Braid action on a fixed-charge fusion space

Fix a unitary braided fusion category $\mathcal{C}$ and the simple-object representative set $\mathrm{Irr}(\mathcal{C})$ used in Multiplicity-aware fusion-tree basis. Let $x,c\in \mathrm{Irr}(\mathcal{C})$, let $n\ge 2$, and suppose $V(x^n\to c)\mathrm{\ne }0$. Use the left-associated fusion-tree basis. For the Artin generator ${\sigma }_i$ that exchanges the $i$th and $(i+1)$st strands, let $T_i$ be the specified sequence of $F$-moves that changes from the fixed tree basis to a basis where those two charges fuse first, and let $R_i$ apply the corresponding $R$-symbol on each pair-charge channel. Define ${\rho }_c({\sigma }_i):=T_i^{-1}{R}_i{T}_i\mathrm{.}$ This defines the generator assignment. The next theorem proves that the assignment descends from words to a well-defined group homomorphism. Word products use matrix multiplication acting right-to-left on column vectors. The abstract group in this definition is the Artin group The braid group by Artin presentation; the positive crossing is the generator convention used for the displayed $R$-symbols. If a motion is described chronologically, its matrix product is written in reverse temporal order.

The diagram depicts the positive adjacent generator on two identical charges. Time runs from top to bottom; the continuous left-input strand passes over the broken right-input strand. The crossing represents $c_{x,x}$, while its pullback on a fusion space uses the inverse in the formula above. This fixes the $R$-matrix orientation; the proof of the Artin relations is categorical.

The geometric configuration convention in Geometric braid classes and the unordered configuration fundamental group uses first-loop-then-second multiplication, while raw slicing reverses stacking. This definition fixes the algebraic Artin action directly and makes no extra assertion that the library's geometric group is identified with the Artin presentation under a stronger axiom branch.