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When I use bilbao crystallographic server recently, I noticed a notation called physically irreducible representation.

Paper says it is a direct sum of two complex conjugate representations (if $\Delta$ is a complex irreducible representation, then a physical irreps is $\Delta \oplus \Delta^*$). (http://journals.aps.org/prb/pdf/10.1103/PhysRevB.18.2391)

For example, in this article http://www.bgcryst.com/symp10/proceeding/02_Aroyo_183-197.pdf (see section 4.1.1 full group irreps part (pg 13)), they gave an example on space group no. 135 for k = T(0.37, 1/2, 1/2). It has 4 allowable irreps while its full group irreps characters are listed after those. e.g.

The little group of the k-vector has  4  allowed irreps.
The matrices, corresponding to all of the little group elements are :

Irrep T_1 ,  dimension 1
      1               2               3               4         
(1.000,  0.0)   (1.000,113.4)   (1.000,  0.0)   (1.000,113.4)   


 Irrep T_2 ,  dimension 1
      1               2               3               4         
(1.000,  0.0)   (1.000,113.4)   (1.000,180.0)   (1.000,293.4)   


 Irrep T_4 ,  dimension 1
  1               2               3               4         
(1.000,  0.0)   (1.000,293.4)   (1.000,  0.0)   (1.000,293.4)   


 Irrep T_3 ,  dimension 1
      1               2               3               4         
(1.000,  0.0)   (1.000,293.4)   (1.000,180.0)   (1.000,113.4)   


General position characters:
Gen Pos:      1             2             3             4             5           6             7             8             9             10             11             12             13             14             15             16             
T_1      (4.000,  0.0) (0.000,  0.0) (0.000,  0.0) (0.000,  0.0) (1.836, 90.0) (1.836, 90.0) (0.000,  0.0) (0.000,  0.0) (0.000,  0.0) (0.000,  0.0) (0.000,  0.0) (0.000,  0.0) (1.836, 90.0) (1.836,270.0) (0.000,  0.0) (0.000,  0.0) 
T_2      (4.000,  0.0) (0.000,  0.0) (0.000,  0.0) (0.000,  0.0) (1.836, 90.0) (1.836, 90.0) (0.000,  0.0) (0.000,  0.0) (0.000,  0.0) (0.000,  0.0) (0.000,  0.0) (0.000,  0.0) (1.836,270.0) (1.836, 90.0) (0.000,  0.0) (0.000,  0.0) 
T_4      (4.000,  0.0) (0.000,  0.0) (0.000,  0.0) (0.000,  0.0) (1.836,270.0) (1.836,270.0) (0.000,  0.0) (0.000,  0.0) (0.000,  0.0) (0.000,  0.0) (0.000,  0.0) (0.000,  0.0) (1.836,270.0) (1.836, 90.0) (0.000,  0.0) (0.000,  0.0) 
T_3      (4.000,  0.0) (0.000,  0.0) (0.000,  0.0) (0.000,  0.0) (1.836,270.0) (1.836,270.0) (0.000,  0.0) (0.000,  0.0) (0.000,  0.0) (0.000,  0.0) (0.000,  0.0) (0.000,  0.0) (1.836, 90.0) (1.836,270.0) (0.000,  0.0)    (0.000,  0.0) 

Then it shows:

Physically-irreducible representations:
  *T_1+*T_4  *T_2+*T_3

Can anyone give me some instructions what it means or some references about its definition and physical meaning?

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  • $\begingroup$ This shows the possible degeneracy due to time reversal symmetry. Whether time reversal symmetry double the degeneracy or not is related with the irreps' reality. $\endgroup$ – Xiaoyu Liu Dec 26 '18 at 12:45
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This means this representation cannot be reduced in real numbers. But it can be split into two irreducible represenations if complex numbers are allowed.

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  • 1
    $\begingroup$ Can you expand this answer to provide a more explicit explanation, or a suitable reference? $\endgroup$ – Emilio Pisanty Apr 19 at 12:45

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