Let G be a compact connected Lie group (nonabelian) and $(A,B)$ be a fixed pair of topological generators. Let $T: G\rightarrow U(d)$ be an irrep of dimension $d>1$.

Then $|1+T(A)+T(B)|<3$, because if $|1+T(A)+T(B)|=3$ it would imply the existence of a common fixed vector for operators $T(A)$ and $T(B)$ and so for the whole group $G$ , which would contradict the irreducibility of $T$.

Question: as $d\rightarrow\infty$ what is the asymptotic $\textit{lower bound}$ for the gap ${\Delta}_d=3-|1+T(A)+T(B)|$?

Let G be a compact connected Lie group (nonabelian) and $(A,B)$ be a fixed pair of topological generators. Let $T: G\rightarrow U(d)$ be an irrep of dimension $d>1$.

Then $|1+T(A)+T(B)|<3$, because if $|1+T(A)+T(B)|=3$ it would imply the existence of a common fixed vector for operators $T(A)$ and $T(B)$ and so for the whole group $G$ , which would contradict the irreducibility of $T$.

Question: as $d\rightarrow\infty$ what is the asymptotic lower bound for the gap ${\Delta}_d=3-|1+T(A)+T(B)|$?