Cluster algebras of finite type via Coxeter elements and principal minors

Bounded Ratios of Products of Principal Minors of Positive Definite Matrices

Considered is the multiplicative semigroup of ratios of products of principal minors bounded over all positive definite matrices. A long history of literature identifies various elements of this semigroup, all of which lie in a sub-semigroup generated by Hadamard-Fischer inequalities. Via cone-theoretic techniques and the patterns of nullity among positive semidefinite matrices, a semigroup containing all bounded ratios is given. This allows the complete determination of the semigroup of bounded ratios for 4-by-4 positive definite matrices, whose 46 generators include ratios not implied by Had…

一个被非负秩一矩阵扰动的奇异M矩阵具有正的主 minors；它是否D稳定？

A singular M-matrix perturbed by a nonnegative rank one matrix has positive principal minors; is it D-stable?

研究奇异M矩阵在被非负秩一扰动后的正稳定性与D稳定性，特别情况下可以证明正稳定性或D稳定性，但一般情况无法成立，通过反例说明。然而，此类矩阵被证明为P矩阵。

On generators of bounded ratios of minors for totally positive matrices

We provide a method for factoring all bounded ratios of the form $$\det A(I_1|I_1')\det A(I_2|I_2')/\det A(J_1|J_1')\det A(J_2|J_2')$$ where $A$ is a totally positive matrix, into a product of more elementary ratios each of which is bounded by 1, thus giving a new proof of Skandera's result. The approach we use generalizes the one employed by Fallat et al. in their work on principal minors. We also obtain a new necessary condition for a ratio to be bounded for the case of non-principal minors.

The module structure of the Solomon-Tits algebra of the symmetric group

Let $(W,S)$ be a finite Coxeter system. Tits defined an associative product on the set $Σ$ of simplices of the associated Coxeter complex. The corresponding semigroup algebra is the Solomon-Tits algebra of $W$. It contains the Solomon algebra of $W$ as the algebra of invariants with respect to the natural action of $W$ on $Σ$. For the symmetric group $S_n$, there is a 1-1 correspondence between $Σ$ and the set of all set compositions (or ordered set partitions) of $\{1,...,n\}$. The product on $Σ$ has a simple combinatorial description in terms of set compositions. We study in detail the repre…

Log-minor distributions and an application to estimating mean subsystem entropy

A common task in physics, information theory, and other fields is the analysis of properties of subsystems of a given system. Given the covariance matrix $M$ of a system of $n$ coupled variables, the covariance matrices of the subsystems are principal submatrices of $M$. The rapid growth with $n$ of the set of principal submatrices makes it impractical to exhaustively study each submatrix for even modestly-sized systems. It is therefore of great interest to derive methods for approximating the distributions of important submatrix properties for a given matrix. Motivated by the importance of di…

The center of the reflection equation algebra via quantum minors

We give simple formulas for the elements $c_k$ appearing in a quantum Cayley-Hamilton formula for the reflection equation algebra (REA) associated to the quantum group $U_q(\mathfrak{gl}_N)$, answering a question of Kolb and Stokman. The $c_k$'s are certain canonical generators of the center of the REA, and hence of $U_q(\mathfrak{gl}_N)$ itself; they have been described by Reshetikhin using graphical calculus, by Nazarov-Tarasov using quantum Yangians, and by Gurevich, Pyatov and Saponov using quantum Schur functions; however no explicit formulas for these elements were previously known. As b…

The Kashaev Equation and Related Recurrences

The hexahedron recurrence was introduced by R. Kenyon and R. Pemantle in the study of the double-dimer model in statistical mechanics. It describes a relationship among certain minors of a square matrix. This recurrence is closely related to the Kashaev equation, which has its roots in the Ising model and in the study of relations among principal minors of a symmetric matrix. Certain solutions of the hexahedron recurrence restrict to solutions of the Kashaev equation. We characterize the solutions of the Kashaev equation that can be obtained by such a restriction. This characterization leads t…

有限域上的保正算子 II

Positivity preservers over finite fields II

本文解决了有限域上逐元素保正算子分类中$n=2$、$q\boldsymbol{\text{≡}}1\boldsymbol{\text{(mod}\text{ }4\text{)}}$且$q$非平方数的遗留情形，结合幂等归约与二次特征性质完成了所有有限域及固定维数的完整分类。

Natural families in evolution algebras

In this paper we introduce the notion of evolution rank and give a decomposition of an evolution algebra into its annihilator plus extending evolution subspaces having evolution rank one. This decomposition can be used to prove that in non-degenerate evolution algebras, any family of natural and orthogonal vectors can be extended to a natural basis. Central results are the characterization of those families of orthogonal linearly independent vectors which can be extended to a natural basis. We also consider ideals in perfect evolution algebras and prove that they coincide with the basic ideals…

A nice involution for multivariable polynomial rings

The principal minors of the Toeplitz matrix $\left( x_{i-j+1}\right)_{1\le i,j,\le n}$, where $x_0=1, x_k=0$ if $k\le -1$, directly determine an involution of the polynomial ring $R[x_1, ... ,x_n]$ over any commutative ring $R$.

How to Detect and Construct N-matrices

N-matrices are real $n\times n$ matrices all of whose principal minors are negative. We provide (i) an $O(2^n)$ test to detect whether or not a given matrix is an N-matrix, and (ii) a characterization of N-matrices, leading to the recursive construction of every N-matrix.