Roots of the affine Cremona group

Let k ^[n] = k[x ₁,…, x _n ] be the polynomial algebra in n variables and let \mathbbAⁿ = \textSpec  \boldk^{[ n ]} {\mathbb{A}^n} = {\text{Spec}}\;{{\bold{k}}^{\left[ n \right]}} . In this note we show that the root vectors of \textAu\textt^*( \mathbbAⁿ ) {\text{Au}}{{\text{t}}^*}\left( {{\mathbb{A}^n}} \right) , the subgroup of volume preserving automorphisms in the affine Cremona group \textAut( \mathbbAⁿ ) {\text{Aut}}\left( {{\mathbb{A}^n}} \right) , with respect to the diagonal torus are exactly the locally nilpotent derivations x ^α (∂/∂x _i ), where x ^α is any monomial not depending on x _i . This answers a question posed by Popov.     

Isometries between function algebras with finite codimensional range

Let A be a function algebra on a compact space X. A linear isometry T of A into A is said to be codimension n or finite codimensional if the range of T has codimension n in A. In this paper we prove that such isometries can be represented as weighted composition mappings on a cofinite subset, (∂A)₀, of the Shilov boundary for A, ∂A. We focus on those finite codimensional isometries for which (∂A)₀=∂A. All the above results, applied to the particular case of codimension 1 linear isometries on C(X), are used to improve the classification provided by Gutek et al. in J. Funct. Anal. 101, 97–119 (1991). Received: 3 June 1998 / Revised version: 22 March 1999     

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.     

Affine \mathbb{T} -varieties of complexity one and locally nilpotent derivations

Let X = Spec A be a normal affine variety over an algebraically closed field k of characteristic 0 endowed with an effective action of a torus \mathbbT \mathbb{T} of dimension n. Let also ∂ be a homogeneous locally nilpotent derivation on the normal affine \mathbbZⁿ {\mathbb{Z}^n} -graded domain A, so that ∂ generates a k ₊-action on X that is normalized by the \mathbbT \mathbb{T} -action.     

Nonsingularity and group invertibility of linear combinations of two k-potent matrices

An n × n complex matrix A is said to be k-potent if A ^k = A. Let T ₁ and T ₂ be k-potent and c ₁ and c ₂ be two nonzero complex numbers. We study the range space, null space, nonsingularity and group invertibility of linear combinations T = c ₁ T ₁ + c ₂ T ₂ of two k-potent matrices T ₁ and T ₂.     

Some results on the cartan matrix of a frobenius algebra

Let F be a field and A a Frobenius algebra over F. The Jacobson radical of A is denoted by J = J(A) and the kth socle of A by S _k (A). Let [Abar]=A/J ^k or A/S _k (A). This article gives new interesting relations between the Cartan matrix of A and that of [Abar]. From these results we prove that the Cartan matrix of A is diagonal if A/Soc(A) is a symmetric algebra. Let G be a finite group. If A is a block of F|G] with the above condition, then the Cartan matrix of A is (n), where n is the order of the defect group of A and the least integer such that Jⁿ (A)=0.     

On isometries of matrix spaces

Let A_kbe the group of isometries of the space of n-by-n matrices over reals (resp. complexes, quaternions) with respect to the Ky Fan k-norm (see the Introduction for the definitions). Let Γ⁰ be the group of transformations of this space consisting of all products of left and right multiplications by the elements of SO(n)(resp. U(n), Sp(n)). It is shown that, except for three particular casesA_k coincides with the normalizer of Γ in Δ group of isometries of the above matrix space with respect to the standard inner product. We also give an alternative treatment of the case D = Rn = 4k = 2 which was studied in detail by Johnson, Laffey, and Li [4].     

On the normal bundle of curves on nodal quartic surfaces

Let k be an algebraically closed field, char k = 0. Let C be an irreducible nonsingular curve such that 2C = S ? F, where S and F are two surfaces and all the singularities of F are rational double points (if any). We prove that C can never pass through rational singularities of types A _2n n∈N, E₆ and E₈. We give conditions for C to pass through rational singularities of types. A _2k+1 k∈Z⁺ D_n n≥4 and E₇, (0.8).     

