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| Main Authors: | , |
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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2303.10682 |
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| _version_ | 1866914963683016704 |
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| author | Bastías, Katherine Ormeño Ryom-Hansen, Steen |
| author_facet | Bastías, Katherine Ormeño Ryom-Hansen, Steen |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_10682 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Seminormal forms for the Temperley-Lieb algebra Bastías, Katherine Ormeño Ryom-Hansen, Steen Representation Theory 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. |
| title | Seminormal forms for the Temperley-Lieb algebra |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2303.10682 |