Small Rank Research Articles

Ulrich bundles on special cubic fourfolds of small discriminants

In this paper, we provide computer-based construction of Ulrich bundles of small ranks over some special cubic fourfolds of small discriminants via deformation theory. First, we construct a simple sheaf whose Chern classes are the same as an Ulrich bundle of the same rank (if exists) as the syzygy sheaf of an Ulrich sheaf on a surface contained in a special cubic fourfold. We observe that a general deformation of this simple sheaf becomes Ulrich in several cases. We also provide some experimental constructions of Ulrich bundles on special cubic fourfolds of small discriminants.

On Donkin's tilting module conjecture II: counterexamples

In this paper we produce infinite families of counterexamples to Jantzen's question posed in 1980 on the existence of Weyl $p$ -filtrations for Weyl modules for an algebraic group and Donkin's tilting module conjecture formulated in 1990. New techniques to exhibit explicit examples are provided along with methods to produce counterexamples in large rank from counterexamples in small rank. Counterexamples can be produced via our methods for all groups other than when the root system is of type ${\rm A}_{n}$ or ${\rm B}_{2}$ .

New and improved bounds on the contextuality degree of multi-qubit configurations

AbstractWe present algorithms and a C code to reveal quantum contextuality and evaluate the contextuality degree (a way to quantify contextuality) for a variety of point-line geometries located in binary symplectic polar spaces of small rank. With this code we were not only able to recover, in a more efficient way, all the results of a recent paper by de Boutray et al. [(2022). Journal of Physics A: Mathematical and Theoretical55 475301], but also arrived at a bunch of new noteworthy results. The paper first describes the algorithms and the C code. Then it illustrates its power on a number of subspaces of symplectic polar spaces whose rank ranges from 2 to 7. The most interesting new results include: (i) non-contextuality of configurations whose contexts are subspaces of dimension 2 and higher, (ii) non-existence of negative subspaces of dimension 3 and higher, (iii) considerably improved bounds for the contextuality degree of both elliptic and hyperbolic quadrics for rank 4, as well as for a particular subgeometry of the three-qubit space whose contexts are the lines of this space, (iv) proof for the non-contextuality of perpsets and, last but not least, (v) contextual nature of a distinguished subgeometry of a multi-qubit doily, called a two-spread, and computation of its contextuality degree. Finally, in the three-qubit polar space we correct and improve the contextuality degree of the full configuration and also describe finite geometric configurations formed by unsatisfiable/invalid constraints for both types of quadrics as well as for the geometry whose contexts are all 315 lines of the space.

Delegation-Relegation for Boolean Matrix Factorization

The Boolean Matrix Factorization (BMF) problem aims to represent a n×m Boolean matrix as the Boolean product of two matrices of small rank k, where the product is computed using Boolean algebra operations. However, finding a BMF of minimum rank is known to be NP-hard, posing challenges for heuristic algorithms and exact approaches in terms of rank found and computation time, particularly as matrix size or the number of entries equal to 1 grows. In this paper, we present a new approach to simplifying the matrix to be factorized by reducing the number of 1-entries, which allows to directly recover a Boolean factorization of the original matrix from its simplified version. We introduce two types of simplification: one that performs numerous simplifications without preserving the original rank and another that performs fewer simplifications but guarantees that an optimal BMF on the simplified matrix yields an optimal BMF on the original matrix. Furthermore, our experiments show that our approach outperforms existing exact BMF algorithms.

Saturating linear sets of minimal rank

Saturating sets are combinatorial objects in projective spaces over finite fields that have been intensively investigated in the last three decades. They are related to the so-called covering problem of codes in the Hamming metric. In this paper, we consider the recently introduced linear version of such sets, which is, in turn, related to the covering problem in the rank metric. The main questions in this context are how small the rank of a saturating linear set can be and how to construct saturating linear sets of small rank. Recently, Bonini, Borello, and Byrne provided a lower bound on the rank of saturating linear sets in a given projective space, which is shown to be tight in some cases. In this paper, we provide construction of saturating linear sets meeting the lower bound and we develop a link between the saturating property and the scatteredness of linear sets. The last part of the paper is devoted to show some parameters for which the bound is not tight.

