Identities of the Semigroup of Upper Triangular Tropical Matrices

A new family of identities satisfied by the semigroups U _n (𝕋) of n × n upper triangular tropical matrices is constructed and an elementary proof is given.     

Linear spaces and preservers of symmetric matrices of bounded rank-two

Let 𝔽 be a field of characteristic two. Let S _n (𝔽) denote the vector space of all n?×?n symmetric matrices over 𝔽. We characterize i. subspaces of S _n (𝔽) all whose elements have rank at most two where n???3,

ii. linear maps from S _m (𝔽) to S _n (𝔽) that sends matrices of rank at most two into matrices of rank at most two where m, n???3 and |𝔽|?≠?2.

     

Involutions for upper triangular matrix algebras

In this paper we describe completely the involutions of the first kind of the algebra UT_n(F) of n×n upper triangular matrices. Every such involution can be extended uniquely to an involution on the full matrix algebra. We describe the equivalence classes of involutions on the upper triangular matrices. There are two distinct classes for UT_n(F) when n is even and a single class in the odd case.Furthermore we consider the algebra UT₂(F) of the 2×2 upper triangular matrices over an infinite field F of characteristic different from 2. For every involution *, we describe the *-polynomial identities for this algebra. We exhibit bases of the corresponding ideals of identities with involution, and compute the Hilbert (or Poincaré) series and the codimension sequences of the respective relatively free algebras.Then we consider the *-polynomial identities for the algebra UT₃(F) over a field of characteristic zero. We describe a finite generating set of the ideal of *-identities for this algebra. These generators are quite a few, and their degrees are relatively large. It seems to us that the problem of describing the *-identities for the algebra UT_n(F) of the n×n upper triangular matrices may be much more complicated than in the case of ordinary polynomial identities.     

Maps preserving square-zero matrices on the algebra consisting of all upper triangular matrices

Let F be a field, T _n (F) (respectively, N _n (F)) the matrix algebra consisting of all n × n upper triangular matrices (respectively, strictly upper triangular matrices) over F. A ∈ T _n (F) is said to be square zero if A ² = 0. In this article, we firstly characterize non-singular linear maps on N _n (F) preserving square-zero matrices in both directions, then by using it we determine non-singular linear maps on T _n (F) preserving square-zero matrices in both directions.     

Real elements in T n (K)

We study the real elements in triangular matrix groups. We describe some classes of elements that are real in T _n (K) – the groups of upper triangular matrices over a commutative field K. From the obtained results there follow some applications for finding real elements in general linear groups – GL _n (K).     

Involutions in triangular groups

We describe involutions, i.e. elements of order 2, in the groups T _n (K) – of upper triangular matrices of dimension n (n?∈??), and T _∞(K) – of upper triangular infinite matrices, where K is a field of characteristic different from 2. Using the obtained result, we give a formula for the number of all involutions in T _n (K) in the case when K is a finite field.     

Nonlinear Lie derivations on upper triangular matrices

Let M _n (𝔸) and T _n (𝔸) be the algebra of all n?×?n matrices and the algebra of all n?×?n upper triangular matrices over a commutative unital algebra 𝔸, respectively. In this note we prove that every nonlinear Lie derivation from T _n (𝔸) into M _n (𝔸) is of the form A?→?AT???TA?+?A _??+?ξ(A)I _n , where T?∈?M _n (𝔸), ??:?𝔸?→?𝔸 is an additive derivation, ξ?:?T _n (𝔸)?→?𝔸 is a nonlinear map with ξ(AB???BA)?=?0 for all A,?B?∈?T _n (𝔸) and A _? is the image of A under???applied entrywise.     

The Semigroup Generated by the Similarity Class of a Singular Matrix

Let A be a singular matrix of M _n (𝕂), where 𝕂 is an arbitrary field. Using canonical forms, we give a new proof that the sub-semigroup of ( _n (𝕂), ×) generated by the similarity class of A is the set of matrices of M _n (𝕂) with a rank lesser than or equal to that of A.     

Adjacency Preserving Bijection Maps of Hermitian Matrices over any Division Ring with an Involution

Let D be any division ring with an involution,Hn （D） be the space of all n × n hermitian matrices over D. Two hermitian matrices A and B are said to be adjacent if rank（A - B） = 1. It is proved that if φ is a bijective map from Hn（D）（n ≥ 2） to itself such that φ preserves the adjacency, then φ^-1 also preserves the adjacency. Moreover, if Hn（D） ≠J3（F2）, then φ preserves the arithmetic distance. Thus, an open problem posed by Wan Zhe-Xian is answered for geometry of symmetric and hermitian matrices.     

A Note on Bruhat Decomposition of GL(n) over Local Principal Ideal Rings

Let A be a local commutative principal ideal ring. We study the double coset space of GL _n (A) with respect to the subgroup of upper triangular matrices. Geometrically, these cosets describe the relative position of two full flags of free primitive submodules of A ⁿ . We introduce some invariants of the double cosets. If k is the length of the ring, we determine for which of the pairs (n,k) the double coset space depends on the ring in question. For n = 3, we give a complete parametrisation of the double coset space and provide estimates on the rate of growth of the number of double cosets.     

