Group Theory 5.2 Partitions and Young Diagrams

Gist

 

Partition of n: A partition \lambda\equiv\{\lambda_1, \lambda_2,\dots,\lambda_r\} of the integer n is a sequence of positive integers \lambda_i, satisfy: \lambda_i\ge\lambda_{i+1}, i=1,\dots,r-1; and \sum_{i=1}^{r}\lambda_i=n.

Young Diagram: A partition \lambda is represented graphically by a Young Diagram which consists of n squares arranged in r rows, the ith one of which contains \lambda_i squares.

Theorem: The number of distinct Young diagrams for any given n is equal to the number of classes of S_n-which is, in turn, equal the number of inequivalent irreducible representations of S_n

Young Tableau: A Young tableau is one in which the number 1, 2, \dots, n 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 1, 2, \dots, n 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 S_n can be characterized by cycle structure as it is said in Section 2.3. If we have v_1 1-cycle, v_2 2-cycle, \dots etc., the relationship between \{\lambda_i\} and \{v_i\} is \lambda_i=v_i+v_{i+1}+\dots, as it is illustrated in the figure.

The relationship between young diagram and cycles

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