Weak identities in the algebra of symmetric matrices of order two

Some Identities involving Two Sets of Basis Vectors and the Metric and Mixed Matrices

Given two sets of basis vectors in n-dimensional space, there exists a relation between their lengths and mutual angles, expressed as relations between the two metric matrices and the mixed matrix. In this paper these relations are given, and their consequences for 2-dimensional and 3-dimensional space are investigated, using a computer algebra program for simplifying expressions.

关于任意域上上三角矩阵的星齐次分次多项式恒等式

On star-homogeneous-graded polynomial identities of upper triangular matrices over an arbitrary field

本文研究任意域上带精细群分次和齐次对合的上三角矩阵代数的分次多项式恒等式，给出基与渐近行为，并计算任意分次与对合下的指数和第二指数。

Polynomial identities with involution for the algebra of 3 $\times$ 3 upper triangular matrices

Let $\mathbb{F}$ be a field of characteristic $p$, and let $UT_n(\mathbb{F})$ be the algebra of $n \times n$ upper triangular matrices over $\mathbb{F}$ with an involution of the first kind. In this paper we describe: the set of all $*$-central polynomials for $UT_n(\mathbb{F})$ when $n\geq 3$ and $p\neq 2$ ; the set of all $*$-polynomial identities for $UT_3(\mathbb{F})$ when $\mathbb{F}$ is infinite and $p>2$.

Graded polynomial identities and central polynomials of matrices over an infinite integral domain

Let $K$ be an infinite integral domain and $M_{n}(K)$ be the algebra of all $n\times n$ matrices over $K$. This paper aims for the following goals: Find a basis for the graded identities for elementary grading in $M_{n}(K)$ when the neutral component and diagonal coincide; Describe the $\mathbb{Z}_{p}$-graded central polynomials of $M_{p}(K)$ when $p$ is a prime number; Describe the $\mathbb{Z}$-graded central polynomials of $M_{n}(K)$.

*-polynomial identities of 4x4 upper triangular matrices with the reflection involution

Let $UT_4(F)$ be $4\times 4$ upper triangular matrix algebra over a field $F$ of characteristic zero and let $\mathcal{A}$ be the subalgebra of $UT_4(F)$ linearly generated by $\{\mathbf{e}_{ij}:1 \leq i\leq j \leq 4 \} \setminus \mathbf{e}_{23}$ where $\{\mathbf{e}_{ij} : 1 \leq i\leq j \leq 4\}$ is the standard basis of $UT_4(F)$. We describe the set of all $*$-polynomial identities for $\mathcal{A}$ with the involution defined by the reflection of second diagonal.

Semigroup identities of supertropical matrices

We prove that, for any $n$, the monoid of all $n \times n$ supertropical matrices extending tropical matrices satisfies nontrivial semigroup identities. These identities are carried over to walks on labeled-weighted digraphs with double arcs.

Graded Polynomial Identities for Matrices with the Transpose Involution over an Infinite Field

Let $F$ be an infinite field, and let $M_{n}(F)$ be the algebra of $n\times n$ matrices over $F$. Suppose that this algebra is equipped with an elementary grading whose neutral component coincides with the main diagonal. In this paper, we find a basis for the graded polynomial identities of $M_{n}(F)$ with the transpose involution. Our results generalize for infinite fields of arbitrary characteristic previous results in the literature which were obtained for the field of complex numbers and for a particular class of elementary G-gradings.

Graded polynomial identities for matrices with the transpose involution

Let $G$ be a group of order $k$. We consider the algebra $M_k(\mathbb{C})$ of $k$ by $k$ matrices over the complex numbers and view it as a crossed product with respect to $G$ by embedding $G$ in the symmetric group $S_k$ via the regular representation and embedding $S_k$ in $M_k(\mathbb{C})$ in the usual way. This induces a natural $G$-grading on $M_k(\mathbb{C})$ which we call a crossed-product grading. We study the graded $*$-identities for $M_k(\mathbb{C})$ equipped with such a crossed-product grading and the transpose involution. To each multilinear monomial in the free graded algebra wit…

不满足$2n-2$次恒等式的$n\times n$矩阵的子代数

On subalgebras of $n\times n$ matrices not satisfying identities of degree $2n-2$

