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Jonathan Gruber

Publications and source records attributed to Jonathan Gruber.

11 recordsLinked to original sources
arXiv2026

Pseudo-centralizers in affine Hecke algebras

We introduce and study a subalgebra $\mathcal{B}$ of the affine Hecke algebra, which arises from a centralizer construction in the double affine Hecke algebra, and which may be regarded as a $v$-deformation of the affine Fomin-Stanley subalgebra introduced by Lam as a combinatorial model for the affine Grassmannian homology ring. In types $\mathsf{A}_n$ and $\mathsf{B}_2$ and $\mathsf{G}_2$, we show that $\mathcal{B}_\mathrm{aff}$ admits a canonical basis indexed by the cosets of the finite Weyl group in the affine Weyl group. We also discuss conjectural positivity properties of the canonical basis and explain how it can be used to study the center of the affine Hecke algebra.

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arXiv2025

Monoidal Ringel duality and monoidal highest weight envelopes

We show that a large class of non-abelian monoidal categories can be realized as subcategories of tilting objects in abelian monoidal categories with a highest weight structure. The construction relies on a monoidal enhancement of Brundan-Stroppel's semi-infinite Ringel duality and applies to many of Sam-Snowden's triangular categories and Knop's tensor envelopes of regular categories. We also explain how monoidal Ringel duality gives rise to monoidal structures on categories of representations of affine Lie algebras at positive levels.

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arXiv2025

Growth problems in diagram categories

In the semisimple case, we derive (asymptotic) formulas for the growth rate of the number of summands in tensor powers of the generating object in diagram/interpolation categories.

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arXiv2024

Multiplicity free and completely reducible tensor products for $\mathrm{SL}_3(\Bbbk)$ and $\mathrm{Sp}_4(\Bbbk)$

Let $G$ be a simple algebraic group over an algebraically closed field $\Bbbk$ of positive characteristic. We consider the questions of when the tensor product of two simple $G$-modules is multiplicity free or completely reducible. We develop tools for answering these questions in general, and we use them to provide complete answers for the groups $G = \mathrm{SL}_3(\Bbbk)$ and $G = \mathrm{Sp}_4(\Bbbk)$.

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arXiv2023

Generic direct summands of tensor products for simple algebraic groups and quantum groups

Let $\mathbf{G}$ be either a simple linear algebraic group over an algebraically closed field of positive characteristic or a quantum group at a root of unity. We define new classes of indecomposable $\mathbf{G}$-modules, which we call generic direct summands of tensor products because they appear generically in Krull-Schmidt decompositions of tensor products of simple $\mathbf{G}$-modules and of Weyl modules. We establish a Steinberg-Lusztig tensor product theorem for generic direct summands of tensor products of simple $\mathbf{G}$-modules and provide examples of generic direct summands for $\mathbf{G}$ of type $\mathrm{A}_1$ and $\mathrm{A}_2$.

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arXiv2023

Linkage and translation for tensor products of representations of simple algebraic groups and quantum groups

Let $\mathbf{G}$ be either a simple linear algebraic group over an algebraically closed field of characteristic $\ell>0$ or a quantum group at an $\ell$-th root of unity. We define a tensor ideal of singular $\mathbf{G}$-modules in the category $\mathrm{Rep}(\mathbf{G})$ of finite-dimensional $\mathbf{G}$-modules and study the associated quotient category $\mathrm{\underline{Re}p}(\mathbf{G})$, called the regular quotient. Our main results are a 'linkage principle' and a 'translation principle' for tensor products: Let $\mathrm{\underline{Re}p}_0(\mathbf{G})$ be the essential image in $\mathrm{\underline{Re}p}(\mathbf{G})$ of the principal block of $\mathrm{Rep}(\mathbf{G})$. We first show that $\mathrm{\underline{Re}p}_0(\mathbf{G})$ is closed under tensor products in $\mathrm{\underline{Re}p}(\mathbf{G})$. Then we prove that the monoidal structure of $\mathrm{\underline{Re}p}(\mathbf{G})$ is governed to a large extent by the monoidal structure of $\mathrm{\underline{Re}p}_0(\mathbf{G})$. These results can be combined to give an external tensor product decomposition $\mathrm{\underline{Re}p}(\mathbf{G}) \cong \mathrm{Ver}(\mathbf{G}) \boxtimes \mathrm{\underline{Re}p}_0(\mathbf{G})$, where $\mathrm{Ver}(\mathbf{G})$ denotes the Verlinde category of $\mathbf{G}$.

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arXiv2022

Tensor ideals for quantum groups via minimal tilting complexes

We use minimal tilting complexes to construct an explicit bijection between the set of thick tensor ideals with the two-out-of-three property in the category of finite-dimensional modules over a quantum group at a root of unity and the set of thick tensor ideals in the subcategory of tilting modules. We also explain why the analogous construction for rational representations of a reductive algebraic group over a field of positive characteristic does not give rise to a bijection.

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arXiv2022

On minimal tilting complexes in highest weight categories

We explain the construction of minimal tilting complexes for objects of highest weight categories and we study in detail the minimal tilting complexes for standard objects and simple objects. For certain categories of representations of complex simple Lie algebras, affine Kac-Moody algebras and quantum groups at roots of unity, we relate the multiplicities of indecomposable tilting objects appearing in the terms of these complexes to the coefficients of Kazhdan-Lusztig polynomials.

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arXiv2021

Coxeter combinatorics for sum formulas in the representation theory of algebraic groups

Let $G$ be a simple algebraic group over an algebraically closed field $\mathbb{F}$ of characteristic $p\geq h$, the Coxeter number of $G$. We observe an easy `recursion formula' for computing the Jantzen sum formula of a Weyl module with $p$-regular highest weight. We also discuss a `duality formula' that relates the Jantzen sum formula to Andersen's sum formula for tilting filtrations and we give two different representation theoretic explanations of the recursion formula. As a corollary, we also obtain an upper bound on the length of the Jantzen filtration of a Weyl module with $p$-regular highest weight in terms of the length of the Jantzen filtration of a Weyl module with highest weight in an adjacent alcove.

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arXiv2020

On complete reducibility of tensor products of simple modules over simple algebraic groups

Let $G$ be a simply connected simple algebraic group over an algebraically closed field $k$ of characteristic $p>0$. The category of rational $G$-modules is not semisimple. We consider the question of when the tensor product of two simple $G$-modules $L(\lambda)$ and $L(\mu)$ is completely reducible. Using some technical results about weakly maximal vectors (i.e. maximal vectors for the action of the Frobenius kernel $G_1$ of $G$) in tensor products, we obtain a reduction to the case where the highest weights $\lambda$ and $\mu$ are $p$-restricted. In this case, we also prove that $L(\lambda)\otimes L(\mu)$ is completely reducible as a $G$-module if and only if $L(\lambda)\otimes L(\mu)$ is completely reducible as a $G_1$-module.

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