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Young diagram

Contents

The idea

Young diagrams are used to describe many objects in algebra and combinatorics, including:

  • integer partitions. For example, the integer partition

    17=5+4+4+2+1+1 17 = 5 + 4 + 4 + 2 + 1 + 1

    is drawn as the Young diagram

<!-- Created with SVG-edit - http://svg-edit.googlecode.com/ --> Young diagram (5,4,4,2,1,1) Layer 1

Young diagram

A Young diagram F λF^\lambda, also called Ferrers diagram, is a graphical representation of an unordered integer partition λ=(λ 1λ 2λ l\lambda = (\lambda_1\ge\lambda_2\ge\ldots\ge\lambda_l). If λn\lambda\vdash n is a partition of nn then the Young diagram has nn boxes. A partition can be addressed as a multiset over \mathbb{N}.

There are two widely used such representations. The English one uses matrix-like indices, and the French one uses Cartesian coordinate-like indices for the boxes x i,jx_{i,j} in the diagram F λF^\lambda.

In the English representation the boxes are adjusted to the north-west in the 4th quadrant of a 2-dimensional Cartesian coordinate system, with the ‘y’-axis being downward oriented. For instance the diagram F (5,4,4,2,1,1)F^{(5,4,4,2,1,1)} representing the partition (5,4,4,2,1,1)(5,4,4,2,1,1) of 1717 is given in the English representation as:

<!-- Created with SVG-edit - http://svg-edit.googlecode.com/ --> Young diagram (5,4,4,2,1,1) Layer 1

Let 𝕐\mathbb{Y} be the set of Young diagrams. Important functions on Young diagrams include:

  • conjugation: denoted by a prime :𝕐𝕐\prime : \mathbb{Y} \rightarrow \mathbb{Y} reflects the Young diagram along its main diagonal (north-west to south-east). In the above example the conjugated partition would be λ =(6,4,3,3,1)\lambda^\prime=(6,4,3,3,1).
  • weight: wt:𝕐wt \colon \mathbb{Y} \rightarrow \mathbb{N} provides the number of boxes.
  • length: :𝕐\ell \colon \mathbb{Y} \rightarrow \mathbb{N} provides the number of rows or equivalently the number of positive parts of the partition λ\lambda. The length of the conjugated diagram gives the number of columns.
  • plus: +:𝕐×𝕐𝕐::(μ,ν)μ+ν=(μ 1+ν 1,,μ l+ν l)+ \colon \mathbb{Y}\times \mathbb{Y} \rightarrow \mathbb{Y} :: (\mu,\nu) \mapsto \mu + \nu = (\mu_1+\nu_1,\ldots,\mu_l+\nu_l)
  • times: ×:𝕐×𝕐𝕐::(μ,ν)(μν) \times \colon \mathbb{Y}\times \mathbb{Y} \rightarrow \mathbb{Y} :: (\mu,\nu) \mapsto (\mu \cup \nu)_{\ge} the unordered union of the multisets. It follows that μ×ν=(μ +ν ) \mu\times \nu =(\mu^\prime + \nu^\prime)^\prime.

A filling of a Young diagram with elements from a set SS is called a Young tableau.

Skew Young diagram

A generalization of a Young diagram is a skew Young diagram. Let μ,ν\mu,\nu be two partitions, and let νμ\nu \le \mu be defined as i:ν iμ i\forall i : \nu_i\le \mu_i (possibly adding trailing zeros). The skew Young diagram F μ/νF^{\mu/\nu} is given by the Young diagram F μF^\mu with all boxes belonging to F νF^\nu when superimposed removed. If μ=(5,4,4,2,1,1)\mu=(5,4,4,2,1,1) and ν=(3,3,2,1)\nu=(3,3,2,1) then F μ/νF^{\mu/\nu} looks like:

<!-- Created with SVG-edit - http://svg-edit.googlecode.com/ --> Skew Young diagram (5,4,4,2,1,1)/(3,3,2,1) Layer 1
  • A skew diagram is called connected if all boxes share an edge.
  • A skew diagram is called a horizontal strip if every column contains at most one box.
  • A skew diagram is called a vertical strip if every row contains at most one box.
  • conjugation, weight, length extend to skew diagrams accordingly.

Young tableau

References

For a quick online introduction to Young diagrams, try:

  • Alexander Yong, What is … a Young Tableau, Notices of the American Mathematical Society 54 (February 2007), 240–241. (pdf)

A nice introduction to Young diagrams can be found here:

More details are in

  • William Fulton, Young Tableaux, with Applications to Representation Theory and Geometry, Cambridge U. Press, 1997.

  • Kazuhiko Koike, Itaru Terada, Young-diagrammatic methods for the representation theory of the classical groups of type B nB_n, C nC_n, D nD_n, Journal of Algebra, Volume 107, Issue 2, May 1987, Pages 466-511

A connection to algebraic geometry:

  • C. de Concini,D. Eisenbud, C. Procesi, Young diagrams and determinantal varieties, Invent. Math. 56 (1980), 129-165.

category: combinatorics

Revised on June 24, 2017 23:53:55 by Toby Bartels (2605:6000:2b44:1100:fcfb:8220:11b9:95bf)