On finite d$d$‐maximal groups

AbstractLet be a positive integer. A finite group is called ‐maximal if it can be generated by precisely elements, whereas its proper subgroups have smaller generating sets. For , the ‐maximal groups have been classified up to isomorphism and only partial results have been proved for larger . In this work, we prove that a ‐maximal group is supersolvable and we give a characterisation of ‐maximality in terms of so‐called maximal ‐pairs. Moreover, we classify the maximal ‐pairs of small rank obtaining, as a consequence, the classification of the isomorphism classes of 3‐maximal finite groups.

The galaxy of Coxeter groups

In this paper we introduce the galaxy of Coxeter groups (of finite rank) – an infinite dimensional, locally finite, ranked simplicial complex which captures isomorphisms between finite rank Coxeter systems. In doing so, we would like to suggest a new framework to study the isomorphism problem for Coxeter groups. We prove some structural results about this space, provide a full characterization in small ranks and propose many questions. In addition we survey known tools, results and conjectures.Along the way we show profinite rigidity of triangle Coxeter groups – a result which is possibly of independent interest.

Chromatic numbers of Cayley graphs of abelian groups: A matrix method

In this paper, we take a modest first step towards a systematic study of chromatic numbers of Cayley graphs on abelian groups. We lose little when we consider these graphs only when they are connected and of finite degree. As in the work of Heuberger and others, in such cases the graph can be represented by an m×r integer matrix, where we call m the dimension and r the rank. Adding or subtracting rows produces a graph homomorphism to a graph with a matrix of smaller dimension, thereby giving an upper bound on the chromatic number of the original graph. In this article we develop the foundations of this method. As a demonstration of its utility, we provide an alternate proof of Payan's theorem, which states that a cubelike graph (i.e., a Cayley graph on the product Z2×⋯×Z2 of the integers modulo 2 with itself finitely many times) cannot have chromatic number 3. In a series of follow-up articles using the method of Heuberger matrices, we completely determine the chromatic number in cases with small dimension and rank, as well as prove a generalization of Zhu's theorem on the chromatic number of 6-valent integer distance graphs.

The radical of binary dimensional dual hyperovals

Let S be a dimensional dual hyperoval of rank n over F2. We introduce and study the radical P(S), which is a subspace of the ambient space U(S) of S invariant under the automorphism group of S. For the vast majority of the known dimensional dual hyperovals we have P(S)=U(S). Interesting is the case of proper radicals, i.e. P(S)≠U(S). Starting point of our investigations is a result of the second author [10, Thm. 1], [7, Thm. 3.6] (Theorem 1.2 below) which characterizes alternating dual hyperovals by the property that S splits over P(S). This Theorem is extended by Theorem 1.3 where we characterize dimensional dual hyperovals S with dim⁡U(S)−dim⁡P(S)=rank(S)−1. Moreover we will show (Theorem 4.6) that a proper radical implies that this dimensional dual hyperoval is a disjoint union of subDHOs of smaller rank. The notion of “disjoint union of subDHOs” has been introduced by Yoshiara [17]. Some theory on dimensional dual hyperovals with proper radicals is developed. Our paper also provides some computational results on dual hyperovals of small rank with a proper radical. These calculations indicate — though dual hyperovals with a proper radical seem to be scarce — that the number of these hyperovals is steadily growing as function of the rank.

New models for some free algebras of small ranks

We give new models of the free abelian dimonoid of rank $2$, the free generalized digroup of rank $1$ and the free commutative doppelsemigroup of rank $1$.

