op}}"] \&[-2.2em] \mathsf{Set}^{\mathrm{op}} \arrow[r, "{\mathrm{Hom}_{\mathsf{Set}}(-,Y)}"] \& \mathsf{Set} \end{tikzcd} \] - The action on $Y^X$ is given explicitly as follows: for all $g\in G$, $f\in Y^X$ and $x\in X$, + The left action on $\mathrm{Hom}_{\mathsf{Set}}(X,Y)$ is given explicitly as follows: for all $g\in G$, $f\in \mathrm{Hom}_{\mathsf{Set}}(X,Y)$ and $x\in X$, \[ (g\cdot f)(x)=f(g^{-1}\cdot x). \] + Equivalently, the right action $\star$ on $\mathrm{Hom}_{\mathsf{Set}}(X,Y)$ is given by + \[ + (f\star g)(x)=f(g\cdot x). + \] + \end{example} @@ -392,6 +402,14 @@ \subsection{Definitions} &=f\left(\left(g_2^{-1} g_1^{-1}\right)\cdot x\right)\\ &=\left(\left(g_1g_2\right)\cdot f\right)(x). \end{align*} + and also check that + \begin{align*} + ((f\star g_1)\star g_2)(x) + &=\left(f\star g_1\right)\left(g_2\cdot x\right)\\ + &=f\left(g_1\cdot\left(g_2\cdot x\right)\right)\\ + &=f\left((g_1g_2)\cdot x\right)\\ + &=\left(f\star (g_1 g_2)\right)(x). + \end{align*} \end{prf} \begin{definition}{Orbit of a Group Action}{} @@ -405,20 +423,20 @@ \subsection{Definitions} \begin{definition}{Orbit Space}{} Let $G$ be a group acting on a set $X$. The \textbf{orbit space} of $G$ acting on $X$ is defined as \[ - X/G=\{ Gx\mid x\in X\}. + G\backslash X=\{ Gx\mid x\in X\}. \] \end{definition} -If $G$ acts on $X$, then $G$ acts on $X/G$ trivially. +If $G$ acts on $X$, then $G$ acts on $G\backslash X$ trivially by $g\cdot Gx=Gx$. \begin{proposition}{Orbit Decomposition}{} Let $G$ be a group acting on a set $X$. We define a equivalence relation $\sim$ on $X$ by \[ x\sim y \iff Gx=Gy. \] - Then the equivalence class of $x$ is exactly $Gx$. The quotient set $X/\sim$ is exactly the orbit space $X/G$. + Then the equivalence class of $x$ is exactly $Gx$. The quotient set $X/\sim$ is exactly the orbit space $G\backslash X$. And we have a partition of $X$ by the orbits of $G$ acting on $X$ \[ - X=\bigsqcup_{Gx\in X/G}Gx. + X=\bigsqcup_{Gx\in G\backslash X}Gx. \] \end{proposition} @@ -427,7 +445,7 @@ \subsection{Definitions} \end{prf} -If $G$ acts on $X$, then $G$ acts on $X/G$ trivially. +If $G$ acts on $X$, then $G$ acts on $G\backslash X$ trivially. \begin{definition}{$G$-invariant element }{} Let $G$ be a group acting on a set $X$. An element $x\in X$ is called \textbf{$G$-invariant} if $Gx=\{ x\}$ or equivalently $|Gx|=1$. The set of all $G$-invariant elements is denoted by $X^G$ \[ @@ -485,7 +503,7 @@ \subsection{Definitions} \begin{example}{$G$ Acts on Orbit $Gx$ Transitively}{} Let $G$ be a group acting on a set $X$ and $x\in X$. Then $G$ acts on the orbit $Gx$ by left multiplication transitively. And we have a $G$-set isomorphism \[ - X\cong \bigsqcup_{Gx\in X/G}Gx, + X\cong \bigsqcup_{Gx\in G\backslash X}Gx, \] which decomposes any $G$-set into coproduct of transitive $G$-sets. \end{example} @@ -526,16 +544,16 @@ \subsection{Coset} \[ gH = H^\circ g = \{ gh\mid h\in H\}. \] - The set of all left cosets of $H$ is denoted by $G/H^L$, called the left coset space of $G$ modulo $H$. $G/H^L$ is the orbit space of $G$ under the right multiplication action of $H$. + The set of all left cosets of $H$ is denoted by $G/H$, called the left coset space of $G$ modulo $H$. $G/H$ is the orbit space of $G$ under the right multiplication action of $H$. \end{definition} -\begin{example}{$G$ Acts on $G/H^L$ Transitively}{} - Let $G$ be a group and $H$ be a subgroup of $G$. $G$ acts on $G/H^L$ through +\begin{example}{$G$ Acts on $G/H$ Transitively}{} + Let $G$ be a group and $H$ be a subgroup of $G$. $G$ acts on $G/H$ through \begin{align*} - G &\longrightarrow \mathrm{Aut}(G/H^L)\\ + G &\longrightarrow \mathrm{Aut}(G/H)\\ g &\longmapsto (xH\longmapsto gxH) \end{align*} - For any $xH,yH\in G/H^L$, we have $yH=gxH$ for some $g=yx^{-1}\in G$. Thus $G$ acts on $G/H^L$ transitively. + For any $xH,yH\in G/H$, we have $yH=gxH$ for some $g=yx^{-1}\in G$. Thus $G$ acts on $G/H$ transitively. \end{example} \begin{definition}{Right Cosets}{} @@ -548,11 +566,11 @@ \subsection{Coset} \[ Hg = \{ hg\mid h\in H\}, \] - which matches notation of orbit. The set of all right cosets of $H$ is denoted by $G/H^R$, called the right coset space of $G$ modulo $H$. + which matches notation of orbit. The set of all right cosets of $H$ is denoted by $H\backslash G$, called the right coset space of $G$ modulo $H$. \end{definition} \begin{definition}{Index of Subgroup}{} - Let $G$ be a group and $H$ be a subgroup of $G$. The \textbf{index} of $H$ in $G$ is defined as the cardinality of $G/H^L$ or $G/H^R$, denoted by $[G:H]$. + Let $G$ be a group and $H$ be a subgroup of $G$. The \textbf{index} of $H$ in $G$ is defined as the cardinality of $G/H$ or $H\backslash G$, denoted by $[G:H]$. \end{definition} \begin{theorem}{Lagrange's Theorem}{} @@ -560,10 +578,10 @@ \subsection{Coset} \end{theorem} -\begin{proposition}{$G$-Set Isomorphism $G/\mathrm{Stab}_G(x)^L\cong Gx$}{iso_stab_orbit} +\begin{proposition}{$G$-Set Isomorphism $G/\mathrm{Stab}_G(x)\cong Gx$}{iso_stab_orbit} Let $G$ be a group acting on a set $X$ and $x\in X$. Then the map \begin{align*} - F:G/\mathrm{Stab}_G(x)^L &\longrightarrow Gx\\ + F:G/\mathrm{Stab}_G(x) &\longrightarrow Gx\\ g\hspace{1pt}\mathrm{Stab}_G(x) &\longmapsto g\cdot x \end{align*} is a $G$-set isomorphism. @@ -587,14 +605,14 @@ \subsection{Coset} \begin{prf} According to \Cref{th:iso_stab_orbit}, we have \[ -\left|Gx\right|=\left|G/\mathrm{Stab}_G(x)^L\right|=|G|/\left|\mathrm{Stab}_G(x)\right|. +\left|Gx\right|=\left|G/\mathrm{Stab}_G(x)\right|=|G|/\left|\mathrm{Stab}_G(x)\right|. \] \end{prf} \begin{theorem}{Burnside's Lemma}{Burnside's_lemma} Let $G$ be a finite group acting on a finite set $X$. Then the number of orbits of $G$ on $X$ is equal to \[ - |X/G|=\frac{1}{|G|}\sum_{g\in G}|X^g| , + |G\backslash X|=\frac{1}{|G|}\sum_{g\in G}|X^g| , \] where $X^g=\{x\in X\mid g\cdot x=x\}$ is the set of fixed points of $g$. \end{theorem} @@ -604,10 +622,10 @@ \subsection{Coset} \sum_{g \in G}\left|X^g\right|&=|\{(g, x) \in G \times X \mid g \cdot x=x\}|\\ &=\sum_{x \in X}\left|\mathrm{Stab}_G(x)\right|\\ &=\sum_{x \in X}\frac{|G|}{\left|G x\right|} \quad \text{by Orbit-Stabilizer Theorem}\\ - &=|G| \sum_{G y \in X / G}\; \sum_{x \in Gy}\frac{1}{\left|G x\right|}\\ - &=|G| \sum_{G y \in X / G} \left|Gy\right|\frac{1}{\left|G y\right|}\\ - &=|G| \sum_{G y \in X / G} 1\\ - &=|G| \cdot|X / G|. + &=|G| \sum_{G y \in G \backslash X}\; \sum_{x \in Gy}\frac{1}{\left|G x\right|}\\ + &=|G| \sum_{G y \in G \backslash X} \left|Gy\right|\frac{1}{\left|G y\right|}\\ + &=|G| \sum_{G y \in G \backslash X} 1\\ + &=|G| \cdot|G \backslash X|. \end{align*} \end{prf} @@ -901,7 +919,7 @@ \subsection{Conjugacy Action} \begin{prf} With the orbit decomposition of $G$ under conjugacy action, then by orbit-stabilizer theorem we have \[ - |G|=\sum_{Gx\in G/G}\left|\mathrm{Cl}(x)\right|=\sum_{Gx\in G/G}\left[G: \mathrm{Stab}_G(x)\right]=|Z(G)|+\sum_{j=1}^m\left[G: Z_G\left(x_j\right)\right]. + |G|=\sum_{Gx\in G\backslash G}\left|\mathrm{Cl}(x)\right|=\sum_{Gx\in G\backslash G}\left[G: \mathrm{Stab}_G(x)\right]=|Z(G)|+\sum_{j=1}^m\left[G: Z_G\left(x_j\right)\right]. \] \end{prf} @@ -958,6 +976,10 @@ \subsection{Conjugacy Action} \section{Symmetric Groups} +\begin{definition}{Symmetric Group}{} + The \textbf{symmetric group} on a set $X$ is the group of all permutations of $X$, denoted by $S_X=\mathrm{Aut}_{\mathsf{Set}}(X)$. If $X=\{1,2,\cdots,n\}$, then we denote $S_X$ by $S_n$. +\end{definition} + \begin{definition}{$k$-Cycle}{} Let $n\ge 2$ and $k\ge 1$ be integers. A \textbf{$k$-cycle} in $S_n$ is a permutation $\sigma\in S_n$ such that there exist a subset of $P_n=\{1,2,\cdots,n\}$, denoted by $A=\{a_1,a_2,\cdots,a_k\}$, satisfying \begin{enumerate}[(i)] @@ -973,30 +995,31 @@ \section{Symmetric Groups} \begin{theorem}{Pólya Enumeration Theorem (Unweighted)}{Polya_enumeration_unweighted} - Let $X, Y$ be finite sets, where $X=\{1,2,\cdots,n\}$ is the set of points to be colored and $Y$ is the set of colors. Suppose a group $G$ acts on $X$ through $\sigma:G\to\mathrm{Aut}_{\mathsf{Set}}(X)$. Then it also acts on $Y^X$ by \Cref{ex:acting_on_functions}. Define that a coloring configuration of $(X,Y,\sigma)$ is an orbit in the $G$-set $Y^X$. Then the number of essentially distinct coloring configurations is + Let $X, Y$ be finite sets, where $X=\{1,2,\cdots,n\}$ is the set of points to be colored and $Y$ is the set of colors. Suppose a group $G$ acts on $X$ through $\sigma:G\to\mathrm{Aut}_{\mathsf{Set}}(X)$. Then it also acts on $Y^X$ by \Cref{ex:acting_on_functions}. Define that a coloring configuration of $(X,Y,\sigma)$ is an orbit in $Y^X/G$. Then the number of essentially distinct coloring configurations is $$ \left|Y^X / G\right|=\frac{1}{|G|} \sum_{g \in G}|Y|^{c(g)}, $$ -where $c(g):=\left|X/\langle \sigma_g\rangle\right|$ denotes the number of cycles in the cycle decomposition of $\sigma_g \in \mathrm{Aut}_{\mathsf{Set}}(X)$. +where $c(g):=\left|\langle g\rangle \backslash X\right|=\left|\langle \sigma_g\rangle \backslash X\right|$ denotes the number of cycles in the cycle decomposition of $\sigma_g \in S_n$. \end{theorem} \begin{prf} We apply Burnside's lemma \ref{th:Burnside's_lemma}, which states that $$ -\left|Y^X / G\right|=\frac{1}{|G|} \sum_{g \in G}\left|\left(Y^X\right)^g\right| . +\left|Y^X / G\right|=\frac{1}{|G|} \sum_{g \in G}\left|\left(Y^X\right)^g\right| $$ -It remains to show that $\left|\left(Y^X\right)^g\right|=|Y|^{c(g)}$. For any map $f:X\to