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The representation ring of the symmetric group S_3 is a fusion ring of rank 3. We wonder its unit group. By direct computation, it needs to solve two inhomogeneous Diophantine equations, but which we can not fix using the function "isolve" in maple.

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The only units are $\pm 1$ and $\pm \epsilon$ where $\epsilon$ is the sign character.

Proof: Let $G$ be any finite group, let $R_G$ be its representation ring and let $g \in G$. Then $V \mapsto \chi_V(g)$ is a map of rings from $R_G$ to the algebraic integers. In the case that the character table is integer valued, as in $S_3$, the map takes values in $\mathbb{Z}$.

Thus, using the three conjugacy classes of $S_3$, we get three ring maps from $R_{S_3}$ to $\mathbb{Z}$. The matrix of this map is simply the transpose of the character table: $$\begin{bmatrix} 1&1&2 \\ 1&-1&0 \\ 1&1&-1 \\ \end{bmatrix}.$$

Any unit in $R_{S_3}$ must be taken to a unit of $\mathbb{Z}^3$, meaning a vector of the form $\begin{bmatrix} \pm 1 \\ \pm 1 \\ \pm 1 \end{bmatrix}$. I simply took these $8$ vectors and checked which of them had integer preimages for the matrix above; four did and four did not.

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