arXiv subjects

Daniel K. Nakano

Publications and source records attributed to Daniel K. Nakano.

At least 19 recordsLinked to original sources

arXiv2026

The homological spectrum for monoidal triangulated categories

The authors develop a notion of homological prime spectrum for an arbitrary monoidal triangulated category, ${\mathbf C}$. Unlike the symmetric case due to Balmer, the homological primes of ${\mathbf C}$ are not defined as the maximal Serre ideals of the small module category ${\sf mod}\text{-}{\mathbf C}$, or via a noncommutative ring theory inspired version of this construction. Instead, the authors work with an extended comparison map from the Serre prime spectrum $\operatorname{Spc} ({\sf mod}\text{-}{\mathbf C})$ to the Balmer spectrum $\operatorname{Spc}{\mathbf C}$, and select the maximal elements of each fiber to define the homological spectrum $\operatorname{Spc}^{\text{h}}{\mathbf C}$. A surjective continuous homological comparison map $\operatorname{Spc}^{\text{h}}{\mathbf C} \to \operatorname{Spc}{\mathbf C}$ is constructed and used to formulate an extended Nerves of Steel Condition, stating that this map is a homeomorphism. It is believed that this condition holds for many naturally arising monoidal triangulated categories. Nerves of Steel is shown to hold under stratification and uniformity conditions on ${\mathbf C}$. The proof is based on a general theorem giving an explicit description of the Balmer--Favi support of the pure-injectives associated to all Serre primes of ${\sf mod}\text{-}{\mathbf C}$. It is shown that the validity of Nerves of Steel carries over from ${\mathbf C}$ to a semidirect product ${\mathbf C} \rtimes G$ with an arbitrary group $G$, and when $G$ is infinite, this provides examples of monoidal triangulated categories satisfying Nerves of Steel and whose thick ideals are not centrally generated. Important cases in which the condition is verified include the stable module categories of the coordinate rings of all finite group schemes and the Benson--Witherspoon Hopf algebras.

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arXiv2026

On Donkin's Tilting Module Conjecture IV: New Frontiers

Let $G$ be a simple, simply connected algebraic group scheme defined over $\mathbb{F}_{p}$, and let $G_{r}$ be the $r$th Frobenius kernel. Donkin's famous Tilting Module Conjecture purports that a given indecomposable injective $G_{r}$-module can be realized as the restriction of a specific tilting module. The conjecture was first stated in 1990 and withstood proof for nearly 30 years. The authors discovered a counterexample in 2019. This paper provides some indication about the rich mathematics surrounding the Tilting Module Conjecture that also discusses an older conjecture by Humphreys and Verma. New versions of the Tilting Module Conjectures are formulated given multiple families of new counterexamples. Finally, through a new calculation, the authors present further evidence that the Tilting Module Conjecture should hold for reductive algebraic groups with an underlying root system of Type $\mathrm{A}_{n}$ for all fields of characteristic $p > 0$.

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arXiv2026

Quantum wreath products and Schur--Weyl duality II

In the first part of this series, the authors introduced the quantum wreath product, providing a unified framework that encompasses numerous results previously addressed only through case-by-case analysis. This paper shifts focus to the fundamental construction of modules over these products, termed wreath modules. Our approach utilizes parabolic induction on tensor products combined with a sophisticated labeling scheme based on multipartitions. While the underlying constructions are technically involved, they offer a transparent realization of several prominent module families. Specifically, these wreath modules recover and unify: Simple modules over the Ariki-Koike algebra; Specht and simple modules over the Hu algebra; (anti)spherical modules and Kashiwara-Miwa-Stern modules over the affine Hecke algebra and its pro-p Iwahori variants. Finally, we demonstrate that these wreath modules for the Hu algebra serve as a critical component in solving the Ginzburg-Guay-Opdam-Rouquier problem. This solution enables a concrete realization of Category O for the rational Cherednik algebra in Type D.

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arXiv2026

On the induction functor from group algebras to distribution algebras

Let $G$ be a reductive algebraic group scheme defined over ${\mathbb F}_{p}$ and $k$ be an algebraically closed field of characteristic $p$. There are two associated families of finite group schemes, the $r$-th Frobenius kernels, denoted by $G_r$, and the fixed points of the iterated Frobenius map, the finite groups of Lie type, denoted by $G(\mathbb{F}_q).$ Bendel, Nakano and Pillen initiated the investigation of the induction functor $\operatorname{ind}_{G(\mathbb{F}_q)}^G-$. Using filtrations and truncation, large amounts of data coming from the algebraic group and the Frobenius kernels can be transferred to the finite group. This paper looks at connections between a fundamental theorem of Chastkofsky and Jantzen and the induction functor via the cohomology and representation theory of $G$.

