In representation theory's terminology, what are the differences between
Branching rule?
Decomposition?
We can take any spin group as examples such as Spin(3) or Spin(8). Are Branching rule vs decomposition the same thing or different?
2026-05-17 05:01:00.1778994060
In representation theory terminology: Branching rule vs decomposition?
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'Decomposition' means that you are given a module and want to write it as a sum of (probably indecomposable) modules, or possibly compute its composition factors. This could be a restricted module, or a tensor product, or anything else.
A branching rule on the other hand is a general description of how to restrict representations to subgroups (or induce from). Usually the group and subgroup are geometrically or combinatorially defined (for example, $S_{n-1}\leq S_n$ or $\mathrm{GL}_{n-1}(q)\leq \mathrm{GL}_n(q)$), and the rule is usually combinatorial in nature. For both of the examples I gave, the ordinary characters and unipotent characters respectively are labelled by partitions, and restriction and Harish-Chandra restriction respectively are given by removing a (removable) box from the partition.