Amitsur-Levitski定理断言$M_n(F)$满足次数为$2n$的多项式恒等式。(Here，$F$是域，$M_n(F)$是$F$上的$n \times n$矩阵的代数）。很容易给出满足低次恒等式的$M_n(F)$的子代数和不满足$\le 2n-2$次多项式恒等式的$M_n(F)$的子代数的例子。本文的目的是给出不满足次数小于$2n$的非零多项式的$n \times n$矩阵的子代数的完全分类。

特征2下$2\times2$矩阵的多线性多项式恒等式

On multilinear polynomial identities for $2\times2$ matrices in characteristic $2$

本文在特征2的无限域上，为$2\times2$矩阵代数$M_2(K)$找到了两个5次多线性恒等式，并与4次标准多项式共同生成所有次数不超过7的多线性恒等式，且证明三个生成元缺一不可。

2-graded polynomial identities for the Jordan algebra of the symmetric matrices of order two

The Jordan algebra of the symmetric matrices of order two over a field $K$ has two natural gradings by $\mathbb{Z}_2$, the cyclic group of order 2. We describe the graded polynomial identities for these two gradings when the base field is infinite and of characteristic different from 2. We exhibit bases for these identities in each of the two cases. In one of the cases we perform a series of computations in order to reduce the problem to dealing with associators while in the other case one employs methods and results from Invariant theory. Moreover we extend the latter grading to a $\mathbb{Z}…

Identities of tropical matrices and plactic monoids

We study semigroup varieties generated by full and upper triangular tropical matrix semigroups and the plactic monoid of rank 4. We prove that the upper triangular tropical matrix semigroup $UT_n(\mathbb{T})$ generates a different semigroup variety for each dimension $n$. We show a weaker version of this fact for the full matrix semigroup: full tropical matrix semigroups of different prime dimensions generate different semigroup varieties. For the plactic monoid of rank 4, $\mathbb{P}_4$, we find a new set of identities satisfied by $\mathbb{P}_4$ shorter than those previously known, and show …

关于带正交辛超对合的$3 \times 3$矩阵代数的恒等式与共特征标

On the identities and cocharacters of the algebra of $3 \times 3$ matrices with orthosymplectic superinvolution

该研究利用群表示论，研究特征零域上带正交辛超对合的3×3矩阵代数$M_{1,2}(F)$的$*$-恒等式，分解其多重线性$*$-恒等式空间并确定了次数不超过3的所有$*$-多项式恒等式。

$\mathbb{Z}_2$-graded polynomial identities for the Jordan algebra of $2\times 2$ upper triangular matrices

Let $K$ be a field (finite or infinite) of char$(K)\neq 2$ and let $UT_n=UT_n(K)$ be the $n\times n$ upper triangular matrix algebra over $K$. If $\cdot $ is the usual product on $UT_n$ then with the new product $a\circ b=(1/2)(a\cdot b +b\cdot a)$ we have that $UT_n$ is a Jordan algebra, denoted by $UJ_n=UJ_n(K)$. In this paper, we describe the set of all $\mathbb{Z}_2$-graded polynomial identities of $UJ_2$ with any nontrivial $\mathbb{Z}_2$-grading. Moreover, we describe a linear basis for the corresponding relatively free $\mathbb{Z}_2$-graded algebra.

Quasi-identities on matrices and the Cayley-Hamilton polynomial

We consider certain functional identities on the matrix algebra $M_n$ that are defined similarly as the trace identities, except that the "coefficients" are arbitrary polynomials, not necessarily those expressible by the traces. The main issue is the question of whether such an identity is a consequence of the Cayley-Hamilton identity. We show that the answer is affirmative in several special cases, and, moreover, for every such an identity $P$ and every central polynomial $c$ with zero constant term there exists $m\in\mathbb{N}$ such that the affirmative answer holds for $c^mP$. In general, h…

Semigroup identities of tropical matrices through matrix ranks

We prove the conjecture that, for any $n$, the monoid of all $n \times n$ tropical matrices satisfies nontrivial semigroup identities. To this end, we prove that the factor rank of a large enough power of a tropical matrix does not exceed the tropical rank of the original matrix.

Identities of the left-symmetric Witt algebras

Let $P_n=k[x_1,x_2,\ldots,x_n]$ be the polynomial algebra over a field $k$ of characteristic zero in the variables $x_1,x_2,\ldots,x_n$ and $\mathscr{L}_n$ be the left-symmetric Witt algebra of all derivations of $P_n$. We describe all right operator identities of $\mathscr{L}_n$ and prove that the set of all algebras $\mathscr{L}_n$, where $n\geq 1$, generates the variety of all left-symmetric algebras. We also describe a class of general (not only right operator) identities for $\mathscr{L}_n$.

