Searcharxiv⌕ Search

arXiv subjects

Robert Muth

Publications and source records attributed to Robert Muth.

At least 19 recordsLinked to original sources
arXiv2026

Kleshchev multipartitions, affine Mirkovi\'c-Vilonen polytopes, and representations of KLR algebras in type ${\tt A}^{(1)}_1$

We construct explicit isomorphisms between three models for the $B(\infty)$ crystal in type ${\tt A}_1^{(1)}$: affine Mirkovi\'c--Vilonen polytopes, Kleshchev multipartitions, and a new model we call upper ledge diagrams. We also present some clarifying results on these crystals, giving a direct method for completing an affine MV polytope from the data of one of its boundary root partitions, and a non-iterative recognition theorem which characterizes Kleshchev multipartitions in type ${\tt A}_1^{(1)}$. We apply these results to the representation theory of KLR algebras, where they yield a combinatorial dictionary between cuspidal- and cellular-theoretic frameworks, along with some augmented branching rules for real root functors of induction and restriction.

math.RT↗
arXiv2025

Superalgebra deformations of web categories: Affine and cyclotomic webs

Let $\mathbb{k}$ be a characteristic zero domain. We define and study a diagrammatic monoidal $\mathbb{k}$-linear supercategory $\mathbf{Web}^{aff}_{A}$ associated to any locally unital Frobenius $\mathbb{k}$-superalgebra $A$. This category can be viewed variously as an affinization of the finite web category $\mathbf{Web}_{A}$ previously defined by the authors and Zhu, as a thickening of the degenerate affine wreath product algebras defined by Savage, or as a Frobenius deformation of affine web categories defined by Song and Wang. We show that there is an asymptotically faithful family of functors from $\mathbf{Web}^{aff}_{A}$ to the monoidal supercategory of endofunctors of $\mathfrak{gl}_n(A)$-modules for every $n \geq 1$, and use this to establish a basis of `decorated double coset diagrams' for morphism spaces in $\mathbf{Web}^{aff}_{A}$. We also define and establish basis results for the cyclotomic quotient category $\mathbf{Web}^{\Lambda}_{A}$ associated with a cyclotomic datum $\Lambda$.

math.RT↗
arXiv2025

An efficient search strategy for hidden ideals in pointed partially ordered sets

We consider a combinatorial question about searching for an unknown ideal $\mu$ within a known pointed poset $\lambda$. Elements of $\lambda$ may be queried for membership in $\mu$, but at most $k$ positive queries are permitted. We provide a general search strategy for this problem, and establish new bounds (based on $k$ and the degree and height of $\lambda$) for the total number of queries required to identify $\mu$. We show that this strategy performs asymptotically optimally on the family of complete $\ell$-ary trees as the height grows.

math.CO↗
arXiv2024

A skew Specht perspective of RoCK blocks and cuspidal systems for KLR algebras in affine type A

Cuspidal systems parameterize KLR algebra representations via root partitions $\pi$, where simple modules $L(\pi)$ arise as heads of proper standard modules. Working in affine type A with an arbitrary convex preorder, we construct explicit skew diagrams $\zeta(\pi)$ such that the skew Specht module $S^{\zeta(\pi)}$ has simple head $L(\pi)$ and a filtration by proper standard modules. A key ingredient in this construction is the development of `core-truncation' functors, which take module categories of level one RoCK blocks to the category of imaginary semicuspidal KLR modules. Every simple imaginary semicuspidal module arises in the image of these functors. This result stems from an in-depth study of the combinatorial interplay between cuspidal systems and RoCK cyclotomic KLR algebras, in which we characterize core blocks and RoCK blocks in arbitrary level via cuspidal tiling properties of multipartitions in these blocks.

