TITLE: Is the Perron-Frobenius dimension of a G-Set given by its cardinality?
QUESTION [5 upvotes]: Given a ring $R$ with finite additive basis $\{e_i\}_{i=1}^{n}$, such that $e_i e_j=\sum c_{ijk}e_k$ with $c_{ijk}\in \mathbb{N}$, we define the Perron-Frobenius dimension $FPDim(e_i)$ of a basis element $e_i$ to be the maximal positive real eigenvalue of matrix $M_{e_i}$, multiplication by $e_i$. This exists by the Perron-Frobenius theorem, and we extend by linearity to all of $R$.
Now take $B(G)$ to be the burnside ring of a finite group $G$, with basis given by isomorphism classes of transitive actions of $G$. One can directly check for $G\cong C_p$ that $FPDim(X)=|X|$. Does this hold in general for finite $G$-sets?
Note that our ring $B(G)$ is not necessarily transitive in the sense of Etingof's Tensor Categories (Definition 3.3.1), so this doesn't seem to follow immediately from the results in there.

REPLY [7 votes]: For $H \subset G$, write $e_H$ for the basis element of $B(G)$ corresponding to the $G$-set $G/H$. Recall that for $K \subset G$, 
$$
e_H\cdot e_K = \sum\limits_{H g K \in H \backslash G / K} e_{H \cap g K g^{-1}}.
$$
Say that $K$ is subconjugate to $H$ if $K$ is conjugate to a subgroup of $H$. This gives rise to a partial order on the set of conjugacy classes of subgroups of $G$, and hence on the basis elements $e_H$. With respect to this partial ordering, $M_{e_H}$ is lower-triangular for each $H \subset G$. The eigenvalues are then the diagonal elements. The diagonal element corresponding to $e_{\{\textrm{id}_G\}}$ is $|G/H|$. The diagonal element corresponding to $e_K$ for $K \subset G$ is bounded above by the number of $(H, K)$-double cosets in $G$, which is bounded above by $|G/H|$.