Graded monomial identities and almost non-degenerate gradings on matrices

Let $F$ be a field of characteristic zero, $G$ be a group and $R$ be the algebra $M_n(F)$ with a $G$-grading. Bahturin and Drensky proved that if $R$ is an elementary and the neutral component is commutative then the graded identities of $R$ follow from three basic types of identities and monomial identities of length $\geq 2$ bounded by a function $f(n)$ of $n$. In this paper we prove the best upper bound is $f(n)=n$, more generally we prove that all the graded monomial identities of an elementary $G$-grading on $M_n(F)$ follow from those of degree at most $n$. We also study gradings which sa…

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.

证明对称合成中恒等式的计算技术

Computational techniques for proving identities in symmetric compositions

我们在这项工作中提出了一个完整的会议在Mathematica笔记本电脑。这本笔记本的目的是检验对称合成中的恒等式。这个笔记本是我们工作的补充[1]，它有所有的显式计算。我们建议读者参阅该论文，该论文可在http://www.uibk.ac.at/mathematik/loos/jordan/index.html上查阅。首先，我们将提出一些命令，以便通过提取标量SOut来简化恒等式。策略的其余部分依赖于使用模式和规则的力量。

Lie identities on symmetric elements of restricted enveloping algebras

Let L be a restricted Lie algebra over a field of characteristic p > 2 and denote by u(L) its restricted enveloping algebra. We determine the conditions under which the set of symmetric elements of u(L) with respect to the principal involution is Lie solvable, Lie nilpotent, or bounded Lie Engel.

A graph theoretic approach to graded identities for matrices

We consider the algebra M_k(C) of k-by-k matrices over the complex numbers and view it as a crossed product with a group G of order k by embedding G in the symmetric group S_k via the regular representation and embedding S_k in M_k(C) in the usual way. This induces a natural G-grading on M_k(C) which we call a crossed product grading. This grading is the so called elementary grading defined by any k-tuple (g_1,g_2,..., g_k) of distinct elements g_i in G. We study the graded polynomial identities for M_k(C) equipped with a crossed product grading. To each multilinear monomial in the free graded…

Skew-symmetric identities of finitely generated alternative algebras

We prove that for every natural number n there exists a natural number N(n) such that every multilinear skew-symmetric polynomial on N(n) or more variables which vanishes in the free associative algebra vanishes as well in any n-generated alternative algebra over a field of characteristic 0. Before this was proved only for a series of such polynomials constructed by the author in [7].

Graded identities of block-triangular matrices

Let $F$ be an infinite field and $UT(d_1,\dots, d_n)$ be the algebra of upper block-triangular matrices over $F$. In this paper we describe a basis for the $G$-graded polynomial identities of $UT(d_1,\dots, d_n)$, with an elementary grading induced by an $n$-tuple of elements of a group $G$ such that the neutral component corresponds to the diagonal of $UT(d_1,\dots,d_n)$. In particular, we prove that the monomial identities of such algebra follow from the ones of degree up to $2n-1$. Our results generalize for infinite fields of arbitrary characteristic, previous results in the literature whi…

Graded Identities and Isomorphisms on Algebras of Upper Block-Triangular Matrices

Let $G$ be an abelian group and $\mathbb{K}$ an algebraically closed field of characteristic zero. A. Valenti and M. Zaicev described the $G$-gradings on upper block-triangular matrix algebras provided that $G$ is finite. We prove that their result holds for any abelian group $G$: any grading is isomorphic to the tensor product $A\otimes B$ of an elementary grading $A$ on an upper block-triangular matrix algebra and a division grading $B$ on a matrix algebra. We then consider the question of whether graded identities $A\otimes B$, where $B$ is an algebra with a division grading, determine $A\o…

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.

2 × 2热带矩阵么半群的半群恒等式

Semigroup identities in the monoid of two-by-two tropical matrices

本文的主要结果证明了所有2 × 2热带矩阵的幺半群满足一个非平凡半群恒等式，这与无限域上矩阵的情形不同。我们利用所有2 × 2热带矩阵的幺半群和双循环幺半群之间的强联系，证明了Adian恒等式的一个变体在我们的例子中成立，我们用热带多项式的一般矩阵证明了我们的结果.

