Homogeneous quantum groups and their easiness level

Abstract

Given a closed subgroup $G\subset U_N^+$ which is homogeneous, in the sense that we have $S_N\subset G\subset U_N^+$, the corresponding Tannakian category $C$ must satisfy $span(\mathcal{NC}_2)\subset C\subset span(P)$. Based on this observation, we construct a certain integer $p\in\mathbb N\cup\{\infty\}$, that we call "easiness level" of $G$. The value $p=1$ corresponds to the case where $G$ is easy, and we explore here, with some theory and examples, the case $p>1$. As a main application, we show that $S_N\subset S_N^+$ and other liberation inclusions, known to be maximal in the easy setting, remain maximal at the easiness level $p=2$ as well.