Seminormal forms for the Temperley-Lieb algebra

Let ${\mathrm{TL}}_n^Q$ be the rational Temperley-Lieb algebra, with loop parameter 2. In the first part of the paper we study the seminormal idempotents $E_t$ for ${\mathrm{TL}}_n^Q$ for $t$ running over two-column standard tableaux. Our main result is here a concrete combinatorial construction of $E_t$ using Jones-Wenzl idempotents ${\mathrm{JW}}_k$ for ${\mathrm{TL}}_k^Q$ where $k\le n$.

In the second part of the paper we consider the Temperley-Lieb algebra ${\mathrm{TL}}_n^{F_p}$ over the finite field $F_p$, where $p>2$. The KLR-approach to ${\mathrm{TL}}_n^{F_p}$ gives rise to an action of a symmetric group $S_m$ on ${\mathrm{TL}}_n^{F_p}$, for some $m<n$. We show that the $E_t$'s from the first part of the paper are simultaneous eigenvectors for the associated Jucys-Murphy elements for $S_m$. This leads to a KLR-interpretation of the p-Jones-Wenzl idempotent ${}^p\mathrm{JW}_n$ for ${\mathrm{TL}}_n^{F_p}$, that was introduced recently by Burull, Libedinsky and Sentinelli.