Seminormal forms for the Temperley-Lieb algebra

Let ${\mathbb{T}\mathbb{L}}_n^{\text{\hspace{0.17em}⁣}\mathbb{Q}}$ 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 ${\mathbb{T}\mathbb{L}}_n^{\text{\hspace{0.17em}⁣}\mathbb{Q}}$ 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 ${\mathbf{J}\mathbf{W}}_{\text{\hspace{0.17em}⁣}k}$ for ${\mathbb{T}\mathbb{L}}_k^{\text{\hspace{0.17em}⁣}\mathbb{Q}}$ where $k\le n$. In the second part of the paper we consider the Temperley-Lieb algebra ${\mathbb{T}\mathbb{L}}_n^{\text{\hspace{0.17em}⁣}{\mathbb{F}}_p}$ over the finite field ${\mathbb{F}}_p$, where $p>2$. The KLR-approach to ${\mathbb{T}\mathbb{L}}_n^{\text{\hspace{0.17em}⁣}{\mathbb{F}}_p}$ gives rise to an action of a symmetric group ${\mathfrak{S}}_m$ on ${\mathbb{T}\mathbb{L}}_n^{\text{\hspace{0.17em}⁣}{\mathbb{F}}_p}$, 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 ${}^p\text{\hspace{0.17em}⁣}{\mathbf{J}\mathbf{W}}_{\text{\hspace{0.17em}⁣}n}$ for ${\mathbb{T}\mathbb{L}}_n^{\text{\hspace{0.17em}⁣}{\mathbb{F}}_p}$, that was introduced recently by Burull, Libedinsky and Sentinelli.