math.GR↗

arXiv2025

Category $\mathcal{O}$ for Lie superalgebras

The authors define a Category $\mathcal{O}$ for any quasi-reductive Lie superalgebra $\mathfrak{g}$ with respect to a triangular decomposition. This much needed approach unifies many important constructions in the existing literature in a rigorous fashion. Our Category $\mathcal{O}$ encompasses all highest weight categories for Lie (super)algebras as well as specific examples which may not be highest weight categories. When the decomposition arises from a principal parabolic subalgebra $\mathfrak{p}$ of $\mathfrak{g}$, the Category $\mathcal{O}$ exhibits rich homological properties. For one, the authors show that in contrast to the case of a semisimple Lie algebra, the Category $\mathcal{O}$ is standardly stratified. Furthermore, the categorical cohomology of $\mathcal{O}$ is a finitely generated ring. This provides a first step towards developing a support variety theory for Category $\mathcal{O}$. It is shown that the complexity of modules in Category $\mathcal{O}$ is finite with an explicit upper bound given by the dimension of the subspace of the odd degree elements in $\mathfrak{g}$. This upgrades results known for $\mathfrak{gl}(m|n)$ to the more general setting. Our arguments are based on foundational connections between the categorical cohomology and the relative Lie superalgebra cohomology as well as the interplay between Category $\mathcal{O}$ for $\mathfrak{g}$ and the Category $\mathcal{O}$ for its corresponding Lie algebra $\mathfrak{g}_{\bar 0}$.

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arXiv2024

On the Hochschild Cohomology for Frobenius Kernels

In this paper the authors investigate the structure of the Hochschild cohomology for Frobenius kernels. The authors first establish some fundamental constructions to compute Hochschild cohomology by using spectral sequences. This enables us to provide a complete description of the $G$-algebra structure of the Hochschild cohomology for the first Frobenius kernel $G_{1}$ where $G=SL_{2}$. This computation heavily relies on the calculation of the adjoint action on the restricted enveloping algebra.

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arXiv2024

Quantum wreath products and Schur-Weyl duality I

In this paper the authors introduce a new notion called the quantum wreath product, which is the algebra $B \wr_Q \mathcal{H}(d)$ produced from a given algebra $B$, a positive integer $d$, and a choice $Q=(R,S,ρ,σ)$ of parameters. Important examples {that arise from our construction} include many variants of the Hecke algebras, such as the Ariki-Koike algebras, the affine Hecke algebras and their degenerate version, Wan-Wang's wreath Hecke algebras, Rosso-Savage's (affine) Frobenius Hecke algebras, Kleshchev-Muth's affine zigzag algebras, and the Hu algebra that quantizes the wreath product $Σ_m \wr Σ_2$ between symmetric groups. In the first part of the paper, the authors develop a structure theory for the quantum wreath products. Necessary and sufficient conditions for these algebras to afford a basis of suitable size are obtained. Furthermore, a Schur-Weyl duality is established via a splitting lemma and mild assumptions on the base algebra $B$. Our uniform approach encompasses many known results which were proved in a case by case manner. The second part of the paper involves the problem of constructing natural subalgebras of Hecke algebras that arise from wreath products. Moreover, a bar-invariant basis of the Hu algebra via an explicit formula for its extra generator is also described.

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arXiv2024

Restricting Rational Modules to Frobenius Kernels

Let $G$ be a connected reductive group over an algebraically closed field of characteristic $p>0$. Given an indecomposable G-module $M$, one can ask when it remains indecomposable upon restriction to the Frobenius kernel $G_r$, and when its $G_r$-socle is simple (the latter being a strictly stronger condition than the former). In this paper, we investigate these questions for $G$ having an irreducible root system of type A. Using Schur functors and inverse Schur functors as our primary tools, we develop new methods of attacking these problems, and in the process obtain new results about classes of Weyl modules, induced modules, and tilting modules that remain indecomposable over $G_r$.

