arXiv subjects

Karthik Ganapathy

Publications and source records attributed to Karthik Ganapathy.

12 recordsLinked to original sources

arXiv2026

Axiomatics for generic categories in positive characteristic

Generic categories arising in representation stability and equivariant commutative algebra are difficult to analyze in positive characteristic, as these categories rarely admit enough (or even any) finitely generated injectives. Our main observation is that they are nonetheless an increasing union of Serre subcategories, each with finitely many simples and enough finite length injectives. We package this into an axiomatic framework which is widely applicable. In particular, we use this to analyze (1) VI-modules in non-describing characteristic, recovering Nagpal's classification of simple generic VI-modules, and (2) GL-equivariant modules over truncated polynomial rings and exterior algebras in infinitely many variables.

math.RT↗

arXiv2026

GL-algebras in positive characteristic III: the divided power algebra

In this paper, we study GL-equivariant modules over the infinite-variable divided power algebra $D = \text{Div}(k^{\infty})$ with $k$ an algebraically closed field of characteristic $p > 0$. Unlike previously analyzed GL-algebras, the divided power algebra is not noetherian or even finitely generated. We show that $D$ is GL-coherent and prove a ``shift theorem'' for finitely presented $D$-modules. Using this, we obtain a (semi-infinite) semi-orthogonal decomposition of its bounded derived category with one piece corresponding to each Frobenius twist $D^{(r)}$ of $D$. Crucial to our approach is the fact that $D$ is a flat colimit of subalgebras which are GL-noetherian.

math.AC↗

arXiv2025

Noncommutative invariants of finite and classical groups

We investigate the structure of the invariant subring of the tensor algebra $T(W)$ of a $G$-representation $W$, viewed as a twisted commutative algebra (tca). For a faithful representation $W$ of a finite group $G$ over a field $k$, we show that if char$(k) \mid \#G$, then $T(W)^G$ is not finitely generated as a tca. In contrast, for a representation $W$ of a classical group $G_{\mathbb{Z}}$, we prove that the invariant subring $T(W_k)^{G_k}$ is finitely generated as a tca when $k$ is algebraically closed of sufficiently large characteristic, provided that $W$ admits a good filtration over $\mathbb{Z}$. Finally, we introduce a categorical variant of the Gelfand--Kirillov dimension and compute its value to be $\binom{n+1}{2}$ for $T(\mathbb{C}^n)$ as a tca. Our key insight is to use the Schur functor to reduce questions about noncommutative invariants to those concerning vector invariants.

math.RA↗

arXiv2025

GL-algebras in positive characteristic II: the polynomial ring

We study GL-equivariant modules over the infinite variable polynomial ring $S = k[x_1, x_2, ..., x_n, ...]$ with $k$ an infinite field of characteristic $p > 0$. We extend many of Sam--Snowden's far-reaching results from characteristic zero to this setting. For example, while the Castelnuovo--Mumford regularity of a finitely generated GL-equivariant $S$-module need not be finite in positive characteristic, we show that the resolution still has finitely many "linear strands of higher slope". The crux of this paper is two technical results. The first is an extension to positive characteristic of Snowden's recent linearization of Draisma's embedding theorem which we use to study the generic category of $S$-modules. The second is a Nagpal-type "shift theorem" about torsion $S$-modules for which we introduce certain categorifications of the Hasse derivative. These two results together allow us to obtain explicit generators for the derived category. In a follow-up paper, we also use these results to prove finiteness results for local cohomology modules.

math.AC↗

arXiv2025

Ideal-theoretic non-noetherianity of polynomial functors in positive characteristic

A long-standing open problem in representation stability is whether every finitely generated commutative algebra in the category of strict polynomial functors satisfies the noetherian property. In this paper, we resolve this problem negatively over fields of positive characteristic using ideas from invariant theory. Specifically, we consider the algebra $P$ of polarizations of elementary symmetric polynomials inside the ring of all multisymmetric polynomials in $p \times \infty$ variables. We show $P$ is not noetherian based on two key facts: (1) the $p$-th power of every multisymmetric polynomial is in $P$ (our main technical result) and (2) the ring of multisymmetric polynomials is Frobenius split.

math.AC↗

arXiv2024

Non-noetherian GL-algebras in characteristic two

Over fields of characteristic two, we construct an infinite ascending chain of GL-stable ideals in the coordinate ring of infinite skew-symmetric matrices. This construction provides the first known example of a non-noetherian GL-algebra, thereby resolving a long-standing open question in the area. Our results build on the work of Draisma, Krasilnikov, and Krone.

