 # Group Theory 5.2 Partitions and Young Diagrams

#### Gist

Partition of n: A partition of the integer n is a sequence of positive integers , satisfy: .

Young Diagram: A partition is represented graphically by a Young Diagram which consists of squares arranged in rows, the th one of which contains squares.

Theorem: The number of distinct Young diagrams for any given n is equal to the number of classes of -which is, in turn, equal the number of inequivalent irreducible representations of Young Tableau: A Young tableau is one in which the number appear in order from left to right and from the top row to the bottom row.

Normal Young Tableau: is one in which the numbers appear in order from left to right and from the top row to the bottom row.

Standard Young Tableau: is one in which the numbers in each row appear increasing (not necessarily in strict order) to the right and those in each column appear increasing to the bottom.

#### Analysis

The classes of group can be characterized by cycle structure as it is said in Section 2.3. If we have 1-cycle, 2-cycle, etc., the relationship between and is , as it is illustrated in the figure. The relationship between young diagram and cycles