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Daniel Tubbenhauer

Publications and source records attributed to Daniel Tubbenhauer.

At least 37 records · Page 2Linked to original sources
arXiv2025

Growth Problems for Representations of Finite Monoids

We give a conjecture for the asymptotic growth rate of the number of indecomposable summands in the tensor powers of representations of finite monoids, expressing it in terms of the (Brauer) character table of the monoid's group of units. We prove it under an additional hypothesis. We also give (exact and asymptotic) formulas for the growth rate of the length of the tensor powers when working over a good characteristic. As examples, we compute the growth rates for the full transformation monoid, the symmetric inverse monoid, and the monoid of 2 by 2 matrices. We also provide code used for our calculation.

math.RT↗
arXiv2025

Cellularity of KLR and weighted KLRW algebras via crystals

We prove that the weighted KLRW algebras of finite type, and their cyclotomic quotients, are cellular algebras. The cellular bases are explicitly described using crystal graphs. As a special case, this proves that the KLR algebras of finite type are cellular. As one application, we give explicit formulas for the graded decomposition numbers of the cyclotomic algebras in level one.

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arXiv2025

On knot detection via picture recognition

Our goal is to one day take a photo of a knot and have a phone automatically recognize it. In this expository work, we explain a strategy to approximate this goal, using a mixture of modern machine learning methods (in particular convolutional neural networks and transformers for image recognition) and traditional algorithms (to compute quantum invariants like the Jones polynomial). We present simple baselines that predict crossing number directly from images, showing that even lightweight CNN and transformer architectures can recover meaningful structural information. The longer-term aim is to combine these perception modules with symbolic reconstruction into planar diagram (PD) codes, enabling downstream invariant computation for robust knot classification. This two-stage approach highlights the complementarity between machine learning, which handles noisy visual data, and invariants, which enforce rigorous topological distinctions.

cs.LG↗
arXiv2025

Orthogonal webs and semisimplification

We define a diagrammatic category that is equivalent to tilting representations for the orthogonal group. Our construction works in characteristic not equal to two. We also describe the semisimplification of this category.

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arXiv2025

Tensor powers of representations of (diagram) monoids

We study tensor powers of representations of finite monoids, focusing on the growth behavior of their composition length and the number of indecomposable summands. Special attention is given to diagram monoids such as the Temperley-Lieb, Motzkin, and Brauer monoids. For these examples, we compute explicit data, including some character tables, and analyze patterns in the decomposition of their tensor powers.

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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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arXiv2025

Idempotents, traces, and dimensions in Hecke categories

We explain how to compute idempotents that correspond to the indecomposable objects in the Hecke category. Closed formulas are provided for some common coefficients that appear in these idempotents. We also explain how to compute categorical dimensions in the asymptotic Hecke category. In many cases, we reduce this to a computation of a partial trace and give recursive formulas for some common partial traces. In the sequel, we apply this technology and perform additional (computer) calculations to complete the description of the asymptotic Hecke category for finite Coxeter groups in all but three cells.

math.RT↗
arXiv2025

Quantum topology without topology

These lecture notes cover 13 sessions and are presented as an e-print, intended to evolve over time. Quantum invariants do more than distinguish topological objects; they build bridges between topology, algebra, number theory and quantum physics helping to transfer ideas, and stimulating mutual development. They also possess deep and intriguing connections to representation theory, particularly through representations of quantum groups. These lecture notes aim to illustrate how categorical algebra provides a framework for studying both algebra and topology. Specifically, they demonstrate how quantum invariants emerge naturally from a mostly categorical perspective.

math.QA↗
arXiv2025

Representation gaps of rigid planar diagram monoids

We define non-pivotal analogs of the Temperley-Lieb, Motzkin, and planar rook monoids, and compute bounds for the sizes of their nontrivial simple representations. From this, we assess the two types of monoids in their relative suitability for use in cryptography by comparing their representation gaps and gap ratios. We conclude that the non-pivotal monoids are generally worse for cryptographic purposes.

math.RT↗
arXiv2025

Big data comparison of quantum invariants

We apply big data techniques, including exploratory and topological data analysis, to investigate quantum invariants. More precisely, our study explores the Jones polynomial's structural properties and contrasts its behavior under four principal methods of enhancement: coloring, rank increase, categorification, and leaving the realm of Lie algebras.

math.GT↗
arXiv2025

On a symplectic quantum Howe duality

We prove a nonsemisimple quantum version of Howe's duality with the rank 2n symplectic and the rank 2 special linear group acting on the exterior algebra of type C. We also discuss the first steps towards the symplectic analog of harmonic analysis on quantum spheres, give character formulas for various fundamental modules, and construct canonical bases of the exterior algebra.

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arXiv2024

Equivariant neural networks and piecewise linear representation theory

Equivariant neural networks are neural networks with symmetry. Motivated by the theory of group representations, we decompose the layers of an equivariant neural network into simple representations. The nonlinear activation functions lead to interesting nonlinear equivariant maps between simple representations. For example, the rectified linear unit (ReLU) gives rise to piecewise linear maps. We show that these considerations lead to a filtration of equivariant neural networks, generalizing Fourier series. This observation might provide a useful tool for interpreting equivariant neural networks.

cs.LG↗
arXiv2024

On rank one 2-representations of web categories

We classify rank one 2-representations of SL2, GL2 and SO3 web categories. The classification is inspired by similar results about quantum groups, given by reducing the problem to the classification of bilinear and trilinear forms, and is formulated such that it can be adapted to other web categories.

math.RT↗
arXiv2024

Sandwich cellularity and a version of cell theory

We explain how the theory of sandwich cellular algebras can be seen as a version of cell theory for algebras. We apply this theory to many examples such as Hecke algebras, and various monoid and diagram algebras.

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arXiv2024

Monoidal categories, representation gap and cryptography

The linear decomposition attack provides a serious obstacle to direct applications of noncommutative groups and monoids (or semigroups) in cryptography. To overcome this issue we propose to look at monoids with only big representations, in the sense made precise in the paper, and undertake a systematic study of such monoids. One of our main tools is Green's theory of cells (Green's relations). A large supply of monoids is delivered by monoidal categories. We consider simple examples of monoidal categories of diagrammatic origin, including the Temperley-Lieb, the Brauer and partition categories, and discuss lower bounds for their representations.

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arXiv2024

Cellularity and subdivision of KLR and weighted KLRW algebras

Weighted KLRW algebras are diagram algebras generalizing KLR algebras. This paper undertakes a systematic study of these algebras culminating in the construction of homogeneous affine cellular bases in affine types A and C, which immediately gives cellular bases for the cyclotomic quotients of these algebras. In addition, we construct subdivision homomorphisms that relate weighted KLRW algebras for different quivers. As an application we obtain new results about the (cyclotomic) KLR algebras of affine type, including (re)proving that the cyclotomic KLR algebras of type A^{(1)}_{e} and C^{(1)}_{e} are graded cellular algebras.

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