Sample document

Patrick Massot

Introduction

This is a sample L^aT_eXdocument intended to show what plasT_eXcan 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.

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

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 $G$.

Proof ▼

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.

	(1)	(12)	(123)
$\chi _\mathrm {triv}$	1	1	1
$\chi _\mathrm {sgn}$	1	-1	1
$\chi _\mathrm {std}$	2	0	-1

Figure 1 Character table for $S_3$

	(1)	(12)	(123)
\(\chi _\mathrm {triv}\)	1	1	1
\(\chi _\mathrm {sgn}\)	1	-1	1
\(\chi _\mathrm {std}\)	2	0	-1