http://publications.mfo.de/handle/mfo/4095 Tue, 07 Apr 2026 23:46:24 GMT 2026-04-07T23:46:24Z http://publications.mfo.de/handle/mfo/4188 Alternating Snake Modules and a Determinantal Formula Brito, Matheus; Chari, Vyjayanthi We introduce a family of modules for the quantum affine algebra which include as very special cases both the snake modules and the modules arising from a monoidal categorification of cluster algebras. We give necessary and sufficient conditions for these modules to be prime and prove a unique factorization result. We also give an explicit formula expressing the module as an alternating sum of Weyl modules. Finally, we give an application of our results to classical questions in the category $\mathcal{ O}(\mathfrak{gl}_r)$. Specifically we apply our results to show that there are a large family of non-regular, non-dominant weights $\mu$ for which the non-zero Kazhdan-Lusztig coefficients $c_{\mu, \nu}$ are $\pm 1$. Tue, 17 Dec 2024 00:00:00 GMT http://publications.mfo.de/handle/mfo/4188 2024-12-17T00:00:00Z Brito, Matheus Chari, Vyjayanthi We introduce a family of modules for the quantum affine algebra which include as very special cases both the snake modules and the modules arising from a monoidal categorification of cluster algebras. We give necessary and sufficient conditions for these modules to be prime and prove a unique factorization result. We also give an explicit formula expressing the module as an alternating sum of Weyl modules. Finally, we give an application of our results to classical questions in the category $\mathcal{ O}(\mathfrak{gl}_r)$. Specifically we apply our results to show that there are a large family of non-regular, non-dominant weights $\mu$ for which the non-zero Kazhdan-Lusztig coefficients $c_{\mu, \nu}$ are $\pm 1$. http://publications.mfo.de/handle/mfo/4187 A CFSG-Free Explicit Jordan’s Theorem over Arbitrary Fields Bajpai, Jitendra; Dona, Daniele We prove a version of Jordan's classification theorem for finite subgroups of $\mathrm{GL}_{n}(K)$ that is at the same time quantitatively explicit, CFSG-free, and valid for arbitrary $K$. This is the first proof to satisfy all three properties at once. Our overall strategy follows Larsen and Pink [24], with explicit computations based on techniques developed by the authors and Helfgott [2, 3], particularly in relation to dimensional estimates. Fri, 29 Nov 2024 00:00:00 GMT http://publications.mfo.de/handle/mfo/4187 2024-11-29T00:00:00Z Bajpai, Jitendra Dona, Daniele We prove a version of Jordan's classification theorem for finite subgroups of $\mathrm{GL}_{n}(K)$ that is at the same time quantitatively explicit, CFSG-free, and valid for arbitrary $K$. This is the first proof to satisfy all three properties at once. Our overall strategy follows Larsen and Pink [24], with explicit computations based on techniques developed by the authors and Helfgott [2, 3], particularly in relation to dimensional estimates. http://publications.mfo.de/handle/mfo/4186 The Congruence Properties of Romik’s Sequence of Taylor Coefficients of Jacobi’s Theta Function $θ_3$ Krattenthaler, Christian; Müller, Thomas W. In [Ramanujan J. 52 (2020), 275-290], Romik considered the Taylor expansion of Jacobi's theta function $\theta_3(q)$ at $q=e^{-\pi}$ and encoded it in an integer sequence $(d(n))_{n\ge0}$ for which he provided a recursive procedure to compute the terms of the sequence. He observed intriguing behaviour of $d(n)$ modulo primes and prime powers. Here we prove (1) that $d(n)$ eventually vanishes modulo any prime power $p^e$ with $p\equiv3$ (mod 4), (2) that $d(n)$ is eventually periodic modulo any prime power $p^e$ with $p\equiv1$ (mod 4), and (3) that $d(n)$ is purely periodic modulo any 2-power $2^e$. Our results also provide more detailed information on period length, respectively from when on the sequence vanishes or becomes periodic. The corresponding bounds may not be optimal though, as computer data suggest. Our approach shows that the above congruence properties hold at a much finer, polynomial level. Fri, 29 Nov 2024 00:00:00 GMT http://publications.mfo.de/handle/mfo/4186 2024-11-29T00:00:00Z Krattenthaler, Christian Müller, Thomas W. In [Ramanujan J. 52 (2020), 275-290], Romik considered the Taylor expansion of Jacobi's theta function $\theta_3(q)$ at $q=e^{-\pi}$ and encoded it in an integer sequence $(d(n))_{n\ge0}$ for which he provided a recursive procedure to compute the terms of the sequence. He observed intriguing behaviour of $d(n)$ modulo primes and prime powers. Here we prove (1) that $d(n)$ eventually