nLab Young diagram

The idea

Young diagrams (also called Ferrers diagrams) are used to describe various kinds of objects in algebra, combinatorics and representation theory, including:

• integer partitions. For example, the integer partition

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

  is drawn as the Young diagram

• conjugacy classes in symmetric groups $S_n$.

• irreducible representations:

  • irreducible representations of the symmetric groups $S_n$ over any field of characteristic zero (see here)

  • irreducible (algebraic) representation of the special linear groups $SL(N,\mathbb{C})$

  • irreducibleunitary representations of the special unitary groups $SU(N)$

• elementary symmetric functions

• Schur functors

• basis vectors for the free lambda-ring on one generator, $\Lambda$

• flag varieties for the special linear groups $SL(N,k)$, where $k$ is any field

• characteristic classes for complex vector bundles: that is, cohomology classes on the classifying spaces of the general linear groups $GL(N,\mathbb{C})$

• characteristic classes for hemitian vector bundles: that is, cohomology classes on the classifying spaces $B U(N)$ of the unitary groups $U(N)$

• finite-dimensional C*-algebras: any such algebra is of the form $M_{n_1}(\mathbb{C}) \oplus \cdots \oplus M_{n_k}(\mathbb{C})$ for some unique list of natural numbers $n_1 \ge n_2 \ge \cdots \ge n_k$.

• finite abelian p-groups: any such group is of the form $\mathbb{Z}/p^{n_1} \oplus \cdots \oplus \mathbb{Z}/p^{n_k}$ for some unique list of natural numbers $n_1 \ge n_2 \ge \cdots \ge n_k$.

• finite commutative semisimple algebras over a prime field $\mathbb{F}_p$: any such algebra is of the form $\mathbb{F}_{p^{n_1}} \oplus \cdots \oplus \mathbb{F}_{p^{n_k}}$ for some unique list of natural numbers $n_1 \ge n_2 \ge \cdots \ge n_k$.

• the trace of the category of finite sets has isomorphism classes of objects corresponding to Young diagrams.

Young diagram

A Young diagram $F^\lambda$, also called Ferrers diagram, is a graphical representation of an unordered integer partition $\lambda = (\lambda_1\ge\lambda_2\ge\ldots\ge\lambda_l$). If $\lambda\vdash n$ is a partition of $n$ then the Young diagram has $n$ 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 $x_{i,j}$ in the diagram $F^\lambda$.

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 $F^{(5,4,4,2,1,1)}$ representing the partition $(5,4,4,2,1,1)$ of $17$ is given in the English representation as:

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

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

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

Skew Young diagram

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

• 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

…

Related concepts

• semistandard Young tableau,

• hook length formula, hook-content formula

• Schur polynomial

• representation theory of the symmetric group

• representation theory of the general linear group

• Schur-Weyl duality

• Schur-Weyl measure

References

Quick introduction

• Alexander Yong: What is… a Young Tableau, Notices of the American Mathematical Society 54 (2007) 240-241 [pdf]

Textbook accounts are in any book on representation theory in general and on the representation theory of the symmetric group in particular; such as:

• William Fulton, Joe Harris, Representation Theory: a First Course, Springer (1991) [pdf]

• Bruce E. Sagan: The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, Springer (2001)

More details:

• Kazuhiko Koike, Itaru Terada: Young-diagrammatic methods for the representation theory of the classical groups of type $B_n$, $C_n$, $D_n$, Journal of Algebra, Volume 107, Issue 2, May 1987, Pages 466-511

• William Fulton, Young Tableaux, with Applications to Representation Theory and Geometry, Cambridge U. Press, 1997 (doi:10.1017/CBO9780511626241)

• Ron M. Adin, Yuval Roichman, Enumeration of Standard Young Tableaux, Chapter 14 in: Miklós Bóna, Handbook of Enumerative Combinatorics, CRC Press 2015 (arXiv:1408.4497, ISBN:9781482220858)

Relation to algebraic geometry:

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

With an eye towards application to (the standard model of) particle physics:

• Howard Georgi, §12 in: Lie Algebras In Particle Physics, Westview Press (1999), CRC Press (2019) [doi:10.1201/9780429499210]

category: combinatorics