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arXiv2024

The Homological Spectrum and Nilpotence Theorems for Lie Superalgebra Representations

Balmer recently showed that there is a general notion of a nilpotence theorem for tensor triangulated categories through the use of homological residue fields and the connection with the homological spectrum. The homological spectrum (like the theory of $π$-points) can be viewed as a topological space that provides an important realization of the Balmer spectrum. Let ${\mathfrak g}={\mathfrak g}_{\bar{0}}\oplus {\mathfrak g}_{\bar{1}}$ be a classical Lie superalgebra over ${\mathbb C}$. In this paper, the authors consider the tensor triangular geometry for the stable category of finite-dimensional Lie superalgebra representations: $\text{stab}({\mathcal F}_{({\mathfrak g},{\mathfrak g}_{\bar{0}})})$, The localizing subcategories for the detecting subalgebra ${\mathfrak f}$ are classified which answers a question of Boe, Kujawa, and Nakano. As a consequence of these results, the authors prove a nilpotence theorem and determine the homological spectrum for the stable module category of ${\mathcal F}_{({\mathfrak f},{\mathfrak f}_{\bar{0}})}$. The authors verify Balmer's ``Nerves of Steel'' Conjecture for ${\mathcal F}_{({\mathfrak f},{\mathfrak f}_{\bar{0}})}$. Let $F$ (resp. $G$) be the associated supergroup (scheme) for ${\mathfrak f}$ (resp. ${\mathfrak g}$). Under the condition that $F$ is a splitting subgroup for $G$, the results for the detecting subalgebra can be used to prove a nilpotence theorem for $\text{stab}({\mathcal F}_{({\mathfrak g},{\mathfrak g}_{\bar{0}})})$, and to determine the homological spectrum in this case. Now using natural assumptions in terms of realization of supports, the authors provide a method to explicitly realize the Balmer spectrum of $\text{stab}({\mathcal F}_{({\mathfrak g},{\mathfrak g}_{\bar{0}})})$, and prove the Nerves of Steel Conjecture in this case.

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arXiv2023

A Chinese remainder theorem and Carlson's theorem for monoidal triangulated categories

In this paper the authors prove fundamental decomposition theorems pertaining to the internal structure of monoidal triangulated categories (M$Δ$Cs). The tensor structure of an M$Δ$C enables one to view these categories like (noncommutative) rings and to attempt to extend the key results for the latter to the categorical setting. The main theorem is an analogue of the Chinese Remainder Theorem involving the Verdier quotients for coprime thick ideals. This result is used to obtain orthogonal decompositions of the extended endomorphism rings of idempotent algebra objects of M$Δ$Cs. The authors also provide topological characterizations on when an M$Δ$C contains a pair of coprime proper thick ideals, and additionally, when the latter are complementary in the sense that their intersection is contained in the prime radical of the category. As an application of the aforementioned results, the authors establish for arbitrary M$Δ$Cs a general version of Carlson's theorem on the connnectedness of supports for indecomposable objects. Examples of our results are given at the end of the paper for the derived category of schemes and for the stable module categories for finite group schemes.

math.CT↗

arXiv2023

Realizing Rings of Regular Functions via the Cohomology of Quantum Groups

Let $G$ be a complex reductive group and $P$ be a parabolic subgroup of $G$. In this paper the authors address questions involving the realization of the $G$-module of the global sections of the (twisted) cotangent bundle over the flag variety $G/P$ via the cohomology of the small quantum group. Our main results generalize the important computation of the cohomology ring for the small quantum group by Ginzburg and Kumar, and provides a generalization of well-known calculations by Kumar, Lauritzen, and Thomsen to the quantum case and the parabolic setting. As an application we answer the question (first posed by Friedlander and Parshall for Frobenius kernels) about the realization of coordinate rings of Richardson orbit closures for complex semisimple groups via quantum group cohomology. Formulas will be provided which relate the multiplicities of simple $G$-modules in the global sections with the dimensions of extension groups over the large quantum group.

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arXiv2023

On the spectrum and support theory of a finite tensor category

Finite tensor categories (FTCs) $\bf T$ are important generalizations of the categories of finite dimensional modules of finite dimensional Hopf algebras, which play a key role in many areas of mathematics and mathematical physics. There are two fundamentally different support theories for them: a cohomological one and a universal one based on the noncommutative Balmer spectra of their stable (triangulated) categories $\underline{\bf T}$. In this paper we introduce the key notion of the categorical center $C^\bullet_{\underline{\bf T}}$ of the cohomology ring $R^\bullet_{\underline{\bf T}}$ of an FTC, $\bf T$. This enables us to put forward a complete and detailed program for determining the exact relationship between the two support theories, based on $C^\bullet_{\underline{\bf T}}$ of the cohomology ring $R^\bullet_{\underline{\bf T}}$ of an FTC, $\bf T$. More specifically, we construct a continuous map from the noncommutative Balmer spectrum of an FTC, $\bf T$, to the $\text{Proj}$ of the categorical center $C^\bullet_{\underline{\bf T}}$, and prove that this map is surjective under a weaker finite generation assumption for $\bf T$ than the one conjectured by Etingof-Ostrik. Under stronger assumptions, we prove that (i) the map is homeomorphism and (ii) the two-sided thick ideals of $\underline{\bf T}$ are classified by the specialization closed subsets of $\text{Proj} C^\bullet_{\underline{\bf T}}$. We conjecture that both results hold for all FTCs. Many examples are presented that demonstrate how in important cases $C^\bullet_{\underline{\bf T}}$ arises as a fixed point subring of $R^\bullet_{\underline{\bf T}}$ and how the two-sided thick ideals of $\underline{\bf T}$ are determined in a uniform fashion. The majority of our results are proved in the greater generality of monoidal triangulated categories.