CP decomposition for tensors via alternating least squares with QR decomposition

AbstractThe CP tensor decomposition is used in applications such as machine learning and signal processing to discover latent low‐rank structure in multidimensional data. Computing a CP decomposition via an alternating least squares (ALS) method reduces the problem to several linear least squares problems. The standard way to solve these linear least squares subproblems is to use the normal equations, which inherit special tensor structure that can be exploited for computational efficiency. However, the normal equations are sensitive to numerical ill‐conditioning, which can compromise the results of the decomposition. In this paper, we develop versions of the CP‐ALS algorithm using the QR decomposition and the singular value decomposition, which are more numerically stable than the normal equations, to solve the linear least squares problems. Our algorithms utilize the tensor structure of the CP‐ALS subproblems efficiently, have the same complexity as the standard CP‐ALS algorithm when the input is dense and the rank is small, and are shown via examples to produce more stable results when ill‐conditioning is present. Our MATLAB implementation achieves the same running time as the standard algorithm for small ranks, and we show that the new methods can obtain lower approximation error.

Analysis and Construction of Zero-Knowledge Proofs for the MinRank Problem

Abstract The MinRank problem is an NP-complete problem that is prevalent in multivariate cryptography and its goal is to find a non-zero linear combination of given a series of matrices over a ring such that the obtained matrix has a small rank. At Asiacrypt 2001, two Zero-Knowledge Proofs of Knowledge (ZKPoK) for the MinRank problem are proposed, and we call them MRZK and MRZK$^{\dagger }$, respectively. The latter is an improved version of the proof size of the former. However, the efficiency of MRZK$^{\dagger }$ has been open and not analyzed. While the MRZK protocol is secure, it must be repeated many times due to the soundness error $2/3$, which leads to the large proof size. For 128-bit security, the MRZK protocol is executed at least 219 iterations and the proof size is about 32 KB. In this paper, we first show that the efficiency of MRZK$^{\dagger }$ is impractical due to unreasonable parameter size. However, when the parameter size is tuned and the efficiency is improved, an imposter can be efficiently constructed. Then, to alleviate the large proof size of MRZK, inspired by the technique designing ZKPoK (Eurocrypt 2020), we propose a sigma protocol with helper to prove the solution to the MinRank problem. Finally, we transform the sigma protocol with helper into a standard ZKPoK (MRZK$^{\sharp }$) by removing the helper. The MRZK$^{\sharp }$ protocol can achieve any small soundness error and enjoy the proof size of about 15 KB (53% improvement over MRZK).

Fano foliations with small algebraic ranks

In this paper we study the algebraic ranks of foliations on Q-factorial normal projective varieties. We start by establishing a Kobayashi-Ochiai's theorem for Fano foliations in terms of algebraic rank. We then investigate the local positivity of the anti-canonical divisors of foliations, obtaining a lower bound for the algebraic rank of a foliation in terms of Seshadri constant. We describe those foliations whose algebraic rank slightly exceeds this bound and classify Fano foliations on smooth projective varieties attaining this bound. Finally we construct several examples to illustrate the general situation, which in particular allow us to answer a question asked by Araujo and Druel on the generalised indices of foliations.

Zonal graphs of small cycle rank

A zonal labeling of a plane graph G is an assignment of the two nonzero elements of the ring Z 3 of integers modulo 3 to the vertices of G such that the sum of the labels of the vertices on the boundary of each region of G is the zero element of Z 3 . A plane graph possessing such a labeling is a zonal graph. There is a connection between zonal labelings of connected bridgeless cubic plane graphs and the Four Color Theorem. Zonal labelings of cycles play a role in this connection. The cycle rank of a connected graph of order n and size m is m − n + 1 . Thus, cycles have cycle rank 1. All zonal connected graphs of cycle rank at most 2 are determined.

