the multiplicity-free sum of all irreducible characters $\sum_{\psi \in {\mathrm{Irr}}(G)} \psi$. Models are interesting because they lead to interesting representations in which the irreducible representations of $G$ live. This is especially the case when the subgroups $H_i$ are taken to be the stabilizers of the orbits of some natural $G$-action.

\[exa2\] Let $G=G(r,n)$ be the group of complex $n\times n$ matrices with exactly one nonzero entry, given by an $r$th root of unity, in each row and column. Assume $r$ is odd. Then $G$ acts on its symmetric elements by $g : X \mapsto gXg^T$, and the distinct orbits of this action are represented by the block diagonal matrices of the form $$X_i {\stackrel{\rm def}{=}}\(\barr{ll} J_{2i} & 0  \\ 0 & I_{n-2i} \earr\),$$ where $J_n$ denotes the $n\times n$ matrix with ones on the anti-diagonal and zeros elsewhere. Write $H_i$ for the stabilizer of $X_i$ in $G$. The elements of $H_i$ preserve the standard copy of ${\mathbb{C}}^{2i} $ in $ {\mathbb{C}}^n$, inducing a map $\pi_i : H_i \to \mathrm{GL}_{2i}({\mathbb{C}})$. If $\lambda_i {\stackrel{\rm def}{=}}\mathrm{det} \circ \pi_i$ then $\{ \lambda_i : H_i \to {\mathbb{C}}\}$ is a model for $G(r,n)$ [@APR Theorem 1.2].

The following definition of Bump and Ginzburg [@BG] captures the 