Grassmannians of 2-Sided Vector Spaces

Let k ? K be an extension of fields, and let A ? M _n (K) be a k-algebra. We study parameter spaces of m-dimensional subspaces of K ⁿ which are invariant under A. The space  _A (m, n), whose R-rational points are A-invariant, free rank m summands of R ⁿ , is well known. We construct a distinct parameter space,  _A (m, n), which is a fiber product of a Grassmannian and the projectivization of a vector space. We then study the intersection  _A (m, n) ∩  _A (m, n), which we denote by  _A (m, n). Under suitable hypotheses on A, we construct affine open subschemes of  _A (m, n) and  _A (m, n) which cover their K-rational points. We conclude by using  _A (m, n),  _A (m, n), and  _A (m, n) to construct parameter spaces of 2-sided subspaces of 2-sided vector spaces.     

Large spaces of symmetric matrices of bounded rank are decomposable ∗

Let k and n be positive integers such that kn. Let S_n (F) denote the space of all n×n symmetric matrices over the field F with char F≠2. A subspace L of S_n (F) is said to be a k-subspace if rank Ak for every A?L.

Now suppose that k is even, and write k=2r. We say a k∥-subspace of S_n (F) is decomposable if there exists in Fⁿ a subspace W of dimension n?r such that x^tAx=0 for every x?W A?L.

We show here, under some mild assumptions on k n and F, that every k∥-subspace of S_n (F) of sufficiently large dimension must be decomposable. This is an analogue of a result obtained by Atkinson and Lloyd for corresponding subspaces of F^m,n .     

Graph decompositions without isolated vertices III

Let G be a graph of order n, and n = Σ^k_i=1 a_i be a partition of n with a_i ≥ 2. In this article we show that if the minimum degree of G is at least 3k−2, then for any distinct k vertices v₁,…, v_k of G, the vertex set V(G) can be decomposed into k disjoint subsets A₁,…, A_k so that |A_i| = a_i,v_iisA_i is an element of A_i and “the subgraph induced by A_i contains no isolated vertices” for all i, 1 ≥ i ≥ k. Here, the bound on the minimum degree is sharp. © 1997 John Wiley & Sons, Inc.     

Solvable line-transitive automorphism groups of finite linear spaces

Let S be a finite linear space, and letG be a group of automorphisms of S. IfG is soluble and line-transitive, then for a givenk but a finite number of pairs of (S, G),S hasv= p ⁿ points andG ⩽AΓL(1,p ⁿ ).     

Tensor Product Representations of the Lie Superalgebra 𝔭(n) and Their Centralizers

Abstract

We construct an associative algebra A _k and show that there is a representation of A _k on V ^?k , where V is the natural 2n-dimensional representation of the Lie superalgebra 𝔭(n). We prove that A _k is the full centralizer of 𝔭(n) on V ^?k , thereby obtaining a “Schur-Weyl duality” for the Lie superalgebra 𝔭(n). This result is used to understand the representation theory of the Lie superalgebra 𝔭(n). In particular, using A _k we decompose the tensor space V ^?k , for k = 2 or 3, and show that V ^?k is not completely reducible for any k ≥ 2.     

On the divisor function of sets with even partition functions

Summary For P∈ F₂[z] with P(0)=1 and deg(P)≧ 1, let A =A(P) be the unique subset of N (cf. [9]) such that Σ_n≧0 p(A,n)zⁿ ≡ P(z) mod 2, where p(A,n) is the number of partitions of n with parts in A. To determine the elements of the set A, it is important to consider the sequence σ(A,n) = Σ _{d|n, d∈A} d, namely, the periodicity of the sequences (σ(A,2^kn) mod 2^k+1)_n≧1 for all k ≧ 0 which was proved in [3]. In this paper, the values of such sequences will be given in terms of orbits. Moreover, a formula to σ(A,2^kn) mod 2^k+1 will be established, from which it will be shown that the weight σ(A₁,2^kz_i) mod 2^k+1 on the orbit <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>z_i$ is moved on some other orbit z_j when A₁ is replaced by A₂ with A₁= A(P₁) and A₂= A(P₂) P₁ and P₂ being irreducible in F₂[z] of the same odd order.     

