Symmetrizers & Basic Trivalent Diagrams

Snippets Page

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

# Source Code

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


## Basis Vectors for Symmetric Tensor Algebra

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

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


## Expansions

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

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

page revision: 3, last edited: 11 Feb 2009 19:59