Y$, if -\[ -f(g\cdot x)=f(x) ,\quad\forall x\in X, -\] -then $f(x)=f(y)\iff Gx=Gy$. By the universal property of the quotient set, there exists a unique map $\overline{f}:X/G\to Y$ such that the following diagram commutes +where +\begin{align*} + \left(Y^X\right)^g&=\left\{ f \in Y^X \mid f(g \cdot x)=f (x)\text{ for all }x \in X\right\}\\ + &=\left\{ f \in Y^X \mid \langle g\rangle x = \langle g\rangle y\implies f(x)=f(y)\right\}. +\end{align*} +Define an equivalence relation $\sim_g$ on $X$ by $x \sim_g y\iff\langle g\rangle x = \langle g\rangle y$. Then we have $X/\sim_g\;=\langle g\rangle \backslash X$. By the universal property of the quotient set, for any map $f:X\to Y$ satisfying $x\sim y\implies f(x)=f(y)$, there exists a unique map $\overline{f}:\langle \sigma_g\rangle \backslash X\to Y$ such that the following diagram commutes \[ \begin{tikzcd}[ampersand replacement=\&] X \arrow[rr, "f"] \arrow[rd,"\pi"'] \& \& Y \\ - \& X/\langle \sigma_g\rangle \arrow[ru, "\overline{f}"'] \& + \& \langle \sigma_g\rangle \backslash X \arrow[ru, "\overline{f}"',dashed] \& \end{tikzcd} \] -Conversely, for any map $h:X/\langle \sigma_g\rangle\to Y$, we can define a map $f:X\to Y$ and checking that $f(g\cdot x)=f(x)$ for all $x\in X$. Hence we have a bijection between $\left(Y^X\right)^g$ and $Y^{X/\langle \sigma_g\rangle}$, which implies that $\left|\left(Y^X\right)^g\right|=|Y|^{c(g)}$. +And we have a natural bijection between $\left(Y^X\right)^g$ and $Y^{\langle g\rangle \backslash X}$, which implies that $\left|\left(Y^X\right)^g\right|=|Y|^{c(g)}$. \end{prf} \begin{definition}{Cycle Index Polynomial}{} @@ -1012,18 +1035,20 @@ \section{Symmetric Groups} \[ q(x_1,\cdots,x_m)=\sum_{y\in Y}x_1^{w_1(y)}x_2^{w_2(y)}\cdots x_m^{w_m(y)}, \] -where the coefficient of the term $x_1^{a_1}x_2^{a_2}\cdots x_m^{a_m}$ is the number of colors with weight $(a_1,a_2,\cdots,a_m)$. For each coloring configuration $f:X\to Y$, define the its weight as $W(f)$, where +where the coefficient of the term $x_1^{a_1}x_2^{a_2}\cdots x_m^{a_m}$ is the number of colors with weight $(a_1,a_2,\cdots,a_m)$. For each coloring map $f:X\to Y$, define the its weight as $W(f)$, where \begin{align*} W:Y^X&\longrightarrow\mathbb{Z}_{\ge 0}^m\\ f&\longmapsto\sum_{x\in X}w(f(x)) \end{align*} -Let $G$ be subgroup of $S_n$. Since for all $g\in G$ and $f\in X^Y$, +Let $G$ be a subgroup of $S_n$. Given any $g\in G$ and $f\in X^Y$, we can check the action of $g$ on $f$ does not change its weight \begin{align*} - W(g\cdot f)=\sum_{x\in X}w((g\cdot f)(x))=\sum_{x\in X}w(f(g^{-1}\cdot x))=\sum_{x\in X}w(f(x))=W(f), + W(f\star g)=\sum_{x\in X}w(( f\star g)(x))=\sum_{x\in X}w(f(g\cdot x))=\sum_{x\in X}w(f(x))=W(f), \end{align*} -we see $G$ acts on $W^{-1}(\omega)$ for each $\omega=(\omega_1,\cdots,\omega_m)\in\mathbb{Z}_{\ge 0}^m$. The generating function for the number of essentially distinct coloring configurations with weight $\omega$ can be expressed as +which implies that for each $\omega=(\omega_1,\cdots,\omega_m)\in\mathbb{Z}_{\ge 0}^m$, the fiber $W^{-1}(\omega)$ is $G$-invariant. + +Define a coloring configuration of $(X,Y,G)$ as an orbit in $Y^X/G$. The