Seminormal forms for the Temperley-Lieb algebra

Let ${\mathbb{TL}_n^{\! \mathbb Q}}$ be the rational Temperley-Lieb algebra, with loop parameter $2$. In the first part of the paper we study the seminormal idempotents $E_{ \mathfrak{t}}$ for ${\mathbb{TL}_n^{\! \mathbb Q}}$ for $\mathfrak{t}$ running over two-column standard tableaux. Our main result is here a concrete combinatorial construction of $E_{\mathfrak{t}}$ using Jones-Wenzl idempotents ${\mathbf{JW}_{\! k}}$ for ${\mathbb{TL}_k^{\! \mathbb Q}}$ where $k \le n$. In the second part of the paper we consider the Temperley-Lieb algebra ${\mathbb{TL}_n^{\! {\mathbb F}_p}}$ over the finite field ${\mathbb F}_p$, where $p>2$. The KLR-approach to ${\mathbb{TL}_n^{\! {\mathbb F}_p}}$ gives rise to an action of a symmetric group $\mathfrak{S}_m$ on ${\mathbb{TL}_n^{\! {\mathbb F}_p}}$, for some $m < n$. We show that the $E_{ \mathfrak{t}}$'s from the first part of the paper are simultaneous eigenvectors for the associated Jucys-Murphy elements for $\mathfrak{S}_m$. This leads to a KLR-interpretation of the $p$-Jones-Wenzl idempotent $^{p}\!{\mathbf{JW}_{\! n}}$ for ${\mathbb{TL}_n^{\! {\mathbb F}_p}}$, that was introduced recently by Burull, Libedinsky and Sentinelli.