arXiv subjects

Théo Pinet

Publications and source records attributed to Théo Pinet.

4 recordsLinked to original sources

arXiv2026

Category $\mathcal{O}$ for truncated shifted Yangians and the bi-infinite Bott-Samelson variety

In this paper, we study the category $\mathcal{O}$ of representations of shifted Yangians associated to a simply-laced simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$. In particular, we prove that the (complexified) Grothendieck ring of this category is isomorphic to the Cox ring of the open bi-infinite Bott-Samelson variety, which is a pro-variety we construct from Bott-Samelson varieties for alternating heaps. Using work of Francone-Leclerc, we prove a conjecture of Hernandez-Zhang by identifying the above Grothendieck ring with a cluster algebra defined by Geiss-Hernandez-Leclerc. Our methods also yield an action of the Langlands dual group $G^{\vee}$ on this Grothendieck ring, and show that the shifted coproducts defined in work of the first and fifth authors with collaborators give rise to coproducts for truncated shifted Yangians. This machinery then allows us to prove further conjectures of Frenkel-Hernandez and Geiss-Hernandez-Leclerc on extended $QQ$-systems, and to obtain a generalization of a duality defined by Hernandez-Leclerc.

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arXiv2024

Inflations for representations of shifted quantum affine algebras

Fix a finite-dimensional simple Lie algebra $\mathfrak{g}$ and let $\mathfrak{g}_J\subseteq\mathfrak{g}$ be a Lie subalgebra coming from a Dynkin diagram inclusion. Then, the corresponding restriction functor is not essentially surjective on finite-dimensional simple $\mathfrak{g}_J$-modules. In this article, we study Finkelberg-Tsymbaliuk's shifted quantum affine algebras $U_q^μ(\mathfrak{g})$ and the associated categories $\mathcal{O}^μ$ (defined by Hernandez). In particular, we introduce natural subalgebras $U_q^ν(\mathfrak{g}_J)\,{\subseteq}\,U_q^μ(\mathfrak{g})$ and obtain a functor $\mathcal{R}_J$ from $\mathcal{O}^{sh}\,{=}\bigoplus_μ\mathcal{O}^μ$ to $\bigoplus_ν(U_q^ν(\mathfrak{g}_J)\text{-Mod})$ using the canonical restriction functors. We then establish that $\mathcal{R}_J$ is essentially surjective on finite-dimensional simple objects by constructing notable preimages that we call inflations. We conjecture that all simple objects in $\mathcal{O}^{sh}_J$ (which is the analog of $\mathcal{O}^{sh}$ for the subalgebras $U_q^ν(\mathfrak{g}_J)$) admit some inflation and prove this for $\mathfrak{g}$ of type A-B or $\mathfrak{g}_J$ a direct sum of copies of $\mathfrak{sl}_2$ and $\mathfrak{sl}_3$. We finally apply our results to deduce certain $R$-matrices and examples of cluster structures over Grothendieck rings.

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arXiv2023

A functor for constructing $R$-matrices in the category $\mathcal{O}$ of Borel quantum loop algebras

We tackle the problem of constructing $R$-matrices for the category $\mathcal{O}$ associated to the Borel subalgebra of an arbitrary untwisted quantum loop algebra $U_q(\mathfrak{g})$. For this, we define an exact functor $\mathcal{F}_q$ from the category $\mathcal{O}$ linked to $U_{q^{-1}}(\mathfrak{g})$ to the one linked to $U_q(\mathfrak{g})$. This functor $\mathcal{F}_q$ is compatible with tensor products, preserves irreducibility and interchanges the subcategories $\mathcal{O}^+$ and $\mathcal{O}^-$ of (D. Hernandez, B. Leclerc, Algebra Number Theory, 2016). We construct $R$-matrices for $\mathcal{O}^+$ by applying $\mathcal{F}_q$ on the braidings already found for $\mathcal{O}^-$ in (D. Hernandez, Rep. Theory, 2022). We also use the factorization of the latter intertwiners in terms of stable maps to deduce an analogous factorization for our new braidings. We finally obtain as byproducts new relations for the Grothendieck ring $K_0(\mathcal{O})$ as well as a functorial interpretation of a remarkable ring isomorphism $K_0(\mathcal{O}^+)\simeq K_0(\mathcal{O}^-)$ of Hernandez--Leclerc.

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arXiv2022

Spin chains as modules over the affine Temperley-Lieb algebra

The affine Temperley-Lieb algebra $\mathsf{a}\hskip-1.8pt\mathsf{TL}_{N}(β)$ is an infinite-dimensional algebra parametrized by a number $β\in \mathbb{C}$ and an integer $N\in \mathbb{N}$. It naturally acts on $(\mathbb{C}^2)^{\otimes N}$ to produce a family of representations labeled by an additional parameter $z\in\mathbb C^\times$. The structure of these representations, which were first introduced by Pasquier and Saleur in their study of spin chains, is here made explicit. They share their composition factors with the cellular $\mathsf{a}\hskip-1.8pt\mathsf{TL}_{N}(β)$-modules of Graham and Lehrer, but differ from the latter representations by the direction of about half of the arrows of their Loewy diagrams. The proof of this statement uses a morphism introduced by Morin-Duchesne and Saint-Aubin as well as new maps that intertwine various $\mathsf{a}\hskip-1.8pt\mathsf{TL}_{N}(β)$-actions on the XXZ chain and generalize applications studied by Deguchi $\textit{et al}$ and after by Morin-Duchesne and Saint-Aubin.

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