---
tag: D17XX5N9VF
kind: concept
name: 6-j symbol
msc: [33C80, 81R05, "22E70"]
sources:
  - url: https://en.wikipedia.org/wiki/6-j_symbol
same_as:
  wikidata: Q2606520
  wikipedia: 6-j symbol
---

## Summary

The Wigner 6-j symbol $\begin{Bmatrix} j_1 & j_2 & j_3 \\ j_4 & j_5 & j_6 \end{Bmatrix}$ is a number attached to six spins.
It compares the two ways to couple three irreducible representations of [[SU(2)|E5P8AG6CK7]]: first the first two, or first the last two.
It is a sum of products of four [[3-j symbols|YJRJVZ998D]], and it is closely related to the Racah W-coefficient and to the [[Racah polynomials|P6JYNDX5DR]].
6-j symbols are used in atomic and nuclear physics, and in the construction of invariants of 3-manifolds.

## Explanations

### L1 human

Three spins can be added in two orders: $(j_1 + j_2) + j_3$ or $j_1 + (j_2 + j_3)$.
Both orders give a basis of the same space, and the change of basis between them is given by 6-j symbols.
So a 6-j symbol is the analogue, for representations, of the associativity of addition.
Its six entries can be drawn on the edges of a tetrahedron, and the symbol has the symmetries of the tetrahedron.