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I want to know how to typeset character tables of point groups correctly. Consider the following two extremes using the $C_{\text{2v}}$ (or is it $\text{C}_{\text{2v}}$?) point group as an example.

Firstly, one where all the symbols (except for the symbols denoting vertical) are cursive (your default LaTeX math mode behaviour):

C2v character table in full math mode

Second, a table where all the symbols are in roman font.

C2v character table fully roman

Which is correct? Or is it a mixture?

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    $\begingroup$ In this book about group theory written for mathematicians and physicists they use upright letters for the irreducible representations and "group names", e.g. $\mathrm{B}_{3\mathrm{u}}$ and $\mathrm{D}_{6\mathrm{h}}$, while the symmetry elements are typeset in italics but subscripts that do not correspond to variables get an upright font, e.g. $\sigma_{\mathrm{d}}$ and $C_{4z}$. $\endgroup$ – Philipp Jun 14 '15 at 15:10
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Notations and conventions used for the description of symmetry in rigid molecules are established in Notations and conventions in molecular spectroscopy: Part 2. Symmetry notation (IUPAC Recommendations 1997):

Symbols for symmetry operators are printed italic, with sub- and super-scripts that are upright except for the variables $k$ and $n$ which are replaced by numbers for a specific operator.

For example:
$E$ – identity operator
$\sigma$ – reflection operator for reflection across a general plane
$\sigma_\mathrm{h}$ – reflection operator for reflection across a horizontal plane
$\sigma_\mathrm{v}$ – reflection operator for reflection across a vertical plane
$C_n{}^k$ – $n$-fold rotation operator for $k$ successive rotations through an angle of $2\pi/n$ about an $n$-fold rotation axis, where $n = 2, 3, \ldots$; $k = 1, 2, \ldots, (n-1)$
$S_n{}^k$ – $n$-fold rotation-reflection operator for $k$ successive rotation-reflections about an $n$-fold rotation-reflection axis for a rotation through an angle of $2\pi/n$ followed by a reflection in a plane perpendicular to the axis, where $n = 2, 3, \ldots$; $k = 1, 2, \ldots, (n-1)$
$i$ – inversion operator through the centre of symmetry

Likewise,

Symbols for symmetry groups are printed italic with upright subscripts, following common usage.

For example:
$C_n$, $S_{2n}$, $D_n$, $D_{n\mathrm{h}}$, $D_{n\mathrm{d}}$, $C_{n\mathrm{v}}$, $C_{n\mathrm{h}}$, $T$, $T_\mathrm{h}$, $T_\mathrm{d}$, $O$, $O_\mathrm{h}$, $I_\mathrm{h}$, $C_{\infty\mathrm{v}}$, $D_{\infty\mathrm{h}}$

However,

Symbols for irreducible representations of point groups, which are usually called symmetry species in spectroscopy, are printed upright.

For example:
$\mathrm{A}$, $\mathrm{B}$, $\mathrm{E}$, $\mathrm{A}_1$, $\mathrm{A}_2$


The example table given in the question should thus be typeset as follows:

C2v character table with correct formatting

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