Categories of partitions and orthogonal easy quantum groups – Summary

Sorting with respect to freeness

Free orthogonal easy quantum groups
- Free orthogonal QG ( $O^+$ )
- Free bistochastic QG ( $B^+$ )
- Freely modified bistochastic QG ( $B^{\# +}$ )
- Free modified bistochastic QG ( $B^{\prime +}$ )
- Free hyperoctahedral QG ( $H^+$ )
- Free symmetric QG ( $S^+$ )
- Free modified symmetric QG ( $S^{\prime +}$ )
Half-liberated orthogonal easy quantum groups
- Half-liberated orthogonal QG ( $O^\ast$ )
- Half-liberated bistochastic QG ( $B^{\# \ast}$ )
- Half-liberated hyperoctahedral QG ( $H^\ast$ )
- Hyperoctahedral series ( $H^{(\,\cdot\,)}$ )
Group-theoretical easy QGs
- Higher hyperoctahedral series ( $H^{[\,\cdot\,]}$ )
Interpolating series ( $H^{[\pi_{(\,\cdot\,)}]}$ )
- Square commuting hyperoctahedral QG ( $H^\diamond$ )
Classical orthogonal matrix groups
- Orthogonal group ( $O$ )
- Bistochastic group ( $B$ )
- Modified bistochastic group ( $B'$ )
- Hyperoctahedral group ( $H$ )
- Symmetric group ( $S$ )
- Modified symmetric group ( $S'$ )

$\vertarr10cm free<->group.$

Categories of non-crossing partitions
- Category of non-crossing pair partitions ( $\mathrm{O}^+$ )
- Category of non-crossing partitions with small blocks ( $\mathrm{B}^+$ )
- Cat. of NC even size part. w/ small blocks, even dist. betw. legs ( $\mathrm{B}^{\#+}$ )
- Category of non-crossing partitions of even size with small blocks ( $\mathrm{B}^{\prime+}$ )
- Category of non-crossing partitions with blocks of even size ( $\mathrm{H}^+$ )
- Category of all non-crossing partitions ( $\mathrm{S}^+$ )
- Category of non-crossing partitions of even size ( $\mathrm{S}^{\prime+}$ )
Half-liberated categories of partitions
- Category of pair partitions with even distances between legs ( $\mathrm{O}^\ast$ )
- Cat. of even size part. w/ small blocks, even dist. between legs ( $\mathrm{B}^{\#\ast}$ )
- Cat. of part. w/ blocks of even size, parity-balanced legs ( $\mathrm{H}^\ast$ )
- Categories of the hyperoctahedral series ( $\mathrm{H}^{(\,\cdot\,)}$ )
Group-theoretical hyperoctahedral categories of partitions
- Categories of the higher hyperoctahedral series ( $\mathrm{H}^{[\,\cdot\,]}$ )
Interpolating series ( $\mathrm{H}^{[\pi_{(\,\cdot\,)}]}$ )
- Square commuting hyperoctahedral category ( ${\rm H}^\diamond$ )
Group categories
- Category of all pair partitions ( $\mathrm{O}$ )
- Category of all partitions with small blocks ( $\mathrm{B}$ )
- Category of partitions of even size with small blocks ( $\mathrm{B}'$ )
- Category of partitions with blocks of even size ( $\mathrm{H}$ )
- Category of all partitions ( $\mathrm{S}$ )
- Category of partitions of even size ( $\mathrm{S}'$ )