arXiv subjects

Deniz Yılmaz

Publications and source records attributed to Deniz Yılmaz.

At least 19 recordsLinked to original sources

arXiv2026

The $D^Δ$-pair biset category

Let $p$ be a prime number. A $D^Δ$-pair is a pair consisting of a finite $p$-group and a $p'$-automorphism of the group. In this paper, we introduce diagonal $D^Δ$-pair bisets and a category whose objects are $D^Δ$-pairs and whose morphism groups are Grothendieck groups of diagonal $D^Δ$-pair bisets. Our main result shows that, over suitable coefficient rings, the category of diagonal $p$-permutation functors is equivalent to the category of linear functors on a natural quotient of this new category. In this way, diagonal $p$-permutation functors can be studied through a category built only from finite $p$-groups and their automorphisms of $p'$-order.

math.RT↗

arXiv2026

RAGTurk: Best Practices for Retrieval Augmented Generation in Turkish

Retrieval-Augmented Generation (RAG) enhances LLM factuality, yet design guidance remains English-centric, limiting insights for morphologically rich languages like Turkish. We address this by constructing a comprehensive Turkish RAG dataset derived from Turkish Wikipedia and CulturaX, comprising question-answer pairs and relevant passage chunks. We benchmark seven stages of the RAG pipeline, from query transformation and reranking to answer refinement, without task-specific fine-tuning. Our results show that complex methods like HyDE maximize accuracy (85%) that is considerably higher than the baseline (78.70%). Also a Pareto-optimal configuration using Cross-encoder Reranking and Context Augmentation achieves comparable performance (84.60%) with much lower cost. We further demonstrate that over-stacking generative modules can degrade performance by distorting morphological cues, whereas simple query clarification with robust reranking offers an effective solution.

cs.CL↗

arXiv2026

OCRTurk: A Comprehensive OCR Benchmark for Turkish

Document parsing is now widely used in applications, such as large-scale document digitization, retrieval-augmented generation, and domain-specific pipelines in healthcare and education. Benchmarking these models is crucial for assessing their reliability and practical robustness. Existing benchmarks mostly target high-resource languages and provide limited coverage for low-resource settings, such as Turkish. Moreover, existing studies on Turkish document parsing lack a standardized benchmark that reflects real-world scenarios and document diversity. To address this gap, we introduce OCRTurk, a Turkish document parsing benchmark covering multiple layout elements and document categories at three difficulty levels. OCRTurk consists of 180 Turkish documents drawn from academic articles, theses, slide decks, and non-academic articles. We evaluate seven OCR models on OCRTurk using element-wise metrics. Across difficulty levels, PaddleOCR achieves the strongest overall results, leading most element-wise metrics except figures and attaining high Normalized Edit Distance scores in easy, medium, and hard subsets. We also observe performance variation by document type. Models perform well on non-academic documents, while slideshows become the most challenging.

cs.CL↗

arXiv2025

FIBER: A Multilingual Evaluation Resource for Factual Inference Bias

Large language models are widely used across domains, yet there are concerns about their factual reliability and biases. Factual knowledge probing offers a systematic means to evaluate these aspects. Most existing benchmarks focus on single-entity facts and monolingual data. We therefore present FIBER, a multilingual benchmark for evaluating factual knowledge in single- and multi-entity settings. The dataset includes sentence completion, question-answering, and object-count prediction tasks in English, Italian, and Turkish. Using FIBER, we examine whether the prompt language induces inference bias in entity selection and how large language models perform on multi-entity versus single-entity questions. The results indicate that the language of the prompt can influence the model's generated output, particularly for entities associated with the country corresponding to that language. However, this effect varies across different topics such that 31% of the topics exhibit factual inference bias score greater than 0.5. Moreover, the level of bias differs across languages such that Turkish prompts show higher bias compared to Italian in 83% of the topics, suggesting a language-dependent pattern. Our findings also show that models face greater difficulty when handling multi-entity questions than the single-entity questions. Model performance differs across both languages and model sizes. The highest mean average precision is achieved in English, while Turkish and Italian lead to noticeably lower scores. Larger models, including Llama-3.1-8B and Qwen-2.5-7B, show consistently better performance than smaller 3B-4B models.

cs.CL↗

arXiv2025

Alperin's weight conjecture, Galois automorphisms, alternating sums, and functorial equivalences

We show that functorial equivalences can offer new insight into the blockwise Galois Alperin weight conjecture (BGAWC). Inspired by Knörr and Robinson's work, we first formulate the BGAWC in terms of alternating sums indexed by chains of $p$-subgroups, and we also give a functorial reformulation in the Grothendieck group of diagonal $p$-permutation functors. We prove that these formulations are equivalent. We further show that if a functorial equivalence between a block with abelian defect group and its Brauer correspondent descends to the minimal field of the block, then the BGAWC holds for that block. Finally, we prove that Galois conjugate blocks are functorially equivalent over an algebraically closed field of characteristic zero.

