This doctoral thesis covers several topics related to the construction and study of maximal determinant matrices with complex entries. The first three chapters are devoted to number-theoretic tools to prove the non-solvability of Gram matrix equations over certain fields, with a focus on combinatorial applications. Chapter 4 gives a survey on Butson-type Hadamard matrices, and shows an improved lower bound on primes $p$ for the existence of $BH(12p, p)$ matrices. Chapter 5 contains the main contributions of the thesis, where the maximal determinant problem for matrices over the m-th roots of unity is discussed, and where new upper and lower bounds, as well as constructions at small orders, are given. Chapter 6 studies maximal determinant matrices over association schemes. Chapter 7 gives an application of design theory to privacy in communications, and it is connected to the rest of the thesis by the use of the theory of quadratic forms.
As an application of linear algebra for enumerative combinatorics, we introduce two new ideas, signed bigrassmannian polynomials and bigrassmannian determinant. First, a signed bigrassmannian polynomial is a variant of the statistic given by the number of bigrassmannian permutations below a permutation in Bruhat order as Reading suggested (2002) and afterward the author developed (2011). Second, bigrassmannian determinant is a $q$-analog of the determinant with respect to our statistic. It plays a key role for a determinantal expression of those polynomials. We further show that bigrassmannian…
We prove a formula which generalizes both Onn's colorful determinantal formula, related to Rota's basis conjecture, and Svrtan's $n!$ formula, related to the Atiyah-Sutcliffe problem. In some cases, our formula allows us to prove some results similar in spirit to the statement of Rota's basis conjecture. We prove such a result using Svrtan's $n!$ formula, generalizing one of Svrtan's arguments to a combinatorial setting.
We give upper bounds for the determinant of an $n\times n$ zero-one matrix containing $kn$ ones for integral $k$. Our results improve upon a result of Ryser for $k=o(n^{1/3})$. For fixed $k\ge 3$ it was an open question whether Hadamard's inequality could be exponentially improved. We answer this in the affirmative. Our results stem from studying matrices with row sums $k$ and bounding their Gram determinants. Our technique allows us to give upper bounds when these matrices are perturbed.
本文研究了元素为{-1，1}的奇阶矩阵的极大行列式的求法。最一般的最大行列式的上界，由于巴尔巴，只能达到时，秩序是两个连续的平方和。它是明确的，在这种情况下，总是达到的界限。除此之外，仅在第3、7、9、11、17和21阶中确立了最大值。在本文中，我们确认了这些订单的结果，并添加到列表中的第15个订单。我们遵循以前的作者在穷举搜索候选革兰氏矩阵的行列式大于或等于平方的一个已知的最大下界。然后，我们尝试将每个候选分解为{-1，1}-矩阵及其转置的乘积。对于15阶，我们找到了四个候选者，都是埃里希块形式，其中两个具有行列式（105*3^5*2^14）^2，另一个具有行列式（108*3^5*2^14）^2。前者之一分解（以一种基本上独特的方式），而其余三个不分解。这个结果证明了W. D. Smith和J. H. E.科恩我们还利用我们的方法计算了29阶、33阶和37阶极大行列式的改进的上界，并建立了9阶和1,1阶{-1，1}-矩阵的行列式函数的值域.
We prove that $\det A\leq 6^\frac{n}{6}$ whenever $A\in\{0,1\}^{n\times n}$ contains at most $2n$ ones. We also prove an upper bound on the determinant of matrices with the $k$-consecutive ones property, a generalisation of the consecutive ones property, where each row is allowed to have up to $k$ blocks of ones. Finally, we prove an upper bound on the determinant of a path-edge incidence matrix in a tree and use that to bound the leaf rank of a graph in terms of its order.
This chapter combines an introduction and research survey about Schubert varieties. The theme is to combinatorially classify their singularities using a family of polynomial ideals generated by determinants.
Hadamard最大行列式问题要求最大的n乘n行列式，元素在{+1，-1}中。当n全等于1（mod 4）时，Farmakis和Kounias的极大过剩构造法是构造大行列式（虽然很少是极大行列式）的最成功的一般方法。然而，对于某些小的n，仍然有更大的决定因素;最近在ArXiv预印本math.CO/0304410中报道了几个新记录。在这里，我们定义了“3-正规化”n × n Hadamard矩阵，并从它们构造n+1阶的大行列式矩阵。我们的构造解释了以前的大部分“小n”记录，并在n=37，49，65，73，77，85，93和97时创造了新的记录，其中大部分都超出了最大超额技术的范围。我们猜想n=37的行列式，即72 x 9^{17} x 2^{36}，达到了全局极大值。
The present paper is devoted to the study of the combinatorics of 216 maximal $C^3$ circular codes --- a particular type of structure arising in the analysis of genomic sequences. Their circularity property is believed to be intimately connected to the protection against the reading frame shift in the process of RNA translation. We present some new observations concerning the internal structure of circular codes, which give a way to construct all of them in a relatively simple manner.