generating function for the number of essentially distinct coloring configurations with weight $\omega$ can be expressed as \begin{align*} - \mathrm{CGF}\left(x_1, \cdots,x_m\right) &=\sum_{\omega \in \mathbb{Z}_{\ge 0}^m}\left|W^{-1}(\omega)/G\right| x_1^{\omega_1} \cdots x_m^{\omega_m}\\ + \mathrm{CGF}\left(x_1, \cdots,x_m\right) &=\sum_{\omega \in \mathbb{Z}_{\ge 0}^m}\left|G\backslash W^{-1}(\omega)\right| x_1^{\omega_1} \cdots x_m^{\omega_m}\\ &= Z\left(q\left(x_1, \cdots,x_m\right), q\left(x_1^2, \cdots,x_m^2\right), \cdots, q\left(x_1^n, \cdots,x_m^n\right);G\right). \end{align*} \end{theorem} @@ -1031,22 +1056,31 @@ \section{Symmetric Groups} \begin{prf} For any $\omega=(\omega_1,\cdots,\omega_m)\in\mathbb{Z}_{\ge 0}^m$, by applying Burnside's lemma \ref{th:Burnside's_lemma} to the $G$-set $W^{-1}(\omega)$, we have \[ - \left|W^{-1}(\omega)/G\right| =\frac{1}{|G|}\sum_{g\in G}\left|W^{-1}(\omega)^g\right|. + \left|G\backslash W^{-1}(\omega)\right| =\frac{1}{|G|}\sum_{g\in G}\left|W^{-1}(\omega)^g\right|. \] Similar to \Cref{th:Polya_enumeration_unweighted}, we have a bijection between \[ - W^{-1}(\omega)^g=\left\{f\in \left(Y^X\right)^g\mid W(f)=\omega\right\}\quad\text{and}\quad\left\{\overline{f}\in Y^{X/\langle g\rangle}\mid W\left(\,\overline{f}\circ\pi\right)=\omega\right\}, + W^{-1}(\omega)^g=\left\{f\in \left(Y^X\right)^g\mid W(f)=\omega\right\}\quad\text{and}\quad\left\{\overline{f}\in Y^{\langle g\rangle\backslash X}\mid W\left(\,\overline{f}\circ\pi\right)=\omega\right\}, \] - which implies coloring points of $X$ essentially distinctly with colors in $Y$ with weight $\omega=(\omega_1,\cdots,\omega_m)$ is equivalent to coloring orbits $g_1,g_2,\cdots,g_r$ of $X/\langle g\rangle$ with colors in $Y$ with weight $\omega=(|g_1|\omega_{v_1},\cdots,|g_r|\omega_{v_r})$. Hence + Suppose $\langle g\rangle\backslash X=\{\langle g\rangle x_1,\cdots,\langle g\rangle x_r\}$. We can give a coloring configuration by $r$ consecutive steps. In the $i$-th step, we just need to choose a color in $y\in Y$ for the $i$-orbit $\langle g\rangle\backslash x_i$, which will contribute a term + $$ + x_1^{|\langle g\rangle x_i| w_1(y)}x_2^{|\langle g\rangle x_i|w_2(y)}\cdots x_m^{|\langle g\rangle x_i| w_m(y)}. + $$ + Thus we have \begin{align*} - \sum_{\omega\in\mathbb{Z}_{\ge 0}^m} \left |W^{-1}(\omega)^g\right |x_1^{\omega_1} \cdots x_m^{\omega_m} &= \prod_{g_i \in X/\langle g\rangle } q\left (x_1^{|g_i|}, x_2^{|g_i|}, \cdots,x_m^{|g_i|}\right )\\ + \sum_{\omega\in\mathbb{Z}_{\ge 0}^m} \left |W^{-1}(\omega)^g\right |x_1^{\omega_1} \cdots x_m^{\omega_m} &= \prod_{\langle g\rangle x_i \in \langle g\rangle\backslash X }\left(\sum_{y\in Y}x_1^{|\langle g\rangle x_i|w_1(y)}x_2^{|\langle g\rangle x_i|w_2(y)}\cdots x_m^{|\langle g\rangle x_i|w_m(y)}\right) \\ + &= \prod_{\langle g\rangle x_i \in \langle g\rangle\backslash X } q\left (x_1^{|\langle g\rangle x_i|}, x_2^{|\langle g\rangle x_i|}, \cdots,x_m^{|\langle g\rangle x_i|}\right )\\ &= q(x_1, \cdots,x_m)^{c_1(g)} q \left (x_1^2, \cdots,x_m^2 \right )^{c_2(g)} \cdots q \left (x_1^n, \cdots,x_m^n\right )^{c_n(g)}. \end{align*} - Therefore, we have + where + $$ + c_k(g)=\sum_{\langle g\rangle x_i \in \langle g\rangle\backslash X}\mathbf{1}_{|\langle g\rangle x_i |=k} + $$ + denotes the number of orbits with size $k$. With this equality, we can rewrite the generating function as \begin{align*} - \mathrm{CGF}\left(x_1, \cdots,x_m\right) &=\sum_{\omega \in \mathbb{Z}_{\ge 0}^m}\left|W^{-1}(\omega)/G\right| x_1^{\omega_1} \cdots x_m^{\omega_m}\\ + \mathrm{CGF}\left(x_1, \cdots,x_m\right) &=\sum_{\omega \in \mathbb{Z}_{\ge 0}^m}\left|G\backslash W^{-1}(\omega)\right| x_1^{\omega_1} \cdots x_m^{\omega_m}\\ &=\sum_{\omega \in \mathbb{Z}_{\ge 0}^m}\frac{1}{|G|}\sum_{g \in G}\left|W^{-1}(\omega)^g\right| x_1^{\omega_1} \cdots x_m^{\omega_m}\\ - &=\frac{1}{|G|}\sum_{g \in G}\sum_{\omega \in \mathbb{Z}_{\ge 0}^m}\left|W^{-1}(\omega)/G\right| x_1^{\omega_1} \cdots x_m^{\omega_m}\\ + &=\frac{1}{|G|}\sum_{g \in G}\sum_{\omega \in \mathbb{Z}_{\ge 0}^m}\left|W^{-1}(\omega)^g\right| x_1^{\omega_1} \cdots x_m^{\omega_m}\\ &=\frac{1}{|G|}\sum_{g \in G}q(x_1, \cdots,x_m)^{c_1(g)} q \left (x_1^2, \cdots,x_m^2 \right )^{c_2(g)} \cdots q \left (x_1^n, \cdots,x_m^n\right )^{c_n(g)}\\ &= Z\left(q\left(x_1, \cdots,x_m\right), q\left(x_1^2, \cdots,x_m^2\right), \cdots, q\left(x_1^n, \cdots,x_m^n\right);G\right). \end{align*} @@ -1070,7 +1104,7 @@ \section{Symmetric Groups} Z(t_1,\cdots,t_6;D_{12})=\frac{1}{12}\left(t_1^6+3 t_1^2 t_2^2+4 t_2^3+2 t_3^2+2 t_6\right) \] -Assigning a weight of (1,0) to H and a weight of (0,1) to Cl, the corresponding generating function is \(q(\text{H},\text{Cl})=\text{H}+\text{Cl}\). Finally, the generating function for the number of essentially distinct colorings is: +Assigning a weight of (1,0) to H and a weight of (0,1) to Cl, the corresponding generating function is \(q(\text{H},\text{Cl})=\text{H}+\text{Cl}\). Finally, the generating function for the number of essentially distinct coloring configurations is: \[ \begin{aligned} diff --git a/topological_group.tex b/topological_group.tex index 8621e74..2f81335 100644 --- a/topological_group.tex +++ b/topological_group.tex @@ -127,6 +127,8 @@ \section{Continuous Topological Group Action} is continuous and has a continuous inverse $\rho(g^{-1})$. Hence $\rho(g)\in \mathrm{Aut}_{\mathsf{Top}}(X)$. \end{prf} +From the definition, we see if $varpho:G\times X\to X$ is continuous topological group action on topological space $X$, it is also a group action on $X$ by homeomorphisms. If $G$ is discrete, then the converse holds. + \begin{proposition}{Discrete Group Acts Continuously on Topological Space $\iff$ Acts by Homeomorphisms}{} Let $G$ be a group acting on the underlying set of a topological space $X$ through a group homomorphism $\rho:G\to \mathrm{Aut}_{\mathsf{Set}}(X)$. Then the following are equivalent: \begin{enumerate}[(i)] @@ -143,24 +145,18 @@ \section{Continuous Topological Group Action} &=\{(g,x)\in G\times X\mid x\in \rho(g)^{-1}(U)\}\\ &=\bigcup_{g\in G}\left(\left\{g\right\}\times \rho(g)^{-1}(U) \right) \end{align*} - Since $\rho(g)$ is a homeomorphism, $\rho(g)^{-1}(U)$ is open for any open set $U$. Since $G$ is discrete, each $\left\{g\right\}\times \rho(g)^{-1}(U)$ is open in $G\times X$. Hence $\varrho^{-1}(U)$ is open in $G\times X$, which implies $\varrho$ is continuous. + Since $\rho(g)$ is a homeomorphism, $\rho(g)^{-1}(U)$ is open for any open set $U$. Since $G$ is discrete, each $\left\{g\right\}\times \rho(g)^{-1}(U)$ is open in $G\times X$. Hence $\varrho^{-1}(U)$ as a union of open sets is open in $G\times X$, which implies $\varrho$ is continuous. \end{prf} - -% \begin{proposition}{}{} -% Let $X$ be a locally compact Hausdorff space and $G$ be a topological group. -% \end{proposition} - - \begin{definition}{Orbit Space}{} - Let $G$ be a group acting on a topological space $X$. The \textbf{orbit space} of $X$ under the action of $G$ is the quotient space $X / G$ obtained by identifying all points in $X$ that are in the same orbit. $X / G$ is equipped with the quotient topology: a subset $U\subseteq X / G$ is open if and only if $\pi^{-1}(U)$ is open in $X$, where $\pi:X\to X / G$ is the quotient map. + Let $G$ be a group acting on a topological space $X$. The \textbf{orbit space} of $X$ under the action of $G$ is the quotient space $G\backslash X $ obtained by identifying all points in $X$ that are in the same orbit. $G\backslash X $ is equipped with the quotient topology: a subset $U\subseteq G\backslash X $ is open if and only if $\pi^{-1}(U)$ is open in $X$, where $\pi:X\to G\backslash X$ is the quotient map. \end{definition} \begin{proposition}{}{} - For any continuous action of a topological group $G$ on a topological space $E$, the quotient map $p: E \rightarrow E / G$ is an open map. + For any continuous action of a topological group $G$ on a topological space $E$, the quotient map $p: E \rightarrow G\backslash E$ is an open map. \end{proposition} \begin{prf} @@ -168,16 +164,11 @@ \section{Continuous Topological Group Action} $$ g \cdot U=\{g \cdot x: x \in U\} . $$ -If $U \subseteq M$ is open, then $\pi^{-1}(\pi(U))$ is equal to the union of all sets of the form $g \cdot U$ as $g$ ranges over $G$. Since $p \mapsto g \cdot p$ is a homeomorphism, each such set is open, and therefore $\pi^{-1}(\pi(U))$ is open in $M$. Becaues $\pi$ is a quotient map, this implies that $\pi(U)$ is open in $M / G$, and therefore $\pi$ is an open map. +If $U \subseteq M$ is open, then $\pi^{-1}(\pi(U))$ is equal to the union of all sets of the form $g \cdot U$ as $g$ ranges over $G$. Since $p \mapsto g \cdot p$ is a homeomorphism, each such set is open, and therefore $\pi^{-1}(\pi(U))$ is open in $M$. Becaues $\pi$ is a quotient map, this implies that $\pi(U)$ is open in $G\backslash M$, and therefore $\pi$ is an open map. \end{prf} -\begin{definition}{Even Action}{} - Let $G$ be a topological group acting continuously on a topological space $E$. The action of $E$ is \textbf{even} if each point $y \in Y$ has some neighborhood $U$ such that $gU\cap U=\varnothing$ for all $g\in G-\{1_G\}$. In other words, each point has some neighborhood $U$ such that $U \cap gU \neq \varnothing$ implies $g=1_G$. -\end{definition} - -If $G$ acts continuously and evenly on a topological space $E$, then each subgroup $H$ of $G$ also acts continuously and evenly on $E$. \section{Topological Ring} \begin{definition}{Topological Ring}{}