math.RT↗
arXiv2023

Superalgebra deformations of web categories: finite webs

Let $\mathbb{k}$ be a characteristic zero domain. For a locally unital $\mathbb{k}$-superalgebra $A$ with distinguished idempotents $I$and even subalgebra $a \subseteq A_{\bar 0}$, we define and study an associated diagrammatic monoidal $\mathbb{k}$-linear supercategory $\mathbf{Web}^{A,a}_I$. This supercategory yields a diagrammatic description of the generalized Schur algebras $T^A_a(n,d)$. We also show there is an asymptotically faithful functor from $\mathbf{Web}^{A,a}_I$ to the monoidal supercategory of $\mathfrak{gl}_n(A)$-modules generated by symmetric powers of the natural module. When this functor is full, the single diagrammatic supercategory $\mathbf{Web}^{A,a}_I$ provides a combinatorial description of this module category for all $n \geq 1$. We also use these results to establish Howe dualities between $\mathfrak{gl}_{m}(A)$ and $\mathfrak{gl}_{n}(A)$ when $A$ is semisimple.

math.RT↗
arXiv2022

Non-Disclosing Credential On-chaining for Blockchain-based Decentralized Applications

Many service systems rely on verifiable identity-related information of their users. Manipulation and unwanted exposure of this privacy-relevant information, however, must at the same time be prevented and avoided. Peer-to-peer blockchain-based decentralization with a smart contract-based execution model and verifiable off-chain computations leveraging zero-knowledge proofs promise to provide the basis for next-generation, non-disclosing credential management solutions. In this paper, we propose a novel credential on-chaining system that ensures blockchain-based transparency while preserving pseudonymity. We present a general model compliant to the W3C verifiable credential recommendation and demonstrate how it can be applied to solve existing problems that require computational identity-related attribute verification. Our zkSNARKs-based reference implementation and evaluation show that, compared to related approaches based on, e.g., CL-signatures, our approach provides significant performance advantages and more flexible proof mechanisms, underpinning our vision of increasingly decentralized, transparent, and trustworthy service systems.

cs.CR↗
arXiv2022

Analysis of Arbitrary Content on Blockchain-Based Systems using BigQuery

Blockchain-based systems have gained immense popularity as enablers of independent asset transfers and smart contract functionality. They have also, since as early as the first Bitcoin blocks, been used for storing arbitrary contents such as texts and images. On-chain data storage functionality is useful for a variety of legitimate use cases. It does, however, also pose a systematic risk. If abused, for example by posting illegal contents on a public blockchain, data storage functionality can lead to legal consequences for operators and users that need to store and distribute the blockchain, thereby threatening the operational availability of entire blockchain ecosystems. In this paper, we develop and apply a cloud-based approach for quickly discovering and classifying content on public blockchains. Our method can be adapted to different blockchain systems and offers insights into content-related usage patterns and potential cases of abuse. We apply our method on the two most prominent public blockchain systems - Bitcoin and Ethereum - and discuss our results. To the best of our knowledge, the presented study is the first to systematically analyze non-financial content stored on the Ethereum blockchain and the first to present a side-by-side comparison between different blockchains in terms of the quality and quantity of stored data.

cs.CR↗
arXiv2021

The configuration space of a robotic arm over a graph

We investigate the configuration space $\mathcal{S}_{G,b,\ell}$ associated with the movement of a robotic arm of length $\ell$ on a grid over an underlying graph $G$, anchored at a vertex $b \in G$. We study an associated PIP (poset with inconsistent pairs) $\text{IP}_{G,b,\ell}$ consisting of indexed paths on $G$. This PIP acts as a combinatorial model for the robotic arm, and we use $\text{IP}_{G,b,\ell}$ to show that the space $\mathcal{S}_{G,b,\ell}$ is a CAT(0) cubical complex, generalizing work of Ardila, Bastidas, Ceballos, and Guo. This establishes that geodesics exist within the configuration space, and yields explicit algorithms for moving the robotic arm between different configurations in an optimal fashion. We also give a tight bound on the diameter of the robotic arm transition graph (the maximal number of moves necessary to change from one configuration to another) and compute this diameter for a large family of underlying graphs $G$.

math.CO↗
arXiv2021

Howe Duality of Type P

We establish classical and categorical Howe dualities between the Lie superalgebras $\mathfrak{p}(m)$ and $\mathfrak{p}(n)$, for $m,n \geq 1$. We also describe a presentation via generators and relations as well as a Kostant $\mathbb{Z}$-form for the universal enveloping superalgebra $U(\mathfrak{p}(m))$.

