Young diagram

of order $ m $

A graphical representation of a partition $\lambda = ( \lambda _ {1} \dots \lambda _ {r} ) $ of a natural number $m$ (where $ \lambda _ {i} \in \mathbf Z $, $ \lambda _ {1} \geq \dots \geq \lambda _ {r} > 0 $, $ \sum \lambda _ {i} = m $). The Young diagram $t_\lambda $ consists of $ m $ cells, arranged in rows and columns in such a way that the $ i $- th row has $ \lambda _ {i} $ cells, where the first cell in each row lies in one (the first) column. E.g., the partition $ ( 6, 5, 4, 4, 1) $ of 20 is represented by the Young diagram (cf. the diagram on the left).

┌┬┬┬┬┬┐
├┼┼┼┼┼┘
├┼┼┼┼┘
├┼┼┼┤
├┼┴┴┘
└┘

$$ Young {,,,,,|,,,,|,,,|,,,| } spd Young {,,,,,|,\times,\times,\times,\times|,\times,,|,\times,,| } spd Young {,,,,,|,,,\times,\times|,,,\times|,\times,\times,\times| } $$

The transposed Young diagram $ t _ \lambda ^ \prime $ corresponds to the conjugate partition $ \lambda ^ \prime = ( \lambda _ {1} ^ \prime \dots \lambda _ {s} ^ \prime ) $, where $ \lambda _ {j} ^ \prime $ is the number of cells in the $ j $- th column of the Young diagram. Thus, in the example given above the conjugate partition will be $ ( 5, 4, 4, 4, 2, 1) $.

Each cell of a Young diagram defines two sets of cells, known as a hook and a skew-hook. Let $ c _ {ij} $ be the cell situated in the $ i $- th row and the $ j $- th column of a given Young diagram. The hook $ h _ {ij} $ corresponding to it is the set consisting of all cells $ c _ {il} $, $ l \geq j $, and $ c _ {kj} $, $ k \geq i $, while the skew-hook is the least connected set of border cells including the last cell of the $ i $- th row and the last cell of the $ j $- th column. E.g. for the Young diagram chosen on the left, the hook and skew-hook corresponding to the cell $ c _ {22} $ have the shape shown in the centre and on the right of the figure.

The length of a hook (respectively, a skew-hook) is understood to be the number of its cells. The length of the hook $ h _ {ij} $ is $ \lambda _ {ij} = \lambda _ {i} + \lambda _ {j} ^ \prime - i - j + 1 $. By removing from a Young diagram a skew-hook of length $ p $ one obtains a Young diagram of order $ m - p $. The height of a hook (respectively, a skew-hook) is understood to be the number of rows over which the hook (skew-hook) is distributed.

The language of Young diagrams and Young tableaux (cf. Young tableau) is applied in the representation of the symmetric groups and in the representation of the classical groups. It was proposed by A. Young (cf. ).

References

[1a]	A. Young, "On quantitative substitutional analysis" Proc. London Math. Soc. , 33 (1901) pp. 97–146 MR0049156 Zbl 32.0157.02
[1b]	A. Young, "On quantitative substitutional analysis" Proc. London Math. Soc. , 34 (1902) pp. 361–397 MR0049156 Zbl 33.0158.03

Comments

A Young diagram is also known as a Ferrers diagram in the West.

References

[a1]	A. Kerber, G.D. James, "The representation theory of the symmetric group" , Addison-Wesley (1981) MR0644144 Zbl 0491.20010
[a2]	A. Kerber, "Algebraic combinatorics via finite group actions" , B.I. Wissenschaftsverlag Mannheim (1991) MR1115208 Zbl 0726.05002
[a3]	G.E. Andrews, "The theory of partitions" , Addison-Wesley (1976) MR0557013 Zbl 0371.10001
[a4]	I.G. Macdonald, "Symmetric functions and Hall polynomials" , Clarendon Press (1979) MR0553598 Zbl 0487.20007