Geometry of rectangular block triangular matrices

Let D be any division ring, and let T（mi,ni,k） be the set of k × k （k ≥ 2） rectangular block triangular matrices over D. For A, B ∈ T（mi,ni,k）, if rank（A - B） = 1, then A and B are said to be adjacent and denoted by A -B. A map T ： T（mi,ni,k） -〉 T（mi,ni,k） is said to be an adjacency preserving map in both directions if A - B if and only if φ（A） φ（B）. Let G be the transformation group of all adjacency preserving bijections in both directions on T（mi,ni,k）. When m1,nk ≥ 2, we characterize the algebraic structure of G, and obtain the fundamental theorem of rectangular block triangular matrices over D.     

The Combinatorics of a Class of Representation Functions

In the study of the irreducible representations of the unitary groupU(n), one encounters a class of polynomials defined onn²indeterminatesz_ij, 1i, jn, which may be arranged into ann×nmatrix arrayZ=(z_ij). These polynomials are indexed by double Gelfand patterns, or equivalently, by pairs of column strict Young tableaux of the same shape. Using the double labeling property, one may define a square matrixD(Z), whose elements are the double-indexed polynomials. These matrices possess the remarkable “group multiplication property”D(XY)=D(X) D(Y) for arbitrary matricesXandY, even though these matrices may be singular. ForZ=UU(n), these matrices give irreducible unitary representations ofU(n). These results are known, but not always fully proved from the extensive physics literature on representation of the unitary groups, where they are often formulated in terms of the boson calculus, and the multiplication property is unrecognized. The generality of the multiplication property is the key to understanding group representation theory from the purview of combinatorics. The combinatorial structure of the general polynomials is expected to be intricate, and in this paper, we take the first step to explore the combinatorial aspects of a special class which can be defined in terms of the set of integral matrices with given row and column sums. These special polynomials are denoted byL_{α, β}(Z), whereαandβare integral vectors representing the row sums and column sums of a class of integral matrices. We present a combinatorial interpretation of the multiplicative properties of these polynomials. We also point out the connections with MacMahon's Master Theorem and Schwinger's inner product formula, which is essentially equivalent to MacMahon's Master Theorem. Finally, we give a formula for the double Pfaffian, which is crucial in the studies of the generating function of the 3n−jcoefficients in angular momentum theory. We also review the background of the general polynomials and give some of their properties.     

On the Representation of Compact Nilpotent Rings by Triangular Matrices

We prove that every compact nilpotent ring R of characteristic p > 0 can be embedded in a ring of upper triangular matrices over a compact commutative ring. Furthermore, we prove that every compact topologically nilpotent ring R of characteristic p > 0, is embedded in a ring of infinite triangular matrices over \mathbbF_p^w(R)\mathbb{F}_{p}^{w(R)}.     

Idempotence preserving maps between matrix spaces

Suppose 𝔽 is an arbitrary field of characteristic not 2 and 𝔽?≠?𝔽₃. Let M _n (𝔽) be the space of all n?×?n full matrices over 𝔽 and P _n (𝔽) the subset of M _n (𝔽) consisting of all n?×?n idempotent matrices and GL _n (𝔽) the subset of M _n (𝔽) consisting of all n?×?n invertible matrices. Let Φ_𝔽(n,?m) denote the set of all maps from M _n (𝔽) to M _m (𝔽) satisfying A???λB?∈?P _n (𝔽)???φ(A)???λφ(B)?∈?P _m (𝔽) for every A,?B?∈?M _n (𝔽) and λ?∈?𝔽, where m and n are integers with 3?≤?n?≤?m. It is shown that if φ?∈?Φ_𝔽(n,?m), then there exists T?∈?GL _m (𝔽) such that φ(A)?=?T?[A???I _p ?⊕?A ^t ???I _q ?⊕?0]T?^?1 for every A?∈?M _n (𝔽), where I ₀?=?0. This improves the results of some related references.     

Sharp estimates for Jacobi matrices and chain sequences, II

Chain sequences are positive sequences {c_n} of the form c_n=g_n(1−g_n−1) for a nonnegative sequence {g_n}. They are very useful in estimating the norms of Jacobi matrices and for localizing the interval of orthogonality for orthogonal polynomials. We give optimal estimates for the chain sequences which are more precise than the ones obtained in the paper (Constructive Approx. 6 (1990) 363) and in our earlier paper (J. Approx. Theory 118 (2002) 94).     

Additivity of Lie multiplicative maps on triangular algebras

Let 𝒜 and ? be unital algebras over a commutative ring ?, and ? be a (𝒜,??)-bimodule, which is faithful as a left 𝒜-module and also as a right ?-module. Let 𝒰?=?Tri(𝒜,??,??) be the triangular algebra and 𝒱 any algebra over ?. Assume that Φ?:?𝒰?→?𝒱 is a Lie multiplicative isomorphism, that is, Φ satisfies Φ(ST???TS)?=?Φ(S)Φ(T)???Φ(T)Φ(S) for all S, T?∈?𝒰. Then Φ(S?+?T)?=?Φ(S)?+?Φ(T)?+?Z _S,T for all S, T?∈?𝒰, where Z _S,T is an element in the centre 𝒵(𝒱) of 𝒱 depending on S and T.