We describe algorithms for computing maximal determinants of binary circulant matrices of small orders. Here "binary matrix" means a matrix whose elements are drawn from $\{0,1\}$ or $\{-1,1\}$. We describe efficient parallel algorithms for the search, using Duval's algorithm for generation of necklaces and the well-known representation of the determinant of a circulant in terms of roots of unity. Tables of maximal determinants are given for orders $\le 53$. Our computations extend earlier results and disprove two plausible conjectures.
Technological University of the Shannon: Midlands Midwest（香农理工大学：米德兰中西部）；Dublin City University（都柏林城市大学）；Mathematical Sciences Publishers, Berkeley, CA, USA（数学科学出版社）；Worcester Polytechnic Institute, MA, USA（伍斯特理工学院）
In a celebrated paper of 1893, Hadamard established the maximal determinant theorem, which establishes an upper bound on the determinant of a matrix with complex entries of norm at most $1$. His paper concludes with the suggestion that mathematicians study the maximum value of the determinant of an $n \times n$ matrix with entries in $\{ \pm 1\}$. This is the Hadamard maximal determinant problem. This survey provides complete proofs of the major results obtained thus far. We focus equally on upper bounds for the determinant (achieved largely via the study of the Gram matrices), and constructiv…
Maximal determinants and saturated D-optimal designs of orders 19 and 37
饱和D最优设计是具有最大行列式的给定阶{+1，-1}方阵。我们搜索了19阶和37阶的饱和D最优设计，发现Smith，Cohn，Orrick和Solomon的已知矩阵是最优的.对于19阶，我们找到了所有具有最大行列式2^30 x 7^2 x 17的不等价饱和D最优设计，并确认这三个已知设计组成了一个完整的集合。对于37阶，我们证明了最大行列式是2^39 x 3^36，并找到了一个不等价的饱和D最优设计的样本。我们的方法是一个扩展的奥里克用来解决以前最小的未知顺序15;和Chadjipantelis，Kounias和Moyssiadis解决订单17和21。该方法是一个两步的计算，首先搜索候选的革兰氏矩阵，然后试图分解它们。使用类似的方法，我们也找到了13阶{+1，-1}矩阵行列式值的完整谱。
We prove the nonsingularity of a class of integer matrices V(n,d), namely power-product matrix, for positive integers n and d. Some technical proofs are mainly based on linear algebra and enumerative combinatorics, particularly the generating function method and involution principle. We will show that the matrix V(n,d) is nonsingular for all positive integers n and d, and often with sparse structure. Special attention is given to the computation of the determinant V(2,d) with positive integer d.
Let $D(n)$ be the maximal determinant for $n \times n$ $\{\pm 1\}$-matrices, and ${\mathcal R}(n) = D(n)/n^{n/2}$ be the ratio of $D(n)$ to the Hadamard upper bound. We give several new lower bounds on ${\mathcal R}(n)$ in terms of $d$, where $n = h+d$, $h$ is the order of a Hadamard matrix, and $h$ is maximal subject to $h \le n$. A relatively simple bound is \[{\mathcal R}(n) \ge \left(\frac{2}{πe}\right)^{d/2} \left(1 - d^2\left(\fracπ{2h}\right)^{1/2}\right) \;\text{ for all }\; n \ge 1.\] An asymptotically sharper bound is \[{\mathcal R}(n) \ge \left(\frac{2}{πe}\right)^{d/2} \exp\left(d\…
Let ${\mathcal D}(n)$ be the maximal determinant for $n \times n$ $\{\pm 1\}$-matrices, and $\mathcal R(n) = {\mathcal D}(n)/n^{n/2}$ be the ratio of ${\mathcal D}(n)$ to the Hadamard upper bound. Using the probabilistic method, we prove new lower bounds on ${\mathcal D}(n)$ and $\mathcal R(n)$ in terms of $d = n-h$, where $h$ is the order of a Hadamard matrix and $h$ is maximal subject to $h \le n$. For example, $\mathcal R(n) > (πe/2)^{-d/2}$ if $1 \le d \le 3$, and $\mathcal R(n) > (πe/2)^{-d/2}(1 - d^2(π/(2h))^{1/2})$ if $d > 3$. By a recent result of Livinskyi, $d^2/h^{1/2} \to 0$ as $n \…
A basis of the center of the 0-Hecke algebra of an arbitrary finite Coxeter group was described by He in 2015. This basis is indexed by certain equivalence classes of the Coxeter group whose explicit description is rather complicated. Even their number is not obvious. We consider case of the symmetric group $\mathfrak{S}_n$. Building on work of Geck, Kim and Pfeiffer, we obtain a complete set of representatives of the equivalence classes. This set is naturally parametrized by certain compositions of $n$ called maximal. It follows that the maximal compositions of $n$ index the basis of the cent…
We consider maximal non-$l$-intertwining collections, which are a higher-dimensional version of the maximal non-crossing collections which give clusters of Plücker coordinates in the Grassmannian coordinate ring, as described by Scott. We extend a method of Scott for producing such collections, which are related to tensor products of higher Auslander algebras of type $A$. We show that a higher preprojective algebra of the tensor product of two $d$-representation-finite algebras has a $d$-precluster-tilting subcategory. Finally we relate mutations of these collections to a form of tilting for t…
