Wedge product of permutations in definition of pairing?

Question

In May's definition of a pairing of operads, he states that a pairing of operads $\tau: (A,O)\rightarrow C$ consists of a collection of maps $\tau: A(i)\times O(j)\rightarrow C(ij)$ that satisfies the condition that for $a\in A(i)$, $o\in O(j)$, and for $\sigma\in\Sigma_i$ and $\mu\in\Sigma_j$, we have $\tau(a\sigma,o\mu)=\tau(a,o)(\sigma\wedge\mu)$, where $\sigma\wedge\mu\in \Sigma_{ij}$.

I know of the block permutations used in the definition of operads, but I've never come across a wedge product of permutations before; so my question is what exactly is the permutation $\sigma\wedge\mu$?

Answer 1

In the Appendix: Pairings of operads (starting on page 57) of Equivariant iterated loop space theory and permutative G-categories by B. Guillou and J. P. May it is written explicitly:

Write $\boldsymbol{j}=\{1,\ldots,j\}$ and let $$\otimes: \Sigma_j \times \Sigma_k \to \Sigma_{jk}$$ be the homomorphism obtained by identifying $\boldsymbol{j}\times \boldsymbol{k}$ with $\boldsymbol{jk}$ by ordering the set of $jk$ elements $(q,r)$, $1 \leq q \leq j$ and $1 \leq r \leq k$ lexicographically. More precisely, let $\lambda_{j,k}: \boldsymbol{jk} \to \boldsymbol{j}\times\boldsymbol{k}$ be the lexicographic ordering. Then, given $\rho \in \Sigma_j$ and $\sigma \in \Sigma_k$, $\rho \otimes \sigma$ is defined by $$\boldsymbol{jk} \xrightarrow{\lambda_{j,k}} \boldsymbol{j}\times\boldsymbol{k} \xrightarrow{\rho\times\sigma} \boldsymbol{j}\times\boldsymbol{k} \xrightarrow{\lambda_{j,k}^{-1}} \boldsymbol{jk}.$$

Here, the notation $\rho \otimes \sigma$ is used instead of your $\rho \wedge \sigma$.