The aim of the lecture is the introduction to the databases and computer tools on representations of point and space groups that can be found on the Bilbao Crystallographic server. Because of the restricted time and practical goal of the school, basic definitions and results of representation theory will be briefly summarized. After a short overview of point-group irreps, the problems of subduced and induced representations will be treated in more details and illustrated by examples and exercises. The second part of the lecture deals with space-group representations. Due to their importance for practical applications, the correlations between irreps of space groups and their direct products are considered separately. The practical exercises are related to the databases and computer tools for representations on the Bilbao server.

Outlines of the lecture:

1. Representations of crystallographic point groups:
   1. General remarks on representations: equivalent representations, reducible and irreducible representations (irreps).
   2. Irreps of Abelian groups: cyclic groups and direct products of cyclic groups; irreps of non-Abelian point groups.
   3. Subduced and induced representations.
2. Representations of space groups
   Irreps of the translation group; little co-group, little group and star of a k-vector; little-group irreps, induction procedure
   1. Correlations between representations of group-subgroup related space groups.
   2. Direct product of representations.
3. Databases of representations of crystallographic groups of the Bilbao Crystallographic Server
   1. Representations of point groups:
   2. Brillouin-zone database: short introduction to the crystallographic classification of wave-vector (k-vector) types; description of the database.
4. Computer tools for space-group representations on the Bilbao Crystallographic Server:
   1. program for irreps of space groups: REPRES.
   2. correlations between irreps of group-subgroup pair: CORREL.
   3. direct product of irreps of space groups: DIRPRO.

Supplementary Material