arXiv subjects

Amit Kuber

Publications and source records attributed to Amit Kuber.

At least 19 recordsLinked to original sources

arXiv2026

On Zeta functions and $\mu$-series of string algebras

Let $\overline\mu_\Lambda(t):=\sum\limits_{m\geq1}\mu_\Lambda(m)t^m$ be the \emph{$\mu$-series} of a finite-dimensional tame algebra $\Lambda$ over an algebraically closed field, where $\mu_\Lambda(m)$ denotes the minimal number of one-parameter families of $\Lambda$-modules with total dimension $m$. When $\Lambda$ is a string algebra with $\mathrm{Ba}(\Lambda)$ as its set of bands up to cyclic permutation, define the \emph{zeta function} $\zeta_\Lambda(t):=\prod\limits_{\mathfrak b\in\mathrm{Ba}(\Lambda)}(1-t^{|\mathfrak b|})^{-1}$, where $|\mathfrak b|$ is the length of $\mathfrak b$. We prove an analogue of the prime number theorem for string algebras and use it to conclude that non-domestic string algebras are of exponential growth. Finally, we show that a string algebra is domestic if and only if its $\mu$-series is rational.

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arXiv2026

An automata-based test for bricks over string algebras

Motivated by the recent work of Deaconu, Mousavand and Paquette on the connection between infinite string bricks for certain gentle algebras and Sturmian words, we develop a decorated version of a deterministic automaton, called a multi-entry inverse automaton (MIA, for short) that accepts pointed words. We then associate an MIA $\mathsf M_{\Lambda\delta}$ over $\{0,1\}$ to a string algebra $\Lambda$, and show that strings over $\Lambda$ can be viewed as certain equivalence classes of the pointed words accepted by $\mathsf M_{\Lambda\delta}$. By defining (weak) brick words over this MIA, we show that a finite/infinite string module (resp. band module) is a brick if and only if every word in the associated equivalence class of pointed binary words is a brick word (resp. a weak brick word) over $\mathsf M_{\Lambda\delta}$. The result of Deaconu et al. follows as an immediate consequence.

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arXiv2025

Model-theoretic $K_1$ for modules over semisimple rings: (weak) Morita invariance

This paper is a sequel to a paper by the same authors, where they defined $K$-groups of model-theoretic structures, and computed $K_1$ of free modules over PIDs. In this paper, we compute $K_1$ of a right $M_q(R)$-module $M$, where $R$ is a division ring, $q\geq1$, and $|M_q(R)|\neq 2$. As a consequence, we obtain a (weak) Morita invariance $K_1(R_R)\cong K_1((M_q(R))_{M_q(R)})$ for all division rings $R$ and $q\geq 1$. Finally, we compute $K_1$ of a module over a semisimple ring by showing that the model-theoretic $K_1$ commutes with finite product of modules. We also show that the algebraic $K_1$ of a finite product of infinite matrix rings embeds into the model-theoretic $K_1$ of their right regular modules.

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arXiv2025

Rooted tree modules

A rooted tree module (RTM) $M:=M(T,F)$ over a zero-relation algebra $\Lambda:=\mathcal KQ/\langle\rho\rangle$ over a field $\mathcal K$ is given by the data of a quiver morphism $F:T\to Q$ from a rooted tree $T$ (either with a source or a sink) taking paths in $T$ to paths in $Q$ not lying in $\langle\rho\rangle$. When $\mathrm{char}(\mathcal K)\neq2$, we provide a checkable combinatorial characterization of the indecomposability of the RTM $M$ in terms of non-existence of idempotent quiver morphisms $\iota:T\to T$ satisfying $F\circ\iota=F$ and $\iota\neq 1_T$. Further, we provide an iterative method to decompose an RTM into indecomposable RTMs as well as a method to recursively construct indecomposable RTMs.

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arXiv2025

Generalised tree modules: Hom-sets and indecomposability

For a zero-relation algebra over a field $\mathcal K$, Crawley-Boevey introduced the concept of a tree module and provided a combinatorial description of a basis for the space of homomorphisms between two tree modules--the basis elements are called graph maps. The indecomposability of tree modules is essentially due to Gabriel. We relax a condition in the definition of a tree module to define generalised tree modules and when $\mathrm{char}(\mathcal K)\neq2$, under a certain condition, provide a combinatorial description of a finite generating set for the space of homomorphisms between two such modules--we call the generators generalised graph maps. As an application, we provide a sufficient condition for the (in)decomposability of certain generalised tree modules. We also show that all indecomposable modules over a Dynkin quiver of type $\mathbf D$ are isomorphic to generalised tree modules--this result also follows from a theorem of Ringel which states that all exceptional modules over the path algebra $\mathcal KQ$ of a finite quiver $Q$ are generalised tree modules.

