Sample document

Patrick Massot


This is a sample LaTeXdocument intended to show what plasTeXcan do. It is made of random excerpts of mathematical texts.

1 Basic typesetting

Of course you can type paragraphs containing some mathematics, such as the following one.

If \(G=\operatorname{GL}_2\) then \({G_{\operatorname{ad}}}=\operatorname{PGL}_2\) and so, as above, \(G_1=\operatorname{SL}_2\). Now \(G_2\) is the set of \((g,h)\in \operatorname{SL}_2\times \operatorname{GL}_2\) with \(g=h\) in \(\operatorname{PGL}_2\), so \(h=\lambda g\) for some unique \(\lambda \in \mathbf{G}_m\) and \(G_2=\operatorname{SL}_2\times \mathbf{G}_m\), with the obvious map to \(\operatorname{GL}_2\) sending \(\mathbf{G}_m\) into the centre (or perhaps its inverse depending on how one is thinking about things, but this doesn’t matter). The subgroup \(\mu _2\) is embedded diagonally of course, because it’s the kernel of the map \(G_2\to G\). Finally we push out via \(\mu _2\to \mathbf{G}_m\) and this gives us \(\operatorname{SL}_2\times \mathbf{G}_m\times \mathbf{G}_m\) modulo the subgroup of order 2 with non-trivial element \((-1,-1,-1)\). But there’s an automorphism of \(\mathbf{G}_m\times \mathbf{G}_m\) sending \((-1,-1)\) to \((-1,1)\) (namely, send \((x,y)\) to \((x,xy)\)) so again \({\tilde{G}}\) is just \(G\times \mathbf{G}_m\).

You can also use displayed formulas such as:

\[ \int _{I \times \Sigma } \Phi ^*\omega = \int _I\left(\int _{\Phi _t(\Sigma )} \iota _X \omega \right)dt. \]

and refer to displayed formulas such as Equation 1 below.

\begin{equation} \label{eq:stokes} \int _M d\omega = \int _{\partial M} \omega \end{equation}

Commutative diagrams using tikz-cd are supported as well.

\begin{tikzcd} [row sep=1cm]
  	{}   & \ker \pi' = T_mM \times \{0_p\} \ar[d, hookrightarrow] \ar["T_m F_p", dr] &  &  \\
    \ker \pi = T_\sigma\Sigma \ar[r] \ar[r] \ar[r] \ar[r, hookrightarrow] &
  	T_m M \times T_p P \ar["\pi", bend right, swap, rr] \ar["\pi'", d]  \ar["T_\sigma F",
  	r, swap]& T_{F(\sigma)} N \ar["\rho", r, twoheadrightarrow]& \nu_{F(\sigma)} A \\
     & T_p P &  &

2 Theorems and proofs

You can state and prove results, and refer to them, for instance Lemma 1 below.

Lemma 1

Splittings of \(0\to \mathbf{G}_m\to \tilde{G}\to G\to 0\) canonically biject with twisting elements for \(G\).


To give a splitting is to give a map \(\tilde{G}\to \mathbf{G}_m\) such that the composite \(\mathbf{G}_m\to {\tilde{G}}\to \mathbf{G}_m\) is the identity; then the induced map \({\tilde{G}}\to G\times \mathbf{G}_m\) is an injection with trivial kernel so is an isomorphism for dimension reasons. If \(\chi :{\tilde{G}}\to \mathbf{G}_m\) is such a character then \(\chi \) gives rise to an element of \(X^*({\tilde{T}})\) which is Galois-stable, whose image in \(\mathbf{Z}\) is 1, and which pairs to zero with each coroot (because \(\chi \) factors through the maximal torus quotient of \({\tilde{G}}\)). Conversely to give such a character is to give a splitting. Now one checks that \(\theta -\chi \) has image in \(\mathbf{Z}\) equal to zero so gives rise to an element of \(X^*(T)\) which is Galois-stable, and pairs with each simple coroot to 1—but this is precisely a twisting element for \(G\). Conversely if \(t\) is a twisting element for \(G\) then \(\theta -t\) gives a splitting of the exact sequence.

3 Enumerations and tables

You can use lists such as:

  • \((\Phi \circ \Psi )_*X=\Phi _*\Psi _*X\)

  • \((\varphi \circ \psi )^*\alpha =\psi ^*\varphi ^*\alpha \)

  • \(\varphi ^*d\alpha =d\varphi ^*\alpha \)

  • \(\Phi ^*(\mathcal{L}_X\alpha )=\mathcal{L}_{\Phi ^{-1}_*X}\Phi ^*\alpha \)

  • \(\Phi ^*(i_X\alpha )=i_{\Phi ^{-1}_*X}\Phi ^*\alpha \)

and tables, possibly inside a figure environment such as Figure 1.





\(\chi _\mathrm {triv}\)




\(\chi _\mathrm {sgn}\)




\(\chi _\mathrm {std}\)




Figure 1 Character table for \(S_3\)