math.AC↗

arXiv2024

Resolutions of symmetric ideals via stratifications of derived categories

We propose a method to unify various stability results about symmetric ideals in polynomial rings by stratifying related derived categories. We execute this idea for chains of $GL_n$-equivariant modules over an infinite field $k$ of positive characteristic. We prove the Le--Nagel--Nguyen--Römer conjectures for such sequences and obtain stability patterns in their resolutions as corollaries of our main result, which is a semiorthogonal decomposition for the bounded derived category of $GL_{\infty}$-equivariant modules over $S = k[x_1, x_2, \ldots, x_n, \ldots]$. Our method relies on finite generation results for certain local cohomology modules. We also outline approaches (i) to investigate Koszul duality for $S$-modules taking the Frobenius homomorphism (of $GL_{\infty}$) into account, and (ii) to recover and extend Murai's results about free resolutions of symmetric monomial ideals.

math.AC↗

arXiv2024

GL-algebras in positive characteristic I: the exterior algebra

We study the category of GL-equivariant modules over the infinite exterior algebra in positive characteristic. Our main structural result is a shift theorem a la Nagpal. Using this, we obtain a Church--Ellenberg type bound for the Castelnuovo--Mumford regularity. We also prove finiteness results for local cohomology.

math.AC↗

arXiv2024

Self-Tuning Network Control Architectures with Joint Sensor and Actuator Selection

We formulate a mathematical framework for designing a self-tuning network control architecture, and propose a computationally-feasible greedy algorithm for online architecture optimization. In this setting, the locations of active sensors and actuators in the network, as well as the feedback control policy are jointly adapted using all available information about the network states and dynamics to optimize a performance criterion. We show that the case with full-state feedback can be solved with dynamic programming, and in the linear-quadratic setting, the optimal cost functions and policies are piecewise quadratic and piecewise linear, respectively. Our framework is extended for joint sensor and actuator selection for dynamic output feedback control with both control performance and architecture costs. For large networks where exhaustive architecture search is prohibitive, we describe a greedy heuristic for actuator selection and propose a greedy swapping algorithm for joint sensor and actuator selection. Via numerical experiments, we demonstrate a dramatic performance improvement of greedy self-tuning architectures over fixed architectures. Our general formulation provides an extremely rich and challenging problem space with opportunities to apply a wide variety of approximation methods from stochastic control, system identification, reinforcement learning, and static architecture design for practical model-based control.

eess.SY↗

arXiv2023

Self-Tuning Network Control Architectures

We formulate a general mathematical framework for self-tuning network control architecture design. This problem involves jointly adapting the locations of active sensors and actuators in the network and the feedback control policy to all available information about the time-varying network state and dynamics to optimize a performance criterion. We propose a general solution structure analogous to the classical self-tuning regulator from adaptive control. We show that a special case with full-state feedback can be solved in principle with dynamic programming, and in the linear quadratic setting the optimal cost functions and policies are piecewise quadratic and piecewise linear, respectively. For large networks where exhaustive architecture search is prohibitive, we describe a greedy heuristic for joint architecture-policy design. We demonstrate in numerical experiments that self-tuning architectures can provide dramatically improved performance over fixed architectures. Our general formulation provides an extremely rich and challenging problem space with opportunities to apply a wide variety of approximation methods from stochastic control, system identification, reinforcement learning, and static architecture design.

math.OC↗

arXiv2022

History Data Driven Distributed Consensus in Networks

The association of weights in a distributed consensus protocol quantify the trust that an agent has on its neighbors in a network. An important problem in such networked systems is the uncertainty in the estimation of trust between neighboring agents, coupled with the losses arising from mistakenly associating wrong amounts of trust with different neighboring agents. We introduce a probabilistic approach which uses the historical data collected in the network, to determine the level of trust between each agent. Specifically, using the finite history of the shared data between neighbors, we obtain a configuration which represents the confidence estimate of every neighboring agent's trustworthiness. Finally, we propose a History-Data-Driven (HDD) distributed consensus protocol which translates the computed configuration data into weights to be used in the consensus update. The approach using the historical data in the context of a distributed consensus setting marks the novel contribution of our paper.

eess.SY↗

arXiv2020

Stillman's question for twisted commutative algebras

Let $\mathbf{A}_{n, m}$ be the polynomial ring $\text{Sym}(\mathbf{C}^n \otimes \mathbf{C}^m)$ with the natural action of $\mathbf{GL}_m(\mathbf{C})$. We construct a family of $\mathbf{GL}_m(\mathbf{C})$-stable ideals $J_{n, m}$ in $\mathbf{A}_{n, m}$, each equivariantly generated by one homogeneous polynomial of degree $2$. Using the Ananyan-Hochster principle, we show that the regularity of this family is unbounded. This negatively answers a question raised by Erman-Sam-Snowden on a generalization of Stillman's conjecture.

math.AC↗