vanishes modulo any prime power $p^e$ with $p\equiv3$ (mod 4), (2) that $d(n)$ is eventually periodic modulo any prime power $p^e$ with $p\equiv1$ (mod 4), and (3) that $d(n)$ is purely periodic modulo any 2-power $2^e$. Our results also provide more detailed information on period length, respectively from when on the sequence vanishes or becomes periodic. The corresponding bounds may not be optimal though, as computer data suggest. Our approach shows that the above congruence properties hold at a much finer, polynomial level. http://publications.mfo.de/handle/mfo/4178 On the Halpern Method with Adaptive Anchoring Parameters Pinto, Pedro; Pischke, Nicholas We establish the convergence of a speed-up version of the Halpern iteration with adaptive anchoring parameters in the general geodesic setting of Hadamard spaces, generalizing a recent result by He, Xu, Dong and Mei from a linear to a nonlinear setting. In particular, our results extend the fast rates of asymptotic regularity obtained by these authors for the first time to a nonlinear setting. Our approach relies on a quantitative study of these previous results in the linear setting, combined with certain optimizations and an elimination of the weak compactness arguments employed crucially in the linear setting, which not only allows for the lift of the result to a nonlinear setting but also streamlines the previous convergence analysis considerably. This work is set in the context of recent developments in proof mining, and as byproduct of our approach, we further obtain quantitative information in the form of highly uniform rates of metastability of low complexity, which are new already in the context of Hilbert spaces. Mon, 21 Oct 2024 00:00:00 GMT http://publications.mfo.de/handle/mfo/4178 2024-10-21T00:00:00Z Pinto, Pedro Pischke, Nicholas We establish the convergence of a speed-up version of the Halpern iteration with adaptive anchoring parameters in the general geodesic setting of Hadamard spaces, generalizing a recent result by He, Xu, Dong and Mei from a linear to a nonlinear setting. In particular, our results extend the fast rates of asymptotic regularity obtained by these authors for the first time to a nonlinear setting. Our approach relies on a quantitative study of these previous results in the linear setting, combined with certain optimizations and an elimination of the weak compactness arguments employed crucially in the linear setting, which not only allows for the lift of the result to a nonlinear setting but also streamlines the previous convergence analysis considerably. This work is set in the context of recent developments in proof mining, and as byproduct of our approach, we further obtain quantitative information in the form of highly uniform rates of metastability of low complexity, which are new already in the context of Hilbert spaces. http://publications.mfo.de/handle/mfo/4172 Local Existence and Conditional Regularity for the Navier-Stokes-Fourier System Driven by Inhomogeneous Boundary Conditions Abbatiello, Anna; Basarić, Danica; Chaudhuri, Nilasis; Feireisl, Eduard We consider the Navier–Stokes–Fourier system with general inhomogeneous Dirichlet–Neumann boundary conditions. We propose a new approach to the local well-posedness problem based on conditional regularity estimates. By conditional regularity we mean that any strong solution belonging to a suitable class remains regular as long as its amplitude remains bounded. The result holds for general Dirichlet-Neumann boundary conditions provided the material derivative of the velocity field vanishes on the boundary of the physical domain. As a corollary of this result we obtain: Blow up criteria for strong solutions; Local existence of strong solutions in the optimal $L^p - L^q$ framework; Alternative proof of the existing results on local well posedness. Wed, 25 Sep 2024 00:00:00 GMT http://publications.mfo.de/handle/mfo/4172 2024-09-25T00:00:00Z Abbatiello, Anna Basarić, Danica Chaudhuri, Nilasis Feireisl, Eduard We consider the Navier–Stokes–Fourier system with general inhomogeneous Dirichlet–Neumann boundary conditions. We propose a new approach to the local well-posedness problem based on conditional regularity estimates. By conditional regularity we mean that any strong solution belonging to a suitable class remains regular as long as its amplitude remains bounded. The result holds for general Dirichlet-Neumann boundary conditions provided the material derivative of the velocity field vanishes on the boundary of the physical domain. As a corollary of this result we