Corners and simpliciality in oriented matroids and partial cubes

Building on a recent characterization of tope graphs of Complexes of Oriented Matroids (COMs), we tackle and generalize several classical problems in Oriented Matroids (OMs), Lopsided Sets (aka ample set systems), and partial cubes via Metric Graph Theory.Our first main result is that every element of an OM from a class introduced by Mandel is incident to a simplicial tope, i.e, such OMs contain no mutation-free elements. This allows us to refute a conjecture of Mandel from 1983, that would have implied the famous Las Vergnas’ simplex conjecture. Further, we show that the mutation graph of uniform OMs of order at most 9 are connected, thus confirming a stronger conjecture of Cordovil-Las Vergnas in this setting.The second main contribution is the introduction of corners of COMs as a natural generalization of corners in Lopsided Sets. Generalizing results of Bandelt and Chepoi, Tracy Hall, and Chalopin et al. we prove that realizable COMs, rank 2 COMs, as well as hypercellular graphs admit corner peelings. Using this, we confirm Las Vergnas’ conjecture for antipodal partial cubes (a class much larger than OMs) of small rank or isometric dimension.

Twisted forms of classical groups

Twisted forms of classical reductive group schemes are described in a unified way. Such group schemes are constructed from algebraic objects of finite rank, excluding some exceptions of small rank. These objects, called the augmented odd form algebras, consist of 2 2 -step nilpotent groups with an action of the underlying commutative ring, hence the basic descent theory for them will be developed. Finally, classical isotropic reductive groups are described as odd unitary groups up to an isogeny.

Toward a Sharp Baer–Suzuki Theorem for the π-Radical: Exceptional Groups of Small Rank

Toward a Sharp Baer–Suzuki Theorem for the π-Radical: Exceptional Groups of Small Rank

Characters of classical groups, Schur-type functions and discrete splines

We study a spectral problem related to finite-dimensional characters of the groups $Sp(2N)$, $SO(2N+1)$ and $SO(2N)$, which form the classical series $\mathcal{C}$, $\mathcal{B}$ and $\mathcal{D}$, respectively. Irreducible characters of these three series are given by $N$-variate symmetric polynomials. The spectral problem in question consists in the decomposition of characters after their restriction to subgroups of the same type but smaller rank $K<N$. The main result of the paper is the derivation of explicit determinantal formulae for the coefficients of the expansion. In fact, first we compute these coefficients in greater generality - for the multivariate symmetric Jacobi polynomials depending on two continuous parameters. Next, we show that the formulae are drastically simplified in the three special cases of Jacobi polynomials corresponding to characters of the series $\mathcal{C}$, $\mathcal{B}$ and $\mathcal{D}$. In particular, we show that then these coefficients are given by piecewise polynomial functions. This is where a link with discrete splines arises. For characters of the series $\mathcal{A}$ (that is, of the unitary groups $U(N)$) similar results were obtained previously by Borodin and this author [5], and then reproved by Petrov [39] by another method. The case of symplectic and orthogonal characters is more intricate. Bibliography: 58 titles.

Three families of toric rings arising from posets or graphs with small class groups

The main objects of the present paper are (i) Hibi rings (toric rings arising from order polytopes of posets), (ii) stable set rings (toric rings arising from stable set polytopes of perfect graphs), and (iii) edge rings (toric rings arising from edge polytopes of graphs satisfying the odd cycle condition). The goal of the present paper is to analyze those three toric rings and to discuss their structures in the case where their class groups have small rank. We prove that the class groups of (i), (ii) and (iii) are torsionfree. More precisely, we give descriptions of their class groups. Moreover, we characterize the posets or graphs whose associated toric rings have rank 1 or 2. By using those characterizations, we discuss the differences of isomorphic classes of those toric rings with small class groups.

Random translation-invariant Hamiltonians and their spectral gaps

We consider random translation-invariant frustration-free quantum spin Hamiltonians on ZD in which the nearest-neighbor interaction in every direction is randomly sampled and then distributed across the lattice. Our main result is that, under a small rank constraint, the Hamiltonians are automatically frustration-free and they are gapped with a positive probability. This extends previous results on 1D spin chains to all dimensions. The argument additionally controls the local gap. As an application, we obtain a 2D area law for a cut-dependent ground state via recent AGSP methods of Anshu-Arad-Gosset.