The quantum characteristic of an invertible Hecke parameter

Let $F$ be a field and $q\in F^{\times }$ an invertible element. The quantum characteristic $e=e(F,q)$ is the least integer $e\ge 2$ such that $1+q+\cdots +q^{e-1}=0$ in $F$, and $e:=\mathrm{\infty }$ when no such integer exists (The $q$-integer, $q$-factorial and $q$-multinomial coefficients, The generic type-A Hecke algebra). Since $[1{]}_q=1\mathrm{\ne }0$, the least positive integer $e$ with $[e{]}_q=0$, when it exists, automatically satisfies $e\ge 2$, so the two formulations coincide.

Concretely: (i) if $q\mathrm{\ne }1$ then $[m{]}_q=(q^m-1)\mathrm{/}(q-1)$ for all $m\ge 0$ — indeed $(q-1)(1+q+\cdots +q^{m-1})=q^m-1$ and $q-1\mathrm{\ne }0$ — so $[e{]}_q=0$ is equivalent to $q^e=1$, and a finite $e$ exists exactly when $q$ is a root of unity and then $e$ is the multiplicative order of $q$ in $F^{\times }$ (which is $\ge 2$ because $q\mathrm{\ne }1$); (ii) if $q=1$ then $[m{]}_1=m$, so $e=\mathrm{char}(F)$ when $\mathrm{char}(F)>0$ (the characteristic is a prime, hence at least $2$, and $m\cdot 1_F\mathrm{\ne }0$ for $1\le m<p$) and $e=\mathrm{\infty }$ when $\mathrm{char}(F)=0$ (Group algebra and finite-field specializations of the generic Hecke algebra).

If $e<\mathrm{\infty }$ then $q\mathrm{\ne }1$ implies $q^e=1$, and hence $[m+e{]}_q=[m{]}_q$ for every integer $m$: for $m\ge 0$ the sum splits as $[m+e{]}_q=[m{]}_q+q^m[e{]}_q,$ so $[m+e{]}_q=[m{]}_q$ whenever $[e{]}_q=0$; for the remaining $m<0$ one extends the closed formula by $[m{]}_q:=(q^m-1)\mathrm{/}(q-1)$ when $q\mathrm{\ne }1$ and reads $[m{]}_1:=m\in F$ for the canonical image of the integer $m$ when $q=1$, in which case $e=\mathrm{char}(F)$ and $[m+e{]}_1=m+e=m$ in $F$. In particular the quantum characteristic, not merely the multiplicative order, is the parameter governing the modular representation theory of the Hecke algebra. No choice principle is used.