上三角矩阵代数的分次、分次恒等式、$*$-恒等式与分次$*$-恒等式

Gradings, graded identities, $*$-identities and graded $*$-identities of an algebra of upper triangular matrices

本文研究上三角矩阵代数子代数 $\mathcal{A}$ 的阿贝尔群分次，证明其为初等分次，并计算其 $\mathbb{Z}_2$-分次恒等式及带分次对合的分次恒等式的基，同时描述其余特征标。

Tensor polynomial identities

Tensor polynomial identities generalize the concept of polynomial identities on $d \times d$ matrices to identities on tensor product spaces. Here we completely characterize a certain class of tensor polynomial identities in terms of their associated Young tableaux. Furthermore, we provide a method to evaluate arbitrary alternating tensor polynomials in $d^2$ variables.

Trace identities and almost polynomial growth

In this paper we study algebras with trace and their trace polynomial identities over a field of characteristic 0. We consider two commutative matrix algebras: $D_2$, the algebra of $2\times 2$ diagonal matrices and $C_2$, the algebra of $2 \times 2$ matrices generated by $e_{11}+e_{22}$ and $e_{12}$. We describe all possible traces on these algebras and we study the corresponding trace codimensions. Moreover we characterize the varieties with trace of polynomial growth generated by a finite dimensional algebra. As a consequence, we see that the growth of a variety with trace is either polynom…

上三角矩阵的超级对合：恒等式与多重线性多项式的像

Upper triangular matrices with superinvolution: identities and images of multilinear polynomials

本文完整刻画了UT_4(F)上由序列(0,1,0,1)诱导分次及超辛超级对合的恒等式，并证明UT_n(F)上多重线性多项式像为向量空间当且仅当n≤3，推进了L'vov-Kaplansky猜想的类比研究。

四元数矩阵：酉相似，同时三角化和一些迹恒等式

Quaternionic matrices: Unitary similarity, simultaneous triangularization and some trace identities

我们构造了6个酉迹不变量的2 × 2四元数矩阵分离酉相似类，这样的矩阵，并证明了这一套是最小的。我们证明了代数闭域上矩阵代数的可三角化子代数的一个著名特征的两个四元数形式。最后考虑了可同时三角化的四元数n × n矩阵对（X，Y）的半代数集的描述问题。即使是我们更详细分析的n=2的情况，仍然没有解决。

群代数中的对称单位与群恒等式。I

Symmetric Units and Group Identities in Group Algebras. I

本文描述了特征不等于2的域上的群代数，其单位关于经典对合对称，满足某种群恒等式。

Totally positive matrices and dilogarithm identities

We show that two involutions on the variety $N_n^+$ of upper triangular totally positive matrices are related, on the one hand, to the tetrahedron equation and, on the other hand, to the action of the symmetric group $S_3$ on some subvariety of $N_n^+$ and on the set of certain functions on $N_n^+$. Using these involutions, we obtain a family of dilogarithm identities involving minors of totally positive matrices. These identities admit a form manifestly invariant under the action of the symmetric group $S_3$.

Weak polynomial identities for a vector space with a symmetric bilinear form

Let $V_k$ be a $k$-dimensional vector space with a non-degenerate symmetric bilinear form over a field $K$ of characteristic 0 and let $C_k$ be the Clifford algebra on $V_k$. We study the weak polynomial identities of the pair $(C_k,V_k)$. We establish that all they follow from $[x_1^2,x_2]=0$ when $k=\infty$ and from $[x_1^2,x_2]=0$ and $S_{k+1}(x_1,\ldots,x_{k+1})=0$ when $k<\infty$. We also prove that the weak identity $[x_1^2,x_2]=0$ satisfies the Specht property. As a consequence we obtain a new proof of the theorem of Razmyslov that the weak Lie polynomial identities of the pair $(M_2(K)…

由行列式的恒等式导出的迹恒等式

Trace identities from identities for determinants

给出了元$A_{i,j}$等于$a_{i,j}$或$a_{i,0}a_{0,j}-a_{i,j}$的矩阵$(A_{i,j})$的行列式的新恒等式，其中$a_{i,j}$是不定式.我们发现，这些身份背后的跟踪身份$SL(2,\Bbb C)$矩阵发现马格努斯在他的研究轨迹代数。

On $\mathbb{Z}_{2}$-graded polynomial identities of $sl_{2}(F)$ over a finite field

Let $F$ be a finite field of $char F > 3$ and $sl_{2}(F)$ be the Lie algebra of traceless $2\times 2$ matrices over $F$. In this paper, we find a basis for the $\mathbb{Z}_{2}$-graded identities of $sl_{2}(F)$.