math.RT↗

arXiv2025

Functorial equivalence classes of $2$-blocks of tame representation type

For any block of a finite group over an algebraically closed field of characteristic $2$ which has dihedral, semidihedral, or generalized quaternion defect groups, we determine explicitly the decomposition of the associated diagonal $p$-permutation functor over an algebraically closed field $\mathbb{F}$ of characteristic $0$ into a direct sum of simple functors. As a consequence we see that two blocks with dihedral, semidihedral, or generalized quaternion defect groups are functorially equivalent over $\mathbb{F}$ if and only if their fusion systems are isomorphic. It is an open question if two blocks (with arbitrary defect groups) that are functorially equivalent over $\mathbb{F}$ must have isomorphic fusion systems. The converse is wrong in general.

math.RT↗

arXiv2025

On Alperin's conjecture and functorial equivalence of blocks

Let $k$ be an algebraically closed field of positive characteristic $p$ and let $\mathbb{F}$ be an algebraically closed field of characteristic 0. We consider Alperin's weight conjecture (over $k$) from the point of view of (stable) functorial equivalence of blocks over $\mathbb{F}$. We formulate a functorial version of Alperin's blockwise weight conjecture, and show that it is equivalent to the original one. We also show that this conjecture holds stably, i.e., in the category of stable diagonal $p$-permutation functors over $\mathbb{F}$.

math.RT↗

arXiv2025

$p$-Bifree biset functors

We introduce and study the category of $p$-bifree biset functors for a fixed prime $p$, defined via bisets whose left and right stabilizers are $p'$-groups. This category naturally lies between the classical biset functors and the diagonal $p$-permutation functors, serving as a bridge between them. Every biset functor and every diagonal $p$-permutation functor restricts to a $p$-bifree biset functor. We classify the simple $p$-bifree biset functors over a field $K$ of characteristic zero, showing that they are parametrized by pairs $(G,V)$, where $G$ is a finite group and $V$ is a simple $K\mathrm{Out}(G)$-module. As key examples, we compute the composition factors of several representation-theoretic functors in the $p$-bifree setting, including the Burnside ring functor, the $p$-bifree Burnside functor, the Brauer character ring functor, and the ordinary character ring functor. We further investigate classical simple biset functors, $S_{C_p \times C_p, \mathbb{C}}$ and $S_{C_q \times C_q, \mathbb{C}}$ for a prime $q\neq p$.

math.RT↗

arXiv2024

Diagonal $p$-permutation functors in characteristic $p$

Let $p$ be a prime number. We consider diagonal $p$-permutation functors over a (commutative, unital) ring $\mathsf{R}$ in which all prime numbers different from $p$ are invertible. We first determine the finite groups $G$ for which the associated essential algebra $\mathcal{E}_\mathsf{R}(G)$ is non zero: These are groups of the form $G=L\rtimes \langle u\rangle$, where $(L,u)$ is a $D^Δ$-pair. When $\mathsf{R}$ is an algebraically closed field $\mathbb{F}$ of characteristic 0 or $p$, this yields a parametrization of the simple diagonal $p$-permutation functors over $\mathbb{F}$ by triples $(L,u,W)$, where $(L,u)$ is a $D^Δ$-pair, and $W$ is a simple $\mathbb{F}\mathrm{Out}(L,u)$-module. Finally, we describe the evaluations of the simple functor $\mathsf{S}_{L,u,W}$ parametrized by the triple $(L,u,W)$. We show in particular that if $G$ is a finite group and $\mathbb{F}$ has characteristic $p$, the dimension of $\mathsf{S}_{L,1,\mathbb{F}}(G)$ is equal to the number of conjugacy classes of $p$-regular elements of $G$ with defect isomorphic to $L$.

math.GR↗

arXiv2024

On p-permutation equivalences between direct products of blocks

We extend the notion of a p-permutation equivalence to an equivalence between direct products of block algebras. We prove that a p-permutation equivalence between direct products of blocks gives a bijection between the factors and induces a p-permutation equivalence between corresponding blocks.

math.GR↗

arXiv2023

Isotypic blocks are functorially equivalent

Let $\mathbb{F}$ be an algebraically closed field of characteristic zero. In this article we show that isotypic blocks of finite groups are functorially equivalent over $\mathbb{F}$.

math.RT↗

arXiv2023

On functorial equivalence classes of blocks of finite groups

Let $k$ be an algebraically closed field of characteristic $p>0$ and let $\mathbb{F}$ be an algebraically closed field of characteristic $0$. Recently, together with Bouc, we introduced the notion of functorial equivalences between blocks of finite groups and proved that given a $p$-group $D$, there is only a finite number of pairs $(G,b)$ of a finite group $G$ and a block $b$ of $kG$ with defect groups isomorphic to $D$, up to functorial equivalence over $\mathbb{F}$. In this paper, we classify the functorial equivalence classes over $\mathbb{F}$ of blocks with cyclic defect groups and $2$-blocks of defects $2$ and $3$. In particular, we prove that for all these blocks, the functorial equivalence classes depend only on the fusion system of the block.

math.GR↗

arXiv2023

A note on blocks of finite groups with TI Sylow $p$-subgroups

Let $\mathbb{F}$ be an algebraically closed field of characteristic zero. Recently, we proved that isotypic blocks are functorially equivalent over $\mathbb{F}$. In this article we provide an example of functorially equivalent blocks which are not perfectly isometric.

