Bilbao Crystallographic Server STENSOR

Co-irreps for non-trivial and coplanar SpPGs

• (Co-)irreps of the non-trivial SpPGs

  The co-irreps for non-trivial spin point groups correspond exactly to the co-irreps of a closely-related black & white point group. A simple way to find the corresponding black & white point group is to take the decorated Litvin symbol for the non-trivial SpPG and turn every decoration corresponding to an improper (antiunitary) element into a prime, while simply erasing the decorations by proper (unitary) elements. For example, then, the non-trivial spin point group with Litvin number 149, with Litvin symbol ${}_{}{}^{1}4{}_{}{}^{\mathrm{-1}}m{}_{}{}^{\mathrm{-1}}m$ is associated with the black & white point group $\mathrm{4m\text{'}m\text{'}}$ (because -1 is the time-reversal element). Similarly, the non-trivial SpPG with Litvin number 156 and Litvin symbol ${}_{}{}^{2}4{}_{}{}^{\mathrm{-1}}m{}_{}{}^{m}m$ also corresponds to the black & white point group $\mathrm{4m\text{'}m\text{'}}$ (since both -1 and $m$ are improper, while 2 is proper). For a complete list of correspondences, see table 7 in [1]. An in depth explanation of this association can also be found there.

• (Co-)irreps of the coplanar SpPGs

  The co-irreps for coplanar spin point groups correspond to the co-irreps of a closely-related grey point group. To determine the appropriate grey point group, one must first find the "parent" point group of the non-trivial spin point group. This can be easily obtained by erasing all decorations from the Litvin symbol of the non-trivial spin point group. This point group, along with the time-reversed coset form the relevant grey group. For more details, see [1].

Co-irreps of collinear spin point groups

• The spin related quantum numbers

  Due to the continuous SO(2) subgroup appearing in the spin-only part of collinear spin point groups, an additional quantum number arises. It is signified in the upper left-hand corner of the (co-)irrep symbol. While the irreps of SO(2) alone are labeled by a single integer, the collinear spin point group co-irreps are labelled by one of the following:

  $\mathrm{• }\sigma \in \mathbb{Z}$	$\mathrm{• }\nu \in \mathbb{N}\backslash \left\{0\right\}$
  $\mathrm{• }\mu \in \mathbb{Z}\backslash \left\{0\right\}$	$• \pm 0$, or simply 0

  All collinear spin point groups whose non-trivial SpPG is unitary are labelled with $\sigma$. The collinear spin point groups whose non-trivial SpPG have antiunitary elements have irreps labelled by 0, $\pm 0$, $\nu$ or $\mu$. Here, $\pm$ refers to the sign under elements of the form $\{2_{\perp }\Vert \}$. For more details, please refer to [1].
  
  

  (Co-)irrep labeling conventions

  Mulliken proposed a naming convention for the irreducible representations of ordinary point groups [2], which have been used by Bradley & Cracknell for naming the (co-)irreps of magnetic point groups [3]. For the single valued point groups, the co-irrep type is always either (a) or (c), as can be determined by the Dimmock indicator [3-4]. Type (a) irreps retain the Mulliken symbol, whereas pairs of type (c) irreps have their symbols joined together for the co-irrep symbol. We adopt a similar convention for the co-irrep labels of the collinear spin point groups, but additionally taking into account the spin-related quantum number described above.
  
  For the collinear SpPGs with antiunitary non-trivial part, the co-irreps are induced from the irreps of the unitary halving subgroup $X_{\mathrm{1/2}}=\mathrm{SO(2)}\times G_{\mathrm{1/2}}+\{2_{\perp }\Vert a_0\} \mathrm{SO(2)}\times G_{\mathrm{1/2}}$ , whose naming convention is shown below, where $\gamma$ symbolizes the name of the irrep in ${\mathrm{G}}_{\mathrm{1/2}}$.
  
