The Center of the Temperley-Lieb Algebra

We compute the dimension of the center of the Temperley–Lieb algebra $\mathrm{}(\delta )$ over a field of characteristic zero for every nonzero value of the parameter $\delta$. The proof uses the cellular filtration by cup number, together with known facts about the representation theory of the Temperley–Lieb algebra, especially the structure of its standard modules and their radicals. Dilation and compression maps compare the induced graded pieces of the center at levels $n$ and $n-2$, giving an upper bound of one for each such piece. A deformation argument gives the matching lower bound, and hence $\dim Z({\mathrm{TL}}_n(\delta ))=1+\lfloor \frac{n}{2}\rfloor$. We also prove that every central element is fixed by the canonical anti-automorphism and by the natural diagram-reflection automorphism. Finally, we give a congruence criterion for the trivial-radical case and record a Gram-matrix computation for leading terms.