上三角矩阵的微分簇

Differential varieties of upper triangular matrices

本文研究上三角矩阵代数的微分恒等式，为微分指数为3的极小$L$-簇分类奠基，证明相关作用可约化，确定$UT_3$的微分恒等式与余维序列，且$k \geq 3$时的$UT_k$生成的$L$-簇必包含带对应作用的$UT_3$。

On the growth of graded polynomial identities of sl_n

Let K be a field of characteristic 0 and L be a G-graded Lie PI-algebra, where G is a finite group. We define the graded Gelfand-Kirillov dimension of L. Then we measure the growth of the Z_n-graded polynomial identities of the Lie algebra of n x n traceless matrices sl_n(K) giving an exact value of its Z_n-graded Gelfand-Kirillov dimension.

On graded polynomial identities of $sl_{2}(F)$ over a finite field

Let $F$ be a finite field of $char F > 3$ and $sl_{2}(F)$ be the Lie algebra of traceless $2\times 2$ matrices over $F$. This paper aims for the following goals: Find a basis for the $\mathbb{Z}_{2}$-graded identities of $sl_{2}(F)$; Find a basis for the $\mathbb{Z}_{3}$-graded identities of $sl_{2}(F)$ when $F$ contains a primitive 3rd root of one; Find a basis for the $\mathbb{Z}_{2}\times \mathbb{Z}_{2}$-graded identities of $sl_{2}(F)$.

有向图的路径代数的多项式恒等式

Polynomial identities for quivers via incidence algebras

研究有向图的路径代数与矩阵代数的多项式恒等式，证明其PI等价性。

Cocharacters for the weak polynomial identities of the Lie algebra of $3\times 3$ skew-symmetric matrices

Let $so_3(K)$ be the Lie algebra of $3\times 3$ skew-symmetric matrices over a field $K$ of characteristic 0. The ideal $I(M_3(K),so_3(K))$ of the weak polynomial identities of the pair $(M_3(K),so_3(K))$ consists of the elements $f(x_1,\ldots,x_n)$ of the free associative algebra $K\langle X\rangle$ with the property that $f(a_1,\ldots,a_n)=0$ in the algebra $M_3(K)$ of all $3\times 3$ matrices for all $a_1,\ldots,a_n\in so_3(K)$. The generators of $I(M_3(K),so_3(K))$ were found by Razmyslov in the 1980's. In this paper the cocharacter sequence of $I(M_3(K),so_3(K))$ is computed. In other wor…

矩阵与算子恒等式的代数证明方法：Hartwig三重反序律的改进

Algebraic proof methods for identities of matrices and operators: improvements of Hartwig's triple reverse order law

本文以Hartwig三重反序律为例，利用代数证明框架和软件包OperatorGB，在带对合的环中改进了广义逆的结果，并讨论了基于非交换多项式计算的计算机辅助证明。

A few remarks on Euler and Bernoulli polynomials and their connections with binomial coefficients and modified Pascal matrices

We prove certain identities involving Euler and Bernoulli polynomials that can be treated as recurrences. We use these and also other known identities to indicate connection of Euler and Bernoulli numbers with entries of inverses of certain lower triangular built of binomial coefficients. Another words we interpret Euler and Bernoulli numbers in terms of modified Pascal matrices.

多参数 Manin 矩阵的量子代数

Quantum algebra of multiparameter Manin matrices

本文利用 R-matrix 方法系统研究多参数 Manin 矩阵的量子代数，导出并证明行列式、积和式的多种经典恒等式及 MacMahon 主方程，并将部分结果推广至多参数 $q$-Yangians。

