compute: $$\begin{aligned}
\delta_{{\varepsilon}_{kl}=0}u_{ik}'u_{jl}'
&=\sum_{pq} u_{ip}u_{jq}\otimes\delta_{{\varepsilon}_{kl}=0}u_{pk}u_{ql}\\
&=\sum_{pq} u_{ip}u_{jq}\otimes\delta_{{\varepsilon}_{pq}=0}u_{pk}u_{ql}\\
&=\sum_{pq} \delta_{{\varepsilon}_{ij}=0}u_{ip}u_{jq}\otimes u_{pk}u_{ql}\\
&=\delta_{{\varepsilon}_{ij}=0}u_{ik}'u_{jl}'\end{aligned}$$

\(b) In Lemma \[LemOnQG\] we proved that the relations $R^{\varepsilon}$ pass from $u_{ij}$ to $u_{ij}':=\sum_k u_{ik}\otimes u_{kj}$. Thus, $G^{\varepsilon}$ is a compact matrix quantum group. As for $\mathring G^{\varepsilon}$, we use (a) and Lemma \[LemREpsAeqRRing\].

By the same argument as in Proposition \[PropNoInterpolation\] we see that whenever ${\varepsilon}\neq {\varepsilon}_{\operatorname{comm}}$ and ${\varepsilon}\neq {\varepsilon}_{\operatorname{free}}$, we have: $$S_n\not\subset G^{\varepsilon}\subset O_n^+$$ This is particularly interesting, since the concept of easy quantum groups, as developed by Banica and Speicher [@BS], provides a powerful approach for defining and studying quantum subgroups $G\subset O_n^+$, see also [@RW]. However, they come with the restriction $S_n\subset G\subset O_n^+$. Thus, the ${\varepsilon}$-versions of easy quantum groups are a further step in the direction of understanding all quantum subgroups of $O_n^+$.

Definition of $S_n^{\varepsilon}$
---------------------------------

For $S_n^+$ the quotient by $R^{\varepsilon}$ coincides with the one by $\mathring R^{\varepsilon}$, by Lemma \[LemREpsAeqRRing\]. Hence, we define the ${\varepsilon}$-symmetric group $S_n^{\varepsilon}$ as follows.

The *${\varepsilon}$-symmetric group* $S_n^{\varepsilon}$ is given by the quotient of $S_n^+$ by the relations $\mathring R^{\varepsilon}$, i.e.: $$C(S_n^{\varepsilon})= C^*(u_{ij}\;|\; u_{ij}=u_{ij}^*=u_{ij}^2,\sum_ku_{ik}=\sum_ku_{kj}=1\;\forall i,j\textnormal{ and } \mathring R^{\varepsilon})$$

Viewing ${\varepsilon}\in M_n(\{0,1\})$ as the adjacency matrix of an undirected graph $\Gamma_{\varepsilon}$, we observe that 