Editing Young tableau (section)

===Restricted representations===

A representation of the symmetric group on {{mvar|''n''}} elements, {{math|''S''<sub>''n''</sub>}} is also a representation of the symmetric group on {{math|''n'' − 1}} elements, {{math|''S''<sub>''n''−1</sub>}}. However, an irreducible representation of {{math|''S''<sub>''n''</sub>}} may not be irreducible for {{math|''S''<sub>''n''−1</sub>}}. Instead, it may be a [[direct sum of representations|direct sum]] of several representations that are irreducible for {{math|''S''<sub>''n''−1</sub>}}. These representations are then called the factors of the [[restricted representation]] (see also [[induced representation]]).

The question of determining this decomposition of the restricted representation of a given irreducible representation of ''S''<sub>''n''</sub>, corresponding to a partition {{mvar|''λ''}} of {{mvar|''n''}}, is answered as follows. One forms the set of all Young diagrams that can be obtained from the diagram of shape {{mvar|''λ''}} by removing just one box (which must be at the end both of its row and of its column); the restricted representation then decomposes as a direct sum of the irreducible representations of {{math|''S''<sub>''n''−1</sub>}} corresponding to those diagrams, each occurring exactly once in the sum.