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arXiv2024

Model-theoretic $K_1$ of free modules over PIDs

Motivated by Kraji\v{c}ek and Scanlon's definition of the Grothendieck ring $K_0(M)$ of a first-order structure $M$, we introduce the definition of $K$-groups $K_n(M)$ for $n\geq0$ via Quillen's $S^{-1}S$ construction. We provide a recipe for the computation of $K_1(M_R)$, where $M_R$ is a free module over a PID $R$, subject to the knowledge of the abelianizations of the general linear groups $GL_n(R)$. As a consequence, we provide explicit computations of $K_1(M_R)$ when $R$ belongs to a large class of Euclidean domains that includes fields with at least $3$ elements and polynomial rings over fields with characteristic $0$. We also show that the algebraic $K_1$ of a PID $R$ embeds into $K_1(R_R)$.

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arXiv2024

Automating the stable rank computation for special biserial algebras

Given a special biserial algebra $\Lambda$ over an algebraically closed field, let $\mathrm{rad}_\Lambda$ denote the radical of its module category. The authors showed with Sinha that the stable rank of a special biserial algebra $\Lambda$, i.e., the least ordinal $\gamma$ satisfying $\mathrm{rad}_\Lambda^\gamma=\mathrm{rad}_\Lambda^{\gamma+1}$, is strictly bounded above by $\omega^2$. We use finite automata to give simple algorithmic proofs, complete with their time complexity analyses, of two key ingredients in the proof of this result--the first one states that certain linear orders called hammocks associated with such algebras are \emph{finite description linear orders}, i.e., they lie in the smallest class of linear orders that contains finite linear orders and $\omega$, and that is closed under isomorphisms, order-reversals, binary sums, co-lexicographic products and finitary shuffles. We also document a complete proof of the result that the class of order types(=order-isomorphism classes) of finite description linear orders coincides with that of languages of finite automata under inorder.

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arXiv2024

Exponentiable linear orders need not be transitive

It is well-known that every transitive linear order is exponentiable. However, is the converse true? This question was posed in Chapter 8 of the textbook titled "Linear Orderings" by Rosenstein. We define the class CTLO of cyclically transitive linear orders that properly contains the class of transitive linear orders, and show that all discrete unbounded orders in CTLO are exponentiable, thereby providing a negative answer to the question. The class CTLO is closely related to the class of transitive cyclic orders introduced by Droste, Giraudet and Macpherson. We also discuss the closure of subclasses of CTLO under products and iterated Hausdorff condensations.

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arXiv2024

Characterisation of band bricks over certain string algebras and a variant of perfectly clustering words

Generalising a recent work of Dequ\^ene et al. on the connection between perfectly clustering words and band bricks over a particular family of gentle algebras, we characterise band bricks over string algebras whose underlying quiver is acyclic in terms of weakly perfectly clustering pairs of words -- a variant of perfectly clustering words. As a consequence, we characterise band semibricks over all such algebras. Furthermore, the combination of our result and a result of Mousavand and Paquette provides an algorithm to determine whether such a string algebra is brick-infinite.

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arXiv2023

On the stable radical of the module category for special biserial algebras

Suppose $\Lambda$ is a special biserial algebra over an algebraically closed field. Schr\"oer showed that if $\Lambda$ is domestic then the radical of the category of finitely generated (left) $\Lambda$-modules is nilpotent, and the least ordinal, denoted $\mathrm{st}(\Lambda)$, where the decreasing sequence of powers of the radical stabilizes satisfies $\mathrm{st}(\Lambda)<\omega^2$. With Gupta and Sardar, the third author conjectured that if $\Lambda$ has at least one band then $\omega\le\mathrm{st}(\Lambda)<\omega^2$ even when $\Lambda$ is non-domestic. In this paper we settle this conjecture in the affirmative. We also describe an algorithm to compute $\mathrm{st}(\Lambda)$ up to a finite error. We also show that for each $\omega\leq\alpha<\omega^2$ there is a finite-dimensional tame representation type algebra $\Gamma$ with $\mathrm{st}(\Gamma)=\alpha$.

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arXiv2023

A note on injectivity of monomial algebras

We show that a monomial algebra $\Lambda$ over an algebraically closed field $K$ is self-injective if and only if each map $\mathrm{soc}(_{\Lambda}\Lambda)\to \ _{\Lambda}\Lambda$ can be extended to an endomorphism of $_{\Lambda}\Lambda$, and provide a complete classification of such algebras. As a consequence, we show that the class of self-injective monomial algebras is a subclass of Nakayama algebras.