ON THE NORMALITY OF MONOMIAL IDEALS OF MIXED PRODUCTS

Let R = K[x, y] be a polynomial ring in two disjoint sets of variables x, y over a field K. We study ideals of mixed products L = I_kJ_r + I_sJ_t such that k + r = s + t, where I_k (resp. J_r ) denotes the ideal of R generated by the square-free monomials of degree k (resp. r) in the x (resp. y ) variables. Our main result is a characterization of when a given ideal L of mixed products is normal.

     

Endpoint estimates for Riesz transforms of magnetic Schrödinger operators

Let $A=-(\nabla-i\vec{a})^2+VLet be a magnetic Schr?dinger operator acting on L ²(R ⁿ ), n≥1, where and 0≤V∈L ¹ _loc. Following [1], we define, by means of the area integral function, a Hardy space H ¹ _A associated with A. We show that Riesz transforms (∂/∂x _k -i a _k )A ^-1/2 associated with A, , are bounded from the Hardy space H ¹ _A into L ¹. By interpolation, the Riesz transforms are bounded on L ^p for all 1<p≤2.     

An Asymptotic Property of Schachermayer's Space under Renorming

Let X be a Banach space with closed unit ball B. Given k , X is said to be k-β, respectively, (k + 1)-nearly uniformly convex ((k + 1)-NUC), if for every ε > 0 there exists δ, 0 < δ < 1, so that for every x B and every ε-separated sequence (x_n) B there are indices (n_i)^k_{i = 1}, respectively, (n_i)^{k + 1}_{i = 1}, such that (1/(k + 1))||x + ∑^k_{i = 1} x_ni|| ≤ 1 − δ, respectively, (1/(k + 1))||∑^{k + 1}_{i = 1} x_ni|| ≤ 1 − δ. It is shown that a Banach space constructed by Schachermayer is 2-β, but is not isomorphic to any 2-NUC Banach space. Modifying this example, we also show that there is a 2-NUC Banach space which cannot be equivalently renormed to be 1-β.     

Characterization of separating couples

Let A, B be two archimedean ℓ-algebras and let U,V be two positive linear maps from A to B. We call that the couple (U,V) is separating with respect to A and B if |a||b| = 0 in A implies |U (a)||V (b)| = 0 in B. In this paper, we prove that if A is an f-algebra with unit elment e, if B is an ℓ-algebra and if (U,V) is a separating couple with respect to A and B then (U ^∼∼,V ^∼∼), where U ^∼∼ (resp V ^∼∼) is the bi-adjoint of U (resp of V), is again a separating couple with respect to the order continuous order biduals (A′)′ _n and (B′)′ _n of A and B respectively furnished with their Arens products respectively. Moreover, in the case where B′ separates the points of B, we give a characterization of any separating couple with respect to A and B.      

Closed Geodesics in Homology Classes for Convex Co-Compact Hyperbolic Manifolds

Let be a convex co-compact, torsion-free, discrete group of isometries of real hyperbolic space H ⁿ⁺¹. We compute the asymptotics of the counting function for closed geodesics in homology classes for the quotient manifold X = \H ⁿ⁺¹, under the assumption that H ¹(X, Z) is infinite. Our results imply asymptotic equipartition of geodesics in distinct homology classes.     

Monotonicity of permanents of direct sums of doubly stochastic matrices

Let Ωn be the set of all n × n doubly stochastic matrices, let J_n be the n × n matrix all of whose entries are 1/n and let σ _k (A) denote the sum of the permanent of all k × k submatrices of A. It has been conjectured that if A ε Ω _n and A ≠ J_J then g_A,k (θ) ? σ _k ((1 θ)J_n 1 θA) is strictly increasing on [0,1] for k = 2,3,…,n. We show that if A = A ₁ ⊕ ⊕A_t (t ≥ 2) is an n × n matrix where A_i for i = 1,2, …,t, and if for each i g_Ai,ki (θ) is non-decreasing on [0.1] for k_t = 2,3,…,n_i , then g_A,k (θ) is strictly increasing on [0,1] for k = 2,3,…,n.