The Center of the Temperley-Lieb Algebra

Abstract

We compute the dimension of the center of the Temperley--Lieb algebra $\operatorname{TL_n}(δ)$ over a field of characteristic zero for every nonzero value of the parameter $δ$. 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(\operatorname{TL}_n(δ))=1+\Bigl\lfloor \frac{n}{2}\Bigr\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.