math.GR↗

arXiv2023

The group of splendid Morita equivalences of principal $2$-blocks with dihedral and generalised quaternion defect groups

Let $k$ be an algebraically closed field of characteristic $2$, let $G$ be a finite group and let $B$ be the principal $2$-block of $kG$ with a dihedral or a generalised quaternion defect group $P$. Let also $\mathcal{T}(B)$ denote the group of splendid Morita auto-equivalences of $B$. We show that \begin{align*} \mathcal{T}(B)\cong \mathrm{Out}_P(A)\rtimes \mathrm{Out}(P,\mathcal{F})\,, \end{align*} where $\mathrm{Out}(P,\mathcal{F})$ is the group of outer automorphisms of $P$ which stabilize the fusion system $\mathcal{F}$ of $G$ on $P$ and $\mathrm{Out}_P(A)$ is the group of algebra automorphisms of a source algebra $A$ of $B$ fixing $P$ modulo inner automorphisms induced by $(A^P)^\times$.

math.GR↗

arXiv2023

Stable functorial equivalence of blocks

Let $k$ be an algebraically closed field of characteristic $p>0$, let $R$ be a commutative ring and let $\mathcal{F}$ be an algebraically closed field of characteristic $0$. We introduce the category $\overline{\mathcal{F}_{Rpp_k}}$ of stable diagonal $p$-permutation functors over $R$. We prove that the category $\overline{\mathcal{F}_{\mathbb{F}pp_k}}$ is semisimple and give a parametrization of its simple objects in terms of the simple diagonal $p$-permutation functors. We also introduce the notion of a stable functorial equivalence over $R$ between blocks of finite groups. We prove that if $G$ is a finite group and if $b$ is a block idempotent of $kG$ with an abelian defect group $D$ and Frobenius inertial quotient $E$, then there exists a stable functorial equivalence over $\mathbb{F}$ between the pairs $(G,b)$ and $(D\rtimes E,1)$.

math.GR↗

arXiv2022

Diagonal $p$-permutation functors, semisimplicity, and functorial equivalence of blocks

Let $k$ be an algebraically closed field of characteristic $p>0$, let $R$ be a commutative ring, and let $\mathbb{F}$ be an algebraically closed field of characteristic 0. We consider the $R$-linear category $\mathcal{F}^Δ_{Rpp_k}$ of diagonal $p$-permutation functors over $R$. We first show that the category $\mathcal{F}^Δ_{\mathbb{F}pp_k}$ is semisimple, and we give a parametrization of its simple objects, together with a description of their evaluations. Next, to any pair $(G,b)$ of a finite group $G$ and a block idempotent $b$ of $kG$, we associate a diagonal $p$-permutation functor $RT^Δ_{G,b}$ in $\mathcal{F}^Δ_{Rpp_k}$. We find the decomposition of the functor $\mathbb{F}T^Δ_{G,b}$ as a direct sum of simple functors in $\mathcal{F}^Δ_{\mathbb{F}pp_k}$. This leads to a characterization of nilpotent blocks in terms of their associated functors in $\mathcal{F}^Δ_{\mathbb{F}pp_k}$. Finally, for such pairs $(G,b)$ of a finite group and a block idempotent, we introduce the notion of functorial equivalence over $R$, which (in the case $R=\mathbb{Z}$) is slightly weaker than $p$-permutation equivalence, and we prove a corresponding finiteness theorem: for a given finite $p$-group $D$, there is only a finite number of pairs $(G,b)$, where $G$ is a finite group and $b$ a block idempotent of $kG$ with defect isomorphic to $D$, up to functorial equivalence over $\mathbb{F}$.

math.GR↗

arXiv2021

Galois descent of equivalences between blocks of $p$-nilpotent groups

We give sufficient conditions on $p$-blocks of $p$-nilpotent groups over $\mathbb{F}_p$ to be splendidly Rickard equivalent and $p$-permutation equivalent to their Brauer correspondents. The paper also contains Galois descent results on $p$-permutation modules and $p$-permutation equivalences that hold for arbitrary groups.

math.GR↗

arXiv2020

The $A$-fibered Burnside ring as $A$-fibered biset functor in characteristic zero

Let $A$ be an abelian group such that $\mathrm{Hom}(G,A)$ is finite for all finite groups $G$, and let $\mathbb{K}$ be a field of characteristic zero containing roots of unity of all orders equal to finite element orders in $A$. In this paper we prove foundational properties of the $A$-fibered Burnside ring functor $B_{\mathbb{K}}^A$ as an $A$-fibered biset functor over $\mathbb{K}$. This includes the determination of the lattice of subfunctors of $B_{\mathbb{K}}^A$ and the determination of the composition factors of $B_{\mathbb{K}}^A$. The results of the paper extend results of Coşkun and Yılmaz for the $A$-fibered Burnside ring functor restricted to $p$-groups and results of Bouc in the case that $A$ is trivial, i.e., the case of the Burnside ring functor over fields of characteristic zero.

math.RT↗