An Explicit Formula for the Number of Solutions of X^2=0 in Triangular Matrices over a Finite Field

Cocharacters of polynomial identities of upper triangular matrices

We give an easy algorithm which calculates the generating function of the cocharacter sequence of the T-ideal of the polynomial identities of the algebra of upper triangular matrices over a field of characteristic zero. Applying this algorithm we have found the explicit form of the multiplicities in two cases: (i) for the "largest" partitions; (ii) for matrices of small size and for all partitions.

Cocharacters of polynomial identities of block triangular matrices

We give an algorithm which calculates the generating function of the cocharacter sequence of the polynomial identities of the algebra of upper block triangular (p+2q) x (p+2q) matrices over a field of characteristic zero with diagonal consisting of p copies of 1 x 1 and q copies of 2 x 2 matrices. We have found the explicit form of the multiplicities and their asymptotic behaviour for small values of p and q.

特殊矩阵的幂与逆的闭式解

Closed forms for powers and inverses of special matrices

受矩阵幂在组合序列中应用的相关研究启发，研究者借助拉格朗日反演公式与部分贝尔多项式，推导得出特殊上三角矩阵的s次幂、逆以及特殊非三角矩阵的s次幂的闭式解。

三角矩阵模 p 秩的波动

The fluctuations of the mod p rank of triangular matrices

本文研究了元素为独立同分布整数的随机下三角矩阵，证明了其余核的 Sylow p-子群具有与已知矩阵乘积相同的常数阶波动，并以此描述了该矩阵在有限域 F_p 上秩的极限波动。

A Combinatorial Interpretation of the LDU Decomposition of Totally Positive Matrices

We study the combinatorial description of the LDU decomposition of totally positive matrices. We give a description of the lower triangular L, the diagonal D, and the upper triangular U matrices of the LDU decomposition of totally positive matrices in terms of the combinatorial structure of essential planar networks described by Zelvinsky and Fomin. Similarly, we find a combinatorial description of the inverses of these matrices. In addition, we provide recursive formulae for computing the L, D, and U matrices of a totally positive matrix.

三角布尔矩阵的恒等式

Identities of triangular Boolean matrices

本文给出上三角布尔矩阵半环中恒等式的组合刻画，并应用于复杂度、有限公理化及可识别语言类的研究。

A note on identities in plactic monoids and monoids of upper-triangular tropical matrices

This paper uses the combinatorics of Young tableaux to prove the plactic monoid of infinite rank does not satisfy a non-trivial identity, by showing that the plactic monoid of rank $n$ cannot satisfy a non-trivial identity of length less than or equal to $n$. A new identity is then proven to hold for the monoid of $n \times n$ upper-triangular tropical matrices. Finally, a straightforward embedding is exhibited of the plactic monoid of rank $3$ into the direct product of two copies of the monoid of $3\times 3$ upper-triangular tropical matrices, giving a new proof that the plactic monoid of ra…

Branching rules in the ring of superclass functions of unipotent upper-triangular matrices

It is becoming increasingly clear that the supercharacter theory of the finite group of unipotent upper-triangular matrices has a rich combinatorial structure built on set-partitions that is analogous to the partition combinatorics of the classical representation theory of the symmetric group. This paper begins by exploring a connection to the ring of symmetric functions in non-commuting variables that mirrors the symmetric group's relationship with the ring of symmetric functions. It then also investigates some of the representation theoretic structure constants arising from the restriction, …

Upper triangular matrices and Billiard Arrays

Fix a nonnegative integer $d$, a field $\mathbb{F}$, and a vector space $V$ over $\mathbb{F}$ with dimension $d+1$. Let $T$ denote an invertible upper triangular matrix in ${\rm Mat}_{d+1}(\mathbb{F})$. Using $T$ we construct three flags on $V$. We find a necessary and sufficient condition on $T$ for these three flags to be totally opposite. In this case, we use these three totally opposite flags to construct a Billiard Array $B$ on $V$. It is known that $B$ is determined up to isomorphism by a certain triangular array of scalar parameters called the $B$-values. We compute these $B$-values in …

Super-character theory and comparison arguments for a random walk on the upper triangular matrices

Consider the random walk on the $n \times n$ upper triangular matrices with ones on the diagonal and elements over $\mathbb{F}_p$ where we pick a row at random and either add it or subtract it from the row directly above it. The main result of this paper is to prove that the dependency of the mixing time on $p$ is $p^2$. This is proven by combining super-character theory and comparison theory arguments.