math.RT↗
arXiv2021

Webs of Type P

This paper introduces type P web supercategories. They are defined as diagrammatic monoidal $k$-linear supercategories via generators and relations. We study the structure of these categories and provide diagrammatic bases for their morphism spaces. We also prove these supercategories provide combinatorial models for the monoidal supercategory generated by the symmetric powers of the natural module and their duals for the Lie superalgebra of type P.

math.RT↗
arXiv2021

Injectively $k$-colored rooted forests

We enumerate injectively $k$-colored rooted forests with a given number of vertices of each color and a given sequence of root colors. We obtain from this result some new multi-parameter distributions of Fuss-Catalan numbers. As an additional application we enumerate triangulations of regular convex polygons according to their proper 3-coloring type.

math.CO↗
arXiv2021

Decomposable Specht modules indexed by bihooks II

Previously, the last two authors found large families of decomposable Specht modules labelled by bihooks, over the Iwahori--Hecke algebra of type $B$. In most cases we conjectured that these were the only decomposable Specht modules labelled by bihooks, proving it in some instances. Inspired by a recent semisimplicity result of Bowman, Bessenrodt and the third author, we look back at our decomposable Specht modules and show that they are often either semisimple, or very close to being so. We obtain their exact structure and composition factors in these cases. In the process, we determine the graded decomposition numbers for almost all of the decomposable Specht modules indexed by bihooks.

math.RT↗
arXiv2020

Cuspidal ribbon tableaux in affine type A

For any convex preorder on the set of positive roots of affine type A, we classify and construct all associated cuspidal and semicuspidal skew shapes. These combinatorial objects correspond to cuspidal and semicuspidal skew Specht modules for the Khovanov-Lauda-Rouquier algebra of affine type A. Cuspidal skew shapes are ribbons, and we show that every skew shape has a unique ordered tiling by cuspidal ribbons. This tiling data provides an upper bound, in the bilexicographic order on Kostant partitions, for labels of simple factors of Specht modules.

math.CO↗
arXiv2020

Searching for quicksand ideals in partially ordered sets

We consider a combinatorial question about searching for an unknown ideal $μ$ within a known poset $λ$. Elements of $λ$ may be queried for membership in $μ$, but at most $k$ positive query results are permitted. The goal is to find a search strategy which guarantees a solution in a minimal total number $m_k(λ)$ of queries. We provide tight bounds for $m_k(λ)$, and construct optimal search strategies for the case where $k=2$ and $λ$ is the product poset of totally ordered finite sets, one of which has cardinality not more than six.

math.CO↗
arXiv2020

Cooperative half-guards in art galleries

In any simple polygonal art gallery with n walls, we show that it is possible to place floor(n/2)-1 guards whose range of vision is 180 degrees in such a way that every interior point of the gallery can be seen by one of them, and such that the mutual visibility graph formed by the guards is connected. This upper bound is tight, in that there exist galleries which require this number of guards, and equals the known result for guards with full 360 degree range of vision. We also show that for orthogonal art galleries, this result may be improved to floor(n/2)-2 guards with 180 degree range of vision.

math.CO↗
arXiv2018

Based quasi-hereditary algebras

A notion of a split quasi-hereditary algebra has been defined by Cline, Parshall and Scott. Du and Rui describe a based approach to split quasi-hereditary algebras. We develop this approach further to show that over a complete local Noetherian ring, one can achieve even stronger basis properties. This is important for `schurifying' quasi-hereditary algebras as developed in our subsequent work. The schurification procedure associates to an algebra $A$ a new algebra, which is the classical Schur algebra if $A$ is a field. Schurification produces interesting new quasi-hereditary and cellular algebras. It is important to work over an integral domain of characteristic zero, taking into account a super-structure on the input algebra $A$. So we pay attention to super-structures on quasi-hereditary algebras and investigate a subtle conforming property of heredity data which is crucial to guarantee that the schurification of $A$ is quasi-hereditary if so is $A$. We establish a Morita equivalence result which allows us to pass to basic quasi-hereditary algebras preserving conformity.

math.RT↗
arXiv2018

Generalized Schur algebras

We define and study a new class of bialgebras, which generalize certain Turner double algebras related to generic blocks of symmetric groups. Bases and generators of these algebras are given. We investigate when the algebras are symmetric, which is relevant to block theory of finite groups. We then establish a double centralizer property related to blocks of Schur algebras.

math.RT↗