Young Tableaux — Mathematics Knowledge System

Overview

Young tableaux are fillings of Young diagrams with positive integers subject to certain monotonicity conditions. Standard Young tableaux (entries 1 to n, strictly increasing along rows and columns) enumerate bases of Specht modules and count the number of paths in certain posets. Semistandard Young tableaux (weakly increasing rows, strictly increasing columns) give bases for GL_m-representations. Young tableaux appear throughout combinatorics, algebraic geometry (Schubert calculus), and physics (tensor products in quantum groups).

Intuition

Fill a Young diagram with numbers. Standard Young tableaux (SYT) require numbers 1 through n, each appearing once, increasing left-to-right in each row and top-to-bottom in each column. The number of SYT of shape λ equals both the dimension of the Specht module S^λ and the number of maximal chains in Young's lattice from ∅ to λ.

Formal Definition

Definition

Let λ = (λ₁ ≥ … ≥ λₖ) be a partition. A Young diagram of shape λ is a collection of cells arranged in rows, the ith row having λᵢ cells, left-justified. A filling T assigns a positive integer T(i,j) to each cell (i,j).

T \text{ is a standard Young tableau (SYT)} \iff T(i,j) < T(i,j+1) \text{ and } T(i,j) < T(i+1,j)

Standard Young tableau: strictly increasing along rows and columns

T \text{ is semistandard (SSYT)} \iff T(i,j) \leq T(i,j+1) \text{ and } T(i,j) < T(i+1,j)

Semistandard Young tableau: weakly along rows, strictly down columns

f^\lambda = |\mathrm{SYT}(\lambda)| = \frac{n!}{\prod_{(i,j)} h(i,j)}

Number of standard Young tableaux = hook-length formula

s_\lambda(x_1,\ldots,x_m) = \sum_{T \in \mathrm{SSYT}(\lambda,m)} x^T

Schur polynomial = generating function for SSYT with entries in {1,...,m}

Notation

Notation	Meaning
$\mathrm{SYT}(\lambda)$	Set of standard Young tableaux of shape λ
$\mathrm{SSYT}(\lambda,m)$	Semistandard Young tableaux of shape λ with entries in {1,…,m}
$s_\lambda$	Schur symmetric function
$c_{\mu\nu}^\lambda$	Littlewood–Richardson coefficient

Properties

RSK and longest increasing subsequences

The length of the first row of P(\sigma) equals the length of the longest increasing subsequence of \sigma

Schur functions from SSYT

The Schur function s_\lambda(x_1,\ldots,x_m) = \sum_{T \in \mathrm{SSYT}(\lambda,m)} \prod_{i} x_i^{|T^{-1}(i)|}

Dimension from SSYT

|\mathrm{SSYT}(\lambda, m)| = \dim V^\lambda_m \quad \text{(dimension of GL}_m(\mathbb{C})\text{-irrep of highest weight }\lambda\text{)}

Theorems

Theorem 1: RSK Correspondence

There is a bijection \sigma \mapsto (P(\sigma), Q(\sigma)) from permutations in S_n to pairs of standard Young tableaux of the same shape \lambda \vdash n.

Theorem 2: Littlewood–Richardson rule

s_\mu \cdot s_\nu = \sum_\lambda c_{\mu\nu}^\lambda\, s_\lambda, \quad \text{where } c_{\mu\nu}^\lambda = |\mathrm{LR}(\lambda/\mu, \nu)|

Theorem 3: Hook-length formula

|\mathrm{SYT}(\lambda)| = \frac{n!}{\prod_{(i,j) \in \lambda} h(i,j)}

Worked Examples

1
T₁ = [[1,2],[3]] and T₂ = [[1,3],[2]].
$T_1 = \begin{array}{ll}1 & 2 \\ 3\end{array},\quad T_2 = \begin{array}{ll}1 & 3 \\ 2\end{array}$

✓ Answer

There are exactly 2 SYT of shape (2,1), confirming dim S^{(2,1)} = 2.

Practice Problems

Mediumfree response

Count the number of SYT of shape (3,2) using the hook-length formula.

Hardfree response

State Schensted's theorem and explain how RSK encodes the length of the longest increasing subsequence.

Common Mistakes

Common Mistake

Confusing standard and semistandard Young tableaux

SYT: entries 1 to n, each once, strictly increasing in both rows and columns. SSYT: entries from {1,…,m}, repeats allowed, weakly increasing along rows, strictly down columns. They count different objects: SYT give dim S^λ; SSYT give coefficients of Schur functions.

Common Mistake

Thinking the LR coefficient c^λ_{μν} is commutative (c^λ_{μν} = c^λ_{νμ})

The Littlewood–Richardson coefficients are symmetric: c^λ_{μν} = c^λ_{νμ}, because both equal the multiplicity of S^λ in S^μ ⊗ S^ν. However this commutativity is a theorem, not a definition.

Quiz

A standard Young tableau of shape λ ⊢ n is a filling with:

Arbitrary positive integersThe numbers 1 to n with strictly increasing rows and columnsThe numbers 1 to n weakly increasing in rows, strictly in columnsRepeated entries in weakly increasing order

The RSK correspondence maps a permutation σ ∈ S_n to:

A single Young diagramA pair of standard Young tableaux of the same shapeA semistandard Young tableauA Schur function

Historical Background

Alfred Young introduced his eponymous tableaux in papers published between 1900 and 1928, developing them as a tool to construct idempotents in the group algebra of S_n. Robinson (1938) and Schensted (1961) independently discovered the RSK correspondence, linking permutations to pairs of standard Young tableaux. Knuth (1970) extended this to the full RSK algorithm for matrices. The connection to symmetric functions was systematised by Littlewood and Richardson, whose rule for computing tensor product multiplicities (Littlewood–Richardson coefficients) is still a cornerstone of modern combinatorics.

1900–1928
Young introduces Young diagrams and tableaux in a series of papers
Alfred Young
1938
Robinson discovers the RSK correspondence
Gilbert de Beauregard Robinson
1961
Schensted independently rediscovers RSK via longest increasing subsequences
C. Schensted
1970
Knuth generalises RSK to matrices and identifies Knuth equivalence
Donald Knuth

Summary

A Young tableau is a filling of a Young diagram; standard (SYT) requires 1 to n strictly increasing in rows and columns.
The number of SYT of shape λ equals dim S^λ, given by the hook-length formula.
RSK correspondence bijects S_n with pairs of SYT of the same shape; longest increasing subsequence = first row length.
SSYT generate the Schur symmetric functions; LR coefficients count LR tableaux.
Young tableaux appear in Schubert calculus, quantum groups, and the theory of symmetric functions.

References

BookSagan, B. — The Symmetric Group, 2nd ed. (2001), Chapters 3–4
BookFulton, W. — Young Tableaux (1997), Parts 1–2
BookStanley, R. — Enumerative Combinatorics Vol. 2 (1999), Chapter 7
WebsiteWikipedia — Young tableau
WebsiteMathWorld — Young Tableau