Complex Temperley–Lieb critical labels and reflection

Let $q\in {\mathbb{C}}^{\times }$. Define $\kappa =\mathrm{ord}(q^2)$ if $q^2$ has finite multiplicative order, and $\kappa =\mathrm{\infty }$ otherwise. For integers $n\ge 0$ and $p$ with $0\le p\le \lfloor n\mathrm{/}2\rfloor$, set the defect label $t=n-2p$; here $p$ counts arcs and $t$ counts defects. The pair $(n,p)$ is critical at $q$ when $\kappa <\mathrm{\infty }$ and $\kappa \mid (t+1)$, equivalently $q^{2(t+1)}=1$. If $\kappa =\mathrm{\infty }$, no pair is critical. When $\kappa =1$ every allowed pair is critical.

For finite $\kappa \ge 2$ and a noncritical pair, write uniquely $t+1=k\kappa +r,k\ge 0,1\le r<\kappa \mathrm{.}$ Its reflection label across the next critical line is $p^{\mathrm{♯}}=p+r-\kappa \mathrm{.}$ This reflected label is called present only when $p^{\mathrm{♯}}\ge 0$; it then automatically satisfies $p^{\mathrm{♯}}\le \lfloor n\mathrm{/}2\rfloor$. If $p^{\mathrm{♯}}<0$, there is no label in range on that side. A critical label is treated as a singleton under reflection. These are arithmetic label conventions; they do not by themselves assert radical, composition-factor, Hom-space, or projective-cover formulas.

The order $\kappa$ is kept separate from the recurrence quantum characteristic $\mathrm{\ell }$ of Temperley–Lieb quantum integers and quantum characteristic. For $q=\pm 1$, $\kappa =1$ while the complex recurrence quantum characteristic is $\mathrm{\infty }$.