右对称代数的极小多项式恒等式

Minimal polynomial identities for right-symmetric algebras

一个代数$A$与乘法$A\times A \to A, (a,b)\mapsto a\circ b$，称为右对称的，如果$a\circ(b\circ c)-(a\circ b)\circ a\circ (c\circ b)-(a\circ c)\circ b,$对任何$a,b,c\in A$。由$u\der_i\circ v\der_j=v\der_j(u)\der_i.$给出右对称Witt代数的乘法$W_n=\{u\der_i: u\in U, U={\cal K}[x_1^{\pm 1},...,x_n^{\pm}$或$={\cal K}[x_1,...,x_n], i=1,...,n\}, p=0,$或$W_n({\bf m)}=\{u\der_i: u\in U, U=O_n({\bf m})\}$，建立了右对称Witt代数的Amitsur-Levitzki定理的类似物。$ satisfy the standard right-symmetric identity of degree $ 2n +1的右对称Witt代数：$ $ \sum_{σ\in Sym_{2n}} sign（σ）a_{σ(1)} \circ（a_{σ(2)} \circ>.（a_{σ(2n)} \circ a_{2n+1}）.）= 0. $ The minimal deg$的左多项式恒等式$W_n^{r…

三角矩阵的定律

Laws for triangular matrices

该研究探讨域上及含0和1的加法幂等半环上三角矩阵幺半群的恒等式，发现部分此类恒等式无有限公理化的新情况。

The polynomial identities for matrix functions

This is a short review of some recent results obtained by the author. These results are related the problem of obtaining polynomial identities (computational formulas) for some matrix functions by means of the known polarization theorem, including the case of noncommutative variables and of determinant of space matrices.

Bound for the cocharacters of the identities of irreducible representations of $\mathfrak{sl}_2(\mathbb{C})$

For each irreducible finite dimensional representation of the Lie algebra $\mathfrak{sl}_2(\mathbb{C})$ of $2\times 2$ traceless matrices, an explicit uniform upper bound is given for the multiplicities in the cocharacter sequence of the polynomial identities satisfied by the given representation.

Matrix identities with forms

Consider the algebra M(n,F) of n x n matrices over an infinite field F of arbitrary characteristic. An identity for M(n,F) with forms is such a polynomial in n x n generic matrices and in σ_k(x), 0<k\leq n, coefficients in the characteristic polynomial of monomials in generic matrices, that is equal to zero matrix. This notion is a characteristic free analogue of identities for M(n,F) with trace. In 1996 Zubkov established an infinite generating set for the T-ideal T(n) of identities for M(n,F) with forms. Namely, for t>n he introduced partial linearizations of σ_t and proved that they togethe…

Generalized Jacobi identities and Jacobi elements of the group ring of the symmetric group

By definition the identities $[x_1, x_2] + [x_2, x_1] = 0$ and $[x_1, x_2, x_3] + [x_2, x_3, x_1] + [x_3, x_1, x_2] = 0$ hold in any Lie algebra. It is easy to check that the identity $[x_1, x_2, x_3, x_4] + [x_2, x_1, x_4, x_3] + [x_3, x_4, x_1, x_2] + [x_4, x_3, x_2, x_1] = 0$ holds in any Lie algebra as well. I. Alekseev in his recent work introduced the notion of Jacobi subset of the symmetric group $S_n$. It is a subset of $S_n$ that gives an identity of this kind. We introduce a notion of Jacobi element of the group ring $\mathbb{Z}[S_n]$ and describe them on the language of equations on…

Tropical representations and identities of plactic monoids

We exhibit a faithful representation of the plactic monoid of every finite rank as a monoid of upper triangular matrices over the tropical semiring. This answers a question first posed by Izhakian and subsequently studied by several authors. A consequence is a proof of a conjecture of Kubat and Okniński that every plactic monoid of finite rank satisfies a non-trivial semigroup identity. In the converse direction, we show that every identity satisfied by the plactic monoid of rank $n$ is satisfied by the monoid of $n \times n$ upper triangular tropical matrices. In particular this implies that …

Unitary and orthogonal equivalence of sets of matrices

Two matrices $A$ and $B$ are called unitary (resp. orthogonal) equivalent if $AU=VB$ for two unitary (resp. orthogonal) matrices $U$ and $V$. Using trace identities, criteria are given for simultaneous unitary, orthogonal or complex orthogonal equivalence between two sets of matrices.

Noncommutative determinants, Cauchy-Binet formulae, and Capelli-type identities. I. Generalizations of the Capelli and Turnbull identities

We prove, by simple manipulation of commutators, two noncommutative generalizations of the Cauchy-Binet formula for the determinant of a product. As special cases we obtain elementary proofs of the Capelli identity from classical invariant theory and of Turnbull's Capelli-type identities for symmetric and antisymmetric matrices.