Remarks on the analyticity of subadditive pressure for products of triangular matrices

We study Falconer's subadditive pressure function with emphasis on analyticity. We begin by deriving a simple closed form expression for the pressure in the case of diagonal matrices and, by identifying phase transitions with zeros of Dirichlet polynomials, use this to deduce that the pressure is piecewise real analytic. We then specialise to the iterated function system setting and use a result of Falconer and Miao to extend our results to include the pressure for systems generated by matrices which are simultaneously triangularisable. Our closed form expression for the pressure simplifies a …

$\mathbb{Z}/m \mathbb{Z}$ 上上三角矩阵的随机游走

The random walk on upper triangular matrices over $\mathbb{Z}/m \mathbb{Z}$

本文研究了 $\mathbb{Z}/m \mathbb{Z}$ 上 $n \times n$ 上三角矩阵的随机游走，通过加减均匀随机行生成步骤，并证明其混合时间为 $O(m^2n \log n+ n^2 m^{o(1)})$，回答了 Stong 等人的问题。

有限域上随机三角矩阵的持续图

Persistence diagrams of random triangular matrices over finite fields

研究有限域上随机下三角矩阵行张成子空间的演化，利用持续图刻画其性质，给出分布显式公式、寿命大数定律及持续Betti数波动描述。

Cut norm discontinuity of triangular truncation of graphons

The space of $L^p$ graphons, symmetric measurable functions $w: [0,1]^2 \to \mathbb{R}$ with finite $p$-norm, features heavily in the study of sparse graph limit theory. We show that the triangular cut operator $M_χ$ acting on this space is not continuous with respect to the cut norm. This is achieved by showing that as $n\to \infty$, the norm of the triangular truncation operator $\mathcal{T}_n$ on symmetric matrices equipped with the cut norm grows to infinity as well. Due to the density of symmetric matrices in the space of $L^p$ graphons, the norm growth of $\mathcal{T}_n$ generalizes to t…

关于反三角块矩阵的Drazin指数

On the Drazin Index of an Anti-Triangular Block Matrix

本文研究反三角块矩阵的Drazin指数，通过von Neumann反演的加法分解，建立指数与最小多项式的关系，并推导出指数的上下界及闭式表达式，应用于有向图邻接矩阵。

由置换位移生成的Toeplitz矩阵与三角核

Toeplitz matrices from permutation displacements and the triangular kernel

本文研究了通过缩放核a_k=f(k/n)生成的Toeplitz矩阵，证明其特征值均值收敛于f的加权积分，并引入置换位移计数的组合构造，最终分析了与置换位移相关的极限谱结构。

Superregular matrices over small finite fields

A trivially zero minor of a matrix is a minor having all its terms in the Leibniz formula equal to zero. A matrix is superregular if all of its minors that are not trivially zero are nonzero. In the area of Coding Theory, superregular matrices over finite fields are connected with codes with optimum distance proprieties. When a superregular matrix has all its entries nonzero, it is called full superregular and these matrices are used to construct Maximum Distance Separable block codes. In the context of convolutional codes, lower triangular Toeplitz superregular matrices are employed to build …

Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models III. Triangular-Lattice Chromatic Polynomial

We study the chromatic polynomial P_G(q) for m \times n triangular-lattice strips of widths m <= 12_P, 9_F (with periodic or free transverse boundary conditions, respectively) and arbitrary lengths n (with free longitudinal boundary conditions). The chromatic polynomial gives the zero-temperature limit of the partition function for the q-state Potts antiferromagnet. We compute the transfer matrix for such strips in the Fortuin--Kasteleyn representation and obtain the corresponding accumulation sets of chromatic zeros in the complex q-plane in the limit n\to\infty. We recompute the limiting cur…