General lower bounds on maximal determinants of binary matrices
本文给出了n乘n {+1，-1}-矩阵的极大行列式的一般下界，包括Hadamard猜想的假设和不假设.我们的界改进了de Launey和Levin（2010）的早期结果，对于n mod 4的某些同余类，改进了Koukouvinos，Mitrouli和Seberry（2000）的结果。在附录中，我们使用Jacobi行列式恒等式给出了Szölllossi（2010）关于Hadamard矩阵的子式的一个结果的新证明。
New Records for the Hadamard Maximal Determinant Problem
California Institute of Technology（加州理工学院）；University of Cambridge（剑桥大学）；London Institute for Mathematical Sciences（伦敦数学科学研究所）；University of Miami（迈阿密大学）；Temple University（天普大学）
This paper presents a geometric model of the Auslander-Reiten quiver of a type A quiver together with a stability function for which all indecomposable modules are stable. We also introduce a new Catalan object which we call a maximal almost rigid representation. We show that its endomorphism algebra is a tilted algebra of type A. We define a partial order on the set of maximal almost rigid representations and use our new geometric model to show that this partial order is a Cambrian lattice.
In this paper we study maximal chains in certain lattices constructed from powers of chains by iterated lax colimits in the $2$-category of posets. Such a study is motivated by the fact that in lower dimensions, we get some familiar combinatorial objects such as Dyck paths and Kreweras walks.
In this paper we study the geometry and combinatorics of the possible rational polyhedral fans with a given set of rays. The main questions we consider are when such fans are projective, complete, or simplicial. To answer these questions we use techniques of invariant theory to characterize the quotients of maximal saturated open sets in torus representations.
We give a combinatorial evaluation of Iwahori Whittaker functions for unramified genuine principal series representations on metaplectic covers of the general linear group over a non-archimedean local field. To describe the combinatorics, we introduce new combinatorial data that we call colored data: colored Lusztig data, colored Gelfand-Tsetlin patterns, and colored lattice models. We show that all three are equivalent. To achieve the result, we give an explicit Iwahori decomposition for the maximal unipotent subgroup of a split reductive group which gives the parametrization of the generaliz…
We use combinatorics of quivers and the corresponding surfaces to study maximal green sequences of minimal length for quivers of type $\mathbb{A}$. We prove that such sequences have length $n+t$, where $n$ is the number of vertices and $t$ is the number of 3-cycles in the quiver. Moreover, we develop a procedure that yields these minimal length maximal green sequences.
The representation theory of finite groups began with Frobenius's factorization of Dedekind's group determinant. In this paper, we consider the case of the semigroup determinant. The semigroup determinant is nonzero if and only if the complex semigroup algebra is Frobenius, and so our results include applications to the study of Frobenius semigroup algebras. We explicitly factor the semigroup determinant for commutative semigroups and inverse semigroups. We recover the Wilf-Lindström factorization of the semigroup determinant of a meet semilattice and Wood's factorization for a finite commutat…
We give some necessary conditions for maximality of $0/1$-determinant. Let ${\bf M}$ be a nondegenerate $0/1$-matrix of order $n$. Denote by $\bf A$ the matrix of order $n+1$ which appears from ${\bf M}$ after adding the $(n+1)$th row $(0,0,\ldots,0,1)$ and the $(n+1)$th column consisting of $1$'s. Suppose ${\bf A}^{-1}=(l_{ij}),$ then for all $i=1,\ldots,n$ we have $\sum_{j=1}^{n+1} |l_{ij}|\geq 2.$ Moreover, if $|\det({\bf M})|$ is equal to the maximum value of a $0/1$-determinant of order $n$, then $\sum_{j=1}^{n+1} |l_{ij}|= 2$ for all $i=1,\ldots,n$. Keywords: maximum 0/1-deteminant, simp…
We introduce a remarkable new family of norms on the space of $n \times n$ complex matrices. These norms arise from the combinatorial properties of symmetric functions, and their construction and validation involve probability theory, partition combinatorics, and trace polynomials in noncommuting variables. Our norms enjoy many desirable analytic and algebraic properties, such as an elegant determinantal interpretation and the ability to distinguish certain graphs that other matrix norms cannot. Furthermore, they give rise to new dimension-independent tracial inequalities. Their potential meri…
