Spencer quantum-characteristic digits and support set

Let $K$ be a field, let $\delta \in K$, and let $\mathrm{\ell }$ be the recurrence quantum characteristic of $(K,\delta )$ from Temperley–Lieb quantum integers and quantum characteristic. Put $P=\mathrm{char}K$ when $\mathrm{char}K>0$, and put $P=\mathrm{\infty }$ when $\mathrm{char}K=0$. In this definition $\mathrm{\ell }$ is Spencer's recurrence parameter; it is not the complex order $\kappa =\mathrm{ord}(q^2)$.

Suppose first that $\mathrm{\ell }<\mathrm{\infty }$. Set $s_0=1$ and $s_i=\mathrm{\ell }{P}^{i-1}$ for each $i\ge 1$ with $P<\mathrm{\infty }$. If $P=\mathrm{\infty }$, set $s_1=\mathrm{\ell }$ and omit all $s_i$ for $i\ge 2$. Every $M\in \mathbb{N}$ has a unique expansion $M={\sum }_{i=0}^r{m}_i{s}_i,$ where $0\le m_0<\mathrm{\ell }$ and $0\le m_i<P$ for $i\ge 1$ when $P<\mathrm{\infty }$. When $P=\mathrm{\infty }$, write $M=m_0+m_1\mathrm{\ell }$ with $0\le m_0<\mathrm{\ell }$ and $m_1\ge 0$; if $m_1=0$, take $r=0$. The highest digit $m_r$ is nonzero unless $M=0$, for which take $r=0$ and $m_0=0$.

For a label $N\ge 0$, let $n_i$ be the digits of $N+1$, so $N+1={\sum }_{i=0}^r{n}_i{s}_i$. Define the Spencer support set ${\mathrm{Supp}}_{\mathrm{\ell },P}(N)=\{{\sum }_{i=0}^r{\epsilon }_i{n}_i{s}_i-1:{\epsilon }_i\in \{-1,1\},{\epsilon }_r=1,{\sum }_{i=0}^r{\epsilon }_i{n}_i{s}_i-1\ge 0\}\mathrm{.}$ If $\mathrm{\ell }=\mathrm{\infty }$, define ${\mathrm{Supp}}_{\mathrm{\ell },P}(N)=\{N\}$.

This item defines the digit expansion and a finite set only. Spencer's theorem identifying these sets with decomposition multiplicities, and the separate Jones–Wenzl existence criterion, are not asserted here; both require the full field hypotheses and their own proofs.