Symmetrizers & Basic Trivalent Diagrams

Snippets Page

Drawing of symmetrizers, anti-symmetrizers, and simple diagrams containing them.

Source Code

\begin{equation}\label{eq:symmetrizer-diagram}
    \tikz[xscale=.7]{
        \foreach\xa in{1,1.3,2.7,3}{\draw(\xa,0)to(\xa,1);}
        \draw[draw=black,fill=gray!10](.8,.35)rectangle(3.2,.75);\node[basiclabel]at(2,.55){$n$};
        \node[basiclabel]at(2,.1){$\cdots$};
    }
    =\frac{1}{n!}\sum_{\sigma\in\Sigma_n}
    \tikz[xscale=.7]{
        \foreach\xa in{1,1.3,2.7,3}{\draw(\xa,0)to(\xa,1);}
        \draw[draw=black,fill=gray!10](.8,.35)rectangle(3.2,.75);\node[basiclabel]at(2,.55){$\sigma$};
        \node[basiclabel]at(2,.1){$\cdots$};
    },
    \end{equation}

Basis Vectors for Symmetric Tensor Algebra

    \begin{equation}\label{eq:symmetric-basis}
    {\sf n}_{n-k}=
    \tikz[xscale=.7]{
        \foreach\xa/\xb in{1/$e_1$,3/$e_1$,4/$e_2$,6/$e_2$}{
            \draw(\xa,0)node[small vector]{\xb}to(\xa,1);
        }
        \draw[draw=black,fill=gray!10](.8,.45)rectangle(6.2,.8);\node[basiclabel]at(3.5,.625){$n$};
        \node[basiclabel]at(2,0){$\cdots$};\node[basiclabel]at(5,0){$\cdots$};
        \draw[decorate,decoration=brace](3.2,-.35)to node[auto,basiclabel]{$n-k$}(.8,-.35);
        \draw[decorate,decoration=brace](6.2,-.35)to node[auto,basiclabel]{$k$}(3.8,-.35);
    }
    \qquad
    {\sf n}^*_{n-k}=
    \tikz[xscale=.7,yscale=-1,shift={(0,-.5)}]{
        \foreach\xa/\xb in{1/$e_1$,3/$e_1$,4/$e_2$,6/$e_2$}{
            \draw(\xa,0)node[small vector]{\xb}to(\xa,1);
        }
        \draw[draw=black,fill=gray!10](.8,.45)rectangle(6.2,.8);\node[basiclabel]at(3.5,.625){$n$};
        \node[basiclabel]at(2,0){$\cdots$};\node[basiclabel]at(5,0){$\cdots$};
        \draw[decorate,decoration=brace](.8,-.35)to node[auto,basiclabel]{$n-k$}(3.2,-.35);
        \draw[decorate,decoration=brace](3.8,-.35)to node[auto,basiclabel]{$k$}(6.2,-.35);
    }.
    \end{equation}

    \begin{equation}\label{eq:thick-strands}
    {\sf n}_i = \tikz[trivalent]{\draw(0,0)node[small vector]{${\sf n}_i$}to(0,1)node[rightlabel]{$n$};}
    \qquad
    {\sf n}^*_i = \tikz[trivalent]{\draw(0,0)node[rightlabel]{$n$}to(0,1)node[small vector]{${\sf n}_i$};}
    \end{equation}

Expansions

    \begin{equation}\label{eq:trivalent-cap}
    \tikz[trivalent]{\draw(0,.7)arc(180:360:.5)node[rightlabel]{$n$};}\equiv
    \tikz{\draw(-.1,1)--(-.1,.8)arc(180:360:.3)--(.5,1);\draw(-.7,1)--(-.7,.8)arc(180:360:.9)--(1.1,1);
        \draw[draw=black,fill=gray!10](.05,.7)rectangle(.35,-.2);\node[basiclabel,scale=.9]at(.2,.25){$n$};
        \node[basiclabel,scale=.7]at(-.375,.8){$\cdots$};\node[basiclabel,scale=.7]at(.825,.8){$\cdots$};}
    \qquad\text{and}\qquad
    \tikz[trivalent]{\draw(0,.2)arc(180:0:.5)node[rightlabel]{$n$};}\equiv
    \tikz[yscale=-1,shift={(0,-.8)}]{\draw(-.1,1)--(-.1,.8)arc(180:360:.3)--(.5,1);\draw(-.7,1)--(-.7,.8)arc(180:360:.9)--(1.1,1);
        \draw[draw=black,fill=gray!10](.05,.7)rectangle(.35,-.2);\node[basiclabel,scale=.9]at(.2,.25){$n$};
        \node[basiclabel,scale=.7]at(-.375,.8){$\cdots$};\node[basiclabel,scale=.7]at(.825,.8){$\cdots$};}
    .
    \end{equation}

    \begin{equation}\label{eq:trivalent-vertex}
    \tikz[trivalent,shift={(0,.5)}]{
        \foreach\xa/\xb in{150/a,30/b,-90/c}{\draw[rotate=\xa](0,0)--(0:.8)(0:1)node[basiclabel]{$\xb$};}}
    =
    \tikz[scale=.6,shift={(0,1)}]{
        \foreach\xa in{0,120,240}{
            \draw[rotate=\xa](28:2.2)to[bend right](-88:2.2)(0:2.5)to[bend right](-60:2.5);}
        \foreach\xa/\xb in{30/b,150/a,270/c}{
            \draw[rotate=\xa,draw=black,fill=gray!10](1.2,1.5)rectangle(1.8,-1.5);
            \draw[rotate=\xa](0:1.52)node[basiclabel]{$\xb$};
            \draw[rotate=\xa](60:1.25)node[basiclabel,rotate=\xa,rotate=60]{$\cdots$};
        }
    }
    \end{equation}