math.CT↗

arXiv2023

On Donkin's Tilting Module Conjecture II: Counterexamples

In this paper we produce infinite families of counterexamples to Jantzen's question posed in 1980 on the existence of Weyl $p$-filtrations for Weyl modules for an algebraic group and Donkin's Tilting Module Conjecture formulated in 1990. New techniques to exhibit explicit examples are provided along with methods to produce counterexamples in large rank from counterexamples in small rank. Counterexamples can be produced via our methods for all groups other than when the root system is of type $\rm{A}_{n}$ or $\rm{B}_{2}$.

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arXiv2023

On Donkin's Tilting Module Conjecture III: New Generic Lower Bounds

In this paper the authors consider four questions of primary interest for the representation theory of reductive algebraic groups: (i) Donkin's Tilting Module Conjecture, (ii) the Humphreys-Verma Question, (iii) whether $\operatorname{St}_r \otimes L(λ)$ is a tilting module for $L(λ)$ an irrreducible representation of $p^{r}$-restricted highest weight, and (iv) whether $\operatorname{Ext}^{1}_{G_{1}}(L(λ),L(μ))^{(-1)}$ is a tilting module where $L(λ)$ and $L(μ)$ have $p$-restricted highest weight. The authors establish affirmative answers to each of these questions with a new uniform bound, namely $p\geq 2h-4$ where $h$ is the Coxeter number. Notably, this verifies these statements for infinitely many more cases. Later in the paper, questions (i)-(iv) are considered for rank two groups where there are counterexamples (for small primes) to these questions.

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arXiv2022

On Donkin's Tilting Module Conjecture I: Lowering the Prime

In this paper the authors provide a complete answer to Donkin's Tilting Module Conjecture for all rank $2$ semisimple algebraic groups and $\text{SL}_{4}(k)$ where $k$ is an algebraically closed field of characteristic $p>0$. In the process, new techniques are introduced involving the existence of $(p,r)$-filtrations, Lusztig's character formula, and the $G_{r}$T-radical series for baby Verma modules.

math.RT↗

arXiv2022

On sheaf cohomology for supergroups arising from simple classical Lie superalgebras

In this paper the authors study the behavior of the sheaf cohomology functors $R^{\bullet}\text{ind}_{B}^{G}(-)$ where $G$ is an algebraic group scheme corresponding to a simple classical Lie superalgebra and $B$ is a BBW parabolic subgroup as defined by D. Grantcharov, N. Grantcharov, Nakano and Wu. We provide a systematic treatment that allows us to study the behavior of these cohomology groups $R^{\bullet}\text{ind}_{B}^{G}L_{\mathfrak f}(λ)$ where $L_{\mathfrak f}(λ)$ is an irreducible representation for the detecting subalgebra ${\mathfrak f}$. In particular, we prove an analog of Kempf's vanishing theorem and the Bott-Borel-Weil theorem for large weights.

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arXiv2021

Noncommutative tensor triangular geometry and the tensor product property for support maps

The problem of whether the cohomological support map of a finite dimensional Hopf algebra has the tensor product property has attracted a lot of attention following the earlier developments on representations of finite group schemes. Many authors have focussed on concrete situations where positive and negative results have been obtained by direct arguments. In this paper we demonstrate that it is natural to study questions involving the tensor product property in the broader setting of a monoidal triangulated category. We give an intrinsic characterization by proving that the tensor product property for the universal support datum is equivalent to complete primeness of the categorical spectrum. From these results one obtains information for other support data, including the cohomological one. Two theorems are proved giving compete primeness and non-complete primeness in certain general settings. As an illustration of the methods, we give a proof of a recent conjecture of Negron and Pevtsova on the tensor product property for the cohomological support maps for the small quantum Borel algebras for all complex simple Lie algebras.

math.CT↗

arXiv2021

Torsion Free Endotrivial Modules for Finite Groups of Lie Type

In this paper we determine the torsion free rank of the group of endotrivial modules for any finite group of Lie type, in both defining and non-defining characteristic. On our way to proving this, we classify the maximal rank $2$ elementary abelian $\ell$-subgroups in any finite group of Lie type, for any prime $\ell$, which may be of independent interest.

math.GR↗