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arXiv2023

Hammocks for non-domestic string algebras

We show that the order type of the simplest version of a hammock for string algebras lies in the class of finite description linear orders--the smallest class of linear orders containing $\mathbf 0$, $\mathbf 1$, and that is closed under isomorphisms, finite order sum, anti-lexicographic product with $\omega$ and $\omega^*$, and shuffle of finite subsets--using condensation (localization) of linear orders as a tool. We also introduce two finite subsets of the set of bands and use them to describe the location of left $\mathbb N$-strings in the completion of hammocks.

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arXiv2022

On the computation of order types of hammocks for domestic string algebras

For the representation-theoretic study of domestic string algebras, Schr\"{o}er introduced a version of hammocks that are bounded discrete linear orders. He introduced a finite combinatorial gadget called the bridge quiver, which we modified in the prequel of this paper to get a variation called the arch bridge quiver. Here we use it as a tool to provide an algorithm to compute the order type of an arbitrary closed interval in such hammocks. Moreover, we characterize the class of order types of these hammocks as the bounded discrete ones amongst the class of finitely presented linear orders--the smallest class of linear orders containing finite linear orders as well as $\omega$, and that is closed under isomorphisms, order reversal, finite order sums and lexicographic products.

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arXiv2022

Euclidean algorithm for a class of linear orders

Borrowing inspiration from Marcone and Mont\'{a}lban's one-one correspondence between the class of signed trees and the equimorphism classes of indecomposable scattered linear orders, we find a subclass of signed trees which has an analogous correspondence with equimorphism classes of indecomposable finite rank discrete linear orders. We also introduce the class of \emph{finitely presented linear orders}-- the smallest subclass of finite rank linear orders containing $\mathbf 1$, $\omega$ and $\omega^*$ and closed under finite sums and lexicographic products. For this class we develop a generalization of the Euclidean algorithm where the \emph{width} of a linear order plays the role of the Euclidean norm. Using this as a tool we classify the isomorphism classes of finitely presented linear orders in terms of an equivalence relation on their presentations using \emph{3-signed trees}.

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arXiv2021

Variations of the bridge quiver for domestic string algebras

In the computation of some representation-theoretic numerical invariants of domestic string algebras, a finite combinatorial gadget introduced by Schr\"{o}er--the \emph{bridge quiver} whose vertices are (representatives of cyclic permutations of) bands and whose arrows are certain band-free strings--has been used extensively. There is a natural but ill-behaved partial binary operation, $\circ$, on the larger set of \emph{weak bridges} such that bridges are precisely the $\circ$-irreducibles. With the goal of computing hammocks up to isomorphism in a later work we equip an even larger set of \emph{weak arch bridges} with another partial binary operation, $\circ_H$, to obtain a finite category. Each weak arch bridge admits a unique $\circ_H$-factorization into \emph{arch bridges}, i.e., the $\circ_H$-irreducibles.

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arXiv2020

On the stable radical of some non-domestic string algebras

We introduce the concept of a prime band in a string algebra $\Lambda$ and use it to associate to $\Lambda$ its finite bridge quiver. Then we introduce a new technique of `recursive systems' for showing that a graph map between finite dimensional string modules lies in its stable radical. Further we study two classes of non-domestic string algebras in terms of some connectedness properties of its bridge quiver. `Meta-$\bigcup$-cyclic' string algebras constitute the first class that is essentially characterized by the statement that each finite string is a substring of a band. Extending this class we have `meta-torsion-free' string algebras that are characterized by a dichotomy statement for ranks of graph maps between string modules--such maps either have finite rank or are in the stable radical. Their stable ranks can only take values from $\{\omega,\omega+1,\omega+2\}$.

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arXiv2019

Aggregating Relational Structures

We generalize the Arrow's impossibility theorem--a key result in social choice theory--to the setting where the arity $k$ of the relation under consideration is greater than $2$. Some special but natural properties of $k$-ary relations are considered, as well as an analogue for such $k$-ary relations of Endriss and Grandi's result on graph aggregation is proved.

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arXiv2018

Definable combinatorics with dense linear orders

We compute the model-theoretic Grothendieck ring, $K_0(\mathcal{Q})$, of a dense linear order (DLO) with or without end points, $\mathcal{Q}=(Q,<)$, as a structure of the signature $\{<\}$, and show that it is a quotient of the polynomial ring over $\mathbb{Z}$ generated by $\mathbb N_+\times(Q\sqcup\{-\infty\})$ by an ideal that encodes multiplicative relations of pairs of generators. As a corollary we obtain that a DLO satisfies the pigeon hole principle (PHP) for definable subsets and definable bijections between them--a property that is too strong for many structures.

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