---
tag: Q6TH9PH2R7
kind: concept
name: Young tableau
msc: ["05E10", 05A17]
sources:
  - url: https://en.wikipedia.org/wiki/Young_tableau
same_as:
  wikidata: Q2166280
  wikipedia: Young tableau
---

## Summary

A Young tableau is a [[Young diagram|096AHMB3P4]] whose boxes are filled with positive integers, with the entries increasing along rows and down columns.
In a standard tableau the entries are $1, \dots, n$, each used once; in a semistandard tableau repetitions are allowed along rows.
Tableaux index bases of the irreducible representations of the symmetric group and of the general linear group.
They are the combinatorial heart of [[Schur polynomials|43VFMJCDYJ]], the [[hook length formula|VCMS1CP91N]] and the [[Robinson–Schensted correspondence|1SCF8X47G1]].

## Explanations

### L1 human

Draw the rows of boxes of a [[partition|0JV30XN73Z]], longest row on top.
Now write a number in each box so that numbers grow when you move right and when you move down.
For the shape with rows of lengths $2$ and $1$, the fillings of $1, 2, 3$ that work are $\begin{smallmatrix} 1 & 2 \\ 3 & \end{smallmatrix}$ and $\begin{smallmatrix} 1 & 3 \\ 2 & \end{smallmatrix}$.
Counting such fillings answers questions in algebra, probability and the study of permutations.