  

  Irrep of $\mathrm{SO(2)}\times G_{\mathrm{1/2}}$	Doubling in PG?	Doubling in spin part?	Irrep(s) of $X_{\mathrm{1/2}}$
  ${}_{''''}{}^0\gamma _{''''}^{''''}$	$✗$	$✗$	${}_{\pm }{}^0\gamma _{''''}^{''''}$
  ${}_{''''}{}^0\gamma _{''''}^{''''}$	$✓$	$✗$	${}_{''''}{}^0\gamma _{''''}^{''''}\stackrel{\sim }{\gamma }$
  ${}_{''''}{}^{\mu }\gamma _{''''}^{''''}$	$✗$	$✓$	${}_{''''}{}^{\nu }\gamma _{''''}^{''''}$
  ${}_{''''}{}^{\mu }\gamma _{''''}^{''''}$	$✓$	$✓$	${}_{''''}{}^{\mu }\gamma _{''''}^{''''}\stackrel{\sim }{\gamma }$

  
  The following table summarizes the naming convention for the co-irreps of the total SpPG based on the Dimmock type of each $X_{\mathrm{1/2}}$ irrep. When the point group irrep symbol $\gamma$ carries upper-left superscripts, and the co-irrep of $X$ has additional left super-script labels but no parentheses, $\gamma$ becomes wrapped in square brackets $''[''\gamma '']''$ for clarity. For an example of the square brackets, see the SpPG N. 13.149.33.L, co-irrep ${}_{}{}^{''0''}''[''{}_{}{}^{''1''}E{}_{}{}^{''2''}E'']''_{''''}^{''''}$.

  Irrep of $X_{\mathrm{1/2}}$	Dimmock Type	Partner Irrep	Co-irrep of $X$
  ${}_{\pm }{}^0\gamma _{''''}^{''''}$	$\mathrm{(a)}$	$-$	${}_{\pm }{}^0\gamma _{''''}^{''''}$
  ${}_{\pm }{}^0\gamma _{''''}^{''''}$	$\mathrm{(c)}$	${}_{\mp }{}^0\gamma _{''''}^{''''}$	${}_{''''}{}^0''(''\gamma '')''_{''''}^{''''}$
  ${}_{\pm }{}^0\gamma _{''''}^{''''}$	$(c)$	${}_{\mp }{}^0\stackrel{\sim }{\gamma }_{''''}^{''''}$	${}_{\pm }{}^0''(''\gamma \stackrel{\sim }{\gamma }'')''_{''''}^{''''}$
  ${}_{}{}^0\gamma \stackrel{\sim }{\gamma }_{''''}^{''''}$	$(a)$	$-$	${}_{}{}^0\gamma \stackrel{\sim }{\gamma }_{''''}^{''''}$
  ${}_{}{}^0\gamma \stackrel{\sim }{\gamma }_{''''}^{''''}$	$(c)$	Not possible because ${}_{}{}^0\gamma \stackrel{\sim }{\gamma }_{''''}^{''''}$ is real
  ${}_{}{}^{\nu }\gamma _{''''}^{''''}$	$(a)$	$-$	${}_{}{}^{\nu }\gamma _{''''}^{''''}$
  ${}_{}{}^{\nu }\gamma _{''''}^{''''}$	$(c)$	${}_{}{}^{\nu }\stackrel{\sim }{\gamma }_{''''}^{''''}$	${}_{}{}^{''(''\nu '')''}''(''\gamma \stackrel{\sim }{\gamma }'')''_{''''}^{''''}$
  ${}_{}{}^{\mu }\gamma \stackrel{\sim }{\gamma }_{''''}^{''''}$	$(a)$	$-$	${}_{}{}^{\mu }\gamma \stackrel{\sim }{\gamma }_{''''}^{''''}$
  ${}_{}{}^{\mu }\gamma \stackrel{\sim }{\gamma }_{''''}^{''''}$	$(c)$	${}_{}{}^{\mathrm{-\mu }}\gamma \stackrel{\sim }{\gamma }_{''''}^{''''}$	${}_{}{}^{''(''\mu '')''}\gamma \stackrel{\sim }{\gamma }_{''''}^{''''}$