obtain: Blow up criteria for strong solutions; Local existence of strong solutions in the optimal $L^p - L^q$ framework; Alternative proof of the existing results on local well posedness. http://publications.mfo.de/handle/mfo/4164 Diameter and Connectivity of Finite Simple Graphs II Hibi, Takayuki; Saeedi Madani, Sara Let $G$ be a finite simple non-complete connected graph on $[n] = \{1, \ldots, n\}$ and $\kappa(G) \geq 1$ its vertex connectivity. Let $f(G)$ denote the number of free vertices of $G$ and $\mathrm{diam}(G)$ the diameter of $G$. The final goal of this paper is to determine all sequences of integers $(n,f,d,k)$ with $n\geq 8$, $f\geq 0$, $d\geq 2$ and $k\geq 1$ for which there exists a finite simple non-complete connected graph on $[n]$ with $f=f(G)$, $d=\mathrm{diam}(G)$ and $k=\kappa(G)$. Thu, 05 Sep 2024 00:00:00 GMT http://publications.mfo.de/handle/mfo/4164 2024-09-05T00:00:00Z Hibi, Takayuki Saeedi Madani, Sara Let $G$ be a finite simple non-complete connected graph on $[n] = \{1, \ldots, n\}$ and $\kappa(G) \geq 1$ its vertex connectivity. Let $f(G)$ denote the number of free vertices of $G$ and $\mathrm{diam}(G)$ the diameter of $G$. The final goal of this paper is to determine all sequences of integers $(n,f,d,k)$ with $n\geq 8$, $f\geq 0$, $d\geq 2$ and $k\geq 1$ for which there exists a finite simple non-complete connected graph on $[n]$ with $f=f(G)$, $d=\mathrm{diam}(G)$ and $k=\kappa(G)$. http://publications.mfo.de/handle/mfo/4160 The Subgroup Structure of Pseudo-Reductive Groups Bate, Michael; Martin, Benjamin; Röhrle, Gerhard; Sercombe, Damian Let $k$ be a field. We investigate the relationship between subgroups of a pseudo-reductive $k$-group $G$ and its maximal reductive quotient $G'$, with applications to the subgroup structure of $G$. Let $k'/k$ be the minimal field of definition for the geometric unipotent radical of $G$, and let $\pi':G_{k'} \to G'$ be the quotient map. We first characterise those smooth subgroups $H$ of $G$ for which $\pi'(H_{k'})=G'$. We next consider the following questions: given a subgroup $H'$ of $G'$, does there exist a subgroup $H$ of $G$ such that $\pi'(H_{k'})=H'$, and if $H'$ is smooth can we find such a $H$ that is smooth? We find sufficient conditions for a positive answer to these questions. In general there are various obstructions to the existence of such a subgroup $H$, which we illustrate with several examples. Finally, we apply these results to relate the maximal smooth subgroups of $G$ with those of $G'$. Wed, 31 Jul 2024 00:00:00 GMT http://publications.mfo.de/handle/mfo/4160 2024-07-31T00:00:00Z Bate, Michael Martin, Benjamin Röhrle, Gerhard Sercombe, Damian Let $k$ be a field. We investigate the relationship between subgroups of a pseudo-reductive $k$-group $G$ and its maximal reductive quotient $G'$, with applications to the subgroup structure of $G$. Let $k'/k$ be the minimal field of definition for the geometric unipotent radical of $G$, and let $\pi':G_{k'} \to G'$ be the quotient map. We first characterise those smooth subgroups $H$ of $G$ for which $\pi'(H_{k'})=G'$. We next consider the following questions: given a subgroup $H'$ of $G'$, does there exist a subgroup $H$ of $G$ such that $\pi'(H_{k'})=H'$, and if $H'$ is smooth can we find such a $H$ that is smooth? We find sufficient conditions for a positive answer to these questions. In general there are various obstructions to the existence of such a subgroup $H$, which we illustrate with several examples. Finally, we apply these results to relate the maximal smooth subgroups of $G$ with those of $G'$. http://publications.mfo.de/handle/mfo/4157 The Alternating Halpern-Mann Iteration for Families of Maps Firmino, Paulo; Pinto, Pedro We generalize the alternating Halpern-Mann iteration to countably infinite families of nonexpansive maps and prove its strong convergence towards a common fixed point in the general nonlinear setting of Hadamard spaces. Our approach is based on a quantitative perspective which allowed to circumvent prevalent troublesome arguments and in the end provide a simple convergence proof. In that sense, discussing both the asymptotic regularity and the strong convergence of the iteration in quantitative terms, we furthermore provide low complexity uniform rates of convergence and of metastability (in the sense of T. Tao). In CAT(0) spaces, we obtain linear and quadratic uniform rates of convergence. Our results are made possible by proof-theoretical insights of the research program proof mining and extend several previous theorems in the literature. Mon, 15 