nLab Young diagram

The idea

Young diagrams (also called Ferrers diagrams) are used to describe many objects in algebra and combinatorics, including:

• integer partitions. For example, the integer partition

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

  is drawn as the Young diagram

• conjugacy classes in $S_n$.

• irreducible representations:

  • irreducible representations of the symmetric groups $S_n$ over any field of characteristic zero

  • 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 (February 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, Berlin, 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)

Connection 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