# 6j symbol The $6j$ symbol is written as$\left\{\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6\end{array}\right\}.$It has the symmetry of column permutations and swapping the first and second rows of any two columns simultaneously, i.e.,$\left\{\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6\end{array}\right\}=\left\{\begin{array}{lll}2 & 1 & 3 \\ 5 & 4 & 6\end{array}\right\}=\left\{\begin{array}{lll}3 & 2 & 1 \\ 6 & 5 & 4\end{array}\right\}=\cdots,$and$\left\{\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6\end{array}\right\}=\left\{\begin{array}{lll}4 & 5 & 3 \\ 1 & 2 & 6\end{array}\right\}=\cdots.$ It can be presented by a tetrahedron. The edges on opposite sides are in the same column. An important example is the $6j$ for the principal series of the modular double of $\mathbb{U}_q \mathfrak{s l}(2 ; \mathbb{R})$, which we will refer to as the $b$-$6j$ symbol. ## Refs - $b$-$6j$ symbol in different regimes - [[2025#Liu, Ming, Sun, Wu, Yang (Nov)]] - convergence of [[0625 Open-closed TQFT|open]] Virasoro TQFT (which is built using the $6j$-symbols) - [[2025#Liu, Ming, Sun, Wu, Yang (Aug)]] - $b$-$6j$ symbol for complex $b$ - [[2026#Meng, Yang]]