Jul 2024 00:00:00 GMT http://publications.mfo.de/handle/mfo/4157 2024-07-15T00:00:00Z Firmino, Paulo Pinto, Pedro We generalize the alternating Halpern-Mann iteration to countably infinite families of nonexpansive maps and prove its strong convergence towards a common fixed point in the general nonlinear setting of Hadamard spaces. Our approach is based on a quantitative perspective which allowed to circumvent prevalent troublesome arguments and in the end provide a simple convergence proof. In that sense, discussing both the asymptotic regularity and the strong convergence of the iteration in quantitative terms, we furthermore provide low complexity uniform rates of convergence and of metastability (in the sense of T. Tao). In CAT(0) spaces, we obtain linear and quadratic uniform rates of convergence. Our results are made possible by proof-theoretical insights of the research program proof mining and extend several previous theorems in the literature. http://publications.mfo.de/handle/mfo/4156 Proof Mining and the Convex Feasibility Problem : the Curious Case of Dykstra's Algorithm Pinto, Pedro In a recent proof mining application, the proof-theoretical analysis of Dykstra's cyclic projections algorithm resulted in quantitative information expressed via primitive recursive functionals in the sense of Gödel. This was surprising as the proof relies on several compactness principles and its quantitative analysis would require the functional interpretation of arithmetical comprehension. Therefore, a priori one would expect the need of Spector’s bar-recursive functionals. In this paper, we explain how the use of bounded collection principles allows for a modified intermediate proof justifying the finitary results obtained, and discuss the approach in the context of previous eliminations of weak compactness arguments in proof mining. Mon, 15 Jul 2024 00:00:00 GMT http://publications.mfo.de/handle/mfo/4156 2024-07-15T00:00:00Z Pinto, Pedro In a recent proof mining application, the proof-theoretical analysis of Dykstra's cyclic projections algorithm resulted in quantitative information expressed via primitive recursive functionals in the sense of Gödel. This was surprising as the proof relies on several compactness principles and its quantitative analysis would require the functional interpretation of arithmetical comprehension. Therefore, a priori one would expect the need of Spector’s bar-recursive functionals. In this paper, we explain how the use of bounded collection principles allows for a modified intermediate proof justifying the finitary results obtained, and discuss the approach in the context of previous eliminations of weak compactness arguments in proof mining. http://publications.mfo.de/handle/mfo/4153 On Overgroups of Distinguished Unipotent Elements in Reductive Groups and Finite Groups of Lie Type Bate, Michael; Böhm, Sören; Martin, Benjamin; Röhrle, Gerhard Suppose $G$ is a simple algebraic group defined over an algebraically closed field of good characteristic $p$. In 2018 Korhonen showed that if $H$ is a connected reductive subgroup of $G$ which contains a distinguished unipotent element $u$ of $G$ of order $p$, then $H$ is $G$-irreducible in the sense of Serre. We present a short and uniform proof of this result using so-called good $A1$ subgroups of $G$, introduced by Seitz. We also formulate a counterpart of Korhonen’s theorem for overgroups of u which are finite groups of Lie type. Moreover, we generalize both results above by removing the restriction on the order of $u$ under a mild condition on $p$ depending on the rank of $G$, and we present an analogue of Korhonen’s theorem for Lie algebras. Tue, 18 Jun 2024 00:00:00 GMT http://publications.mfo.de/handle/mfo/4153 2024-06-18T00:00:00Z Bate, Michael Böhm, Sören Martin, Benjamin Röhrle, Gerhard Suppose $G$ is a simple algebraic group defined over an algebraically closed field of good characteristic $p$. In 2018 Korhonen showed that if $H$ is a connected reductive subgroup of $G$ which contains a distinguished unipotent element $u$ of $G$ of order $p$, then $H$ is $G$-irreducible in the sense of Serre. We present a short and uniform proof of this result using so-called good $A1$ subgroups of $G$, introduced by Seitz. We also formulate a counterpart of Korhonen’s theorem for overgroups of u which are finite groups of Lie type. Moreover, we generalize both results above by removing the restriction on the order of $u$ under a mild condition on $p$ depending on the rank of $G$, and we present an analogue of Korhonen’s theorem for Lie algebras.