---- > [!definition] Definition. ([[group-equivariant map]]) > Let $G$ be a [[group]] [[group action|acting]] on a set $S$ via $\cdot$ as well as on a set $Y$ via $\star$. A map $f: S \to Y$ is **$G$-equivariant** if it is compatible with the [[group action]]: $f(g \cdot s)=g \star f(s)$ > for all $g \in G$ and $s \in S$. > When $\star$ is the trivial action, we call $f$ **$G$-invariant**. ([[group-invariant function]]) ^13a1d7 ```tikz \usepackage{tikz-cd} \usepackage{amsmath} \usepackage{amsfonts} \usepackage[mathscr]{euscript} \usepackage{tikzit} \usetikzlibrary{shapes.geometric} \begin{document} % TiKZ style file generated by TikZiT. You may edit this file manually, % but some things (e.g. comments) may be overwritten. To be readable in % TikZiT, the only non-comment lines must be of the form: % \tikzstyle{NAME}=[PROPERTY LIST] % Node styles \tikzstyle{red dot}=[fill={rgb,255: red,191; green,0; blue,64}, draw=black, shape=circle] \tikzstyle{green dot}=[fill={rgb,255: red,0; green,128; blue,128}, draw=black, shape=circle] \tikzstyle{rectangle}=[fill=white, draw=black, shape=rectangle] \tikzstyle{triangle}=[fill=white, draw=black, regular polygon sides=3, regular polygon, shape=regular polygon, tikzit shape=regular polygon, shape border rotate=180, minimum size=5mm, inner sep=0pt] \tikzstyle{bullet}=[fill=black, draw=black, shape=circle, minimum size=2mm, inner sep=0pt] \tikzstyle{noborder}=[fill=none, draw=none, shape=circle] \tikzstyle{widerectangle}=[fill=white, draw=black, shape=rectangle, minimum width=16mm, minimum height=4mm, inner sep=1pt, tikzit shape=rectangle] % Edge styles \tikzstyle{new edge style 0}=[-] %% paste tikzpicture here % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAHEACAHW7wFt4nABogAvqXSZc+QigCM5KrUYs27XgKEBNcZJAZseAkTLzl9Zq0QhREqUdlFF56pbU3dY5TCgBzeCJQADMAJwh+JAAmahwIJABmN1VrEGCQagY6ACMYBgAFaWM5EFCsPwALHD0Q8MjEMhA4pEUVKzZeCDQYULo40LA6QWAsKDEePixBOE50+zS6pEbmxBi2jxBeRjQKugB9UUycvMLHExsyyur5sIiW2PjEJPXUrYYd-a8KMSA \begin{tikzcd} G \times X \arrow[r, "\operatorname{id} \times f"] \arrow[d, "\alpha_X"'] & G\times Y \arrow[d, "\alpha_Y"] \\ X \arrow[r, "f"'] & Y \end{tikzcd} %% end paste \end{document} ``` > [!note] Categorical Remark. > [[group action#^equivalence|Recall]] that the actions $\cdot, \star$ of $G$ on $S,Y$ is are each [[covariant functor|functors]] $G \to \mathsf{Set}$. A [[natural transformation]] between them is, by definition, the datum of a single (since $G$ is a one-object [[category]]) set-function $f:S \to Y$ that commutes with the actions, i.e., a $G$-equivariant map. ^note > [!basicexample] > In signal processing, we care a lot about systems which are time-~~in~~equivariant, i.e., functions $H$ of signals $x$ such that if $H\big(x(t)\big)=y(t)$ then $H\big(x(t-\tau)\big)=y(t-\tau)$. A time-shift in the input corresponds to the output, time-shifted by the same amount. ^f83edd ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```