Schur–Weyl duality over finite fields

We prove a version of the Schur–Weyl duality over finite fields. We prove that for any field k, if k has at least r + 1 elements, the Schur–Weyl duality holds for the rth tensor power of a finite dimensional vector space V. Moreover, if the dimension of V is at least r + 1, the natural map ${{k\mathfrak{S}_r \to \mathsf{End}_{{\rm GL}(V)}(V^{\otimes r})}}We prove a version of the Schur–Weyl duality over finite fields. We prove that for any field k, if k has at least r + 1 elements, the Schur–Weyl duality holds for the rth tensor power of a finite dimensional vector space V. Moreover, if the dimension of V is at least r + 1, the natural map k\mathfrakS_r ? End_GL(V)(V^?r){{k\mathfrak{S}_r \to \mathsf{End}_{{\rm GL}(V)}(V^{\otimes r})}} is an isomorphism. This isomorphism may fail if dim _k V is not strictly larger than r.     

On Framed Instanton Bundles and Their Deformations

We consider a compact twistor space P and assume that there is a surface S ⊂ P, which has degree one on twistor fibres and contains a twistor fibre F, e.g. P a LeBrun twistor space ([20], [18]). Similar to [6] and [5] we examine the restriction of an instanton bundle V equipped with a fixed trivialization along F to a framed vector bundle over (S, F). First we develope inspired by [13] a suitable deformation theory for vector bundles over an analytic space framed by a vector bundle over a subspace of arbitrary codimension. In the second section we describe the restriction as a smooth natural transformation into a fine moduli space. By considering framed U(r)‐instanton bundles as a real structure on framed instanton bundles over P, we show that the bijection between isomorphism classes of framed U(r)‐instanton bundles and isomorphism classes of framed vector bundles over (S, F) due to [5] is actually an isomorphism of moduli spaces.     

The incidence structure of subspaces with well-scaled frames

In this paper, the incidence structure of classes of subspaces that generalize the regular (unimodular) subspaces of rational coordinate spaces is studied. Let F the a field and S - F β {0}. A subspace, V, of a coordinate space over F is S-regular if every elementary vector of V can be scaled by an element of F β {0} so that all of its non-zero entries are elements of S. A subspace that is {−1, +1 }-regular over the rational field is regular.Associated with a subspace, V, over an arbitrary (respectively, ordered) field is a matroid (oriented matroid) having as circuits (signed circuits) the set of supports (signed supports) of elementary vectors of V. Fundamental representation properties are established for the matroids that arise from certain classes of subspaces. Matroids that are (minor) minimally non-representable by various classes of subspaces are identified. A unique representability results is established for the oriented matroids of subspaces that are dyadic (i.e., {±2⁰, ±2¹, ±2², …}-regular) over the rationals. A self-dual characterization is established for the matroids of S-regular subspaces which generalizes Minty's characterization of regular spaces as digraphoids.     

A recursive construction for universal cycles of 2-subspaces

We prove that universal cycles of 2-dimensional subspaces of vector spaces over any finite field F exist, i.e., if V is a finite-dimensional vector space over F, there is a cycle of vectors v₁,v₂,…,v_n such that each 2-dimensional subspace of V occurs exactly once as the span of consecutive vectors.     

The Erd?s-Ko-Rado properties of set systems defined by double partitions

Let F be a family of subsets of a finite set V. The star ofFatv∈V is the sub-family {A∈F:v∈A}. We denote the sub-family {A∈F:|A|=r} by F^(r).A double partitionP of a finite set V is a partition of V into large sets that are in turn partitioned into small sets. Given such a partition, the family F(P)induced byP is the family of subsets of V whose intersection with each large set is either contained in just one small set or empty.Our main result is that, if one of the large sets is trivially partitioned (that is, into just one small set) and 2r is not greater than the least cardinality of any maximal set of F(P), then no intersecting sub-family of F(P)^(r) is larger than the largest star of F(P)^(r). We also characterise the cases when every extremal intersecting sub-family of F(P)^(r) is a star of F(P)^(r).     

Supersoluble and Related Cofinitary Groups

Let D be a division ring (possibly commutative) and V an infinite-dimensional left vector space over D. We consider irreducible subgroups G of GL(V) containing an element whose fixed-point set in V is non-zero but finite dimensional (over D). We then derive conclusions about cofinitary groups, an element of GL(V) being cofinitary if its fixed-point set is finite dimensional and a subgroup G of GL(V) being cofinitary if all its non-identity elements are confinitary. In particular we show that an irreducible cofinitary subgroup G of GL(V) is usually imprimitive if G is supersoluble and is frequently imprimitive if G is hypercyclic. The latter includes the case where G is hypercentral, which apparently is also new.     

On the chromatic number of the general Kneser-graph

Given integers k, n, 2 < k < n, let us define a graph with vertex set V = {F ?{1, 2, …, n}: ∩F = k}, and (F, F') is an edge if |F ∩ F′| ≤ 1. We show that for n > n₀(k) the chromatic number of this graph is (k - 1)() + rs, where n = (k - 1)s + r, 0 ≤ r < k - 1.     

Modular Lie powers and the Solomon descent algebra

Let V be an r-dimensional vector space over an infinite field F of prime characteristic p, and let L_n(V) denote the nth homogeneous component of the free Lie algebra on V. We study the structure of L_n(V) as a module for the general linear group GL_r(F) when n=pk and k is not divisible by p and where r≥n. Our main result is an explicit 1-1 correspondence, multiplicity-preserving, between the indecomposable direct summands of L_k(V) and the indecomposable direct summands of L_n(V) which are not isomorphic to direct summands of V^⊗n. Our approach uses idempotents of the Solomon descent algebras, and in addition a correspondence theorem for permutation modules of symmetric groups. Second author supported by Deutsche Forschungsgemeinschaft (DFG-Scho 799).     

Unitary units in modular group algebras of 2-groups

We study the unitary subgroupV _?(F₂ G) in the group algebras F₂ Gof 2-groups of maximal class over the field F₂of two elements. We show that there does not exist a normal complement to Gin V _?(F₂ G) if and only if Gis the dihedral group of order 2ⁿ(n≥ 5) or the semidihedral group of order 2ⁿ(n≥5). We also describe V _?(F₂ G) when the subgroup Ghas order 8 and 16 and in this case there exist a normal complement of Gin V _?(F₂ G).     

A Construction of Uniquely C4-free colourable Graphs

Abstract

An F-free colouring of a graph G is a partition {V₁,V₂,…,V_n} of the vertex set V(G) of G such that F is not an induced subgraph of G[V_i] for each i. A graph is uniquely F-free colourable if any two .F-free colourings induce the same partition of V(G). We give a constructive proof that uniquely C₄-free colourable graphs exist.     

Smooth and path connected Banach submanifold Σ r of B(E,F) and a dimension formula in B(ℝ n ,ℝ m )

Given two Banach spaces E,F, let B(E,F) be the set of all bounded linear operators from E into F, Σ _r the set of all operators of finite rank r in B(E,F), and Σ _r ^# the number of path connected components of Σ _r . It is known that Σ _r is a smooth Banach submanifold in B(E,F) with given expression of its tangent space at each A ∈ Σ _r . In this paper,the equality Σ _r ^# = 1 is proved. Consequently, the following theorem is obtained: for any nonnegative integer r, Σ _r is a smooth and path connected Banach submanifold in B(E,F) with the tangent space T _A Σ _r = {B ∈ B(E,F): BN(A) ⊂ R(A)} at each A ∈ Σ _r if dim F = ∞. Note that the routine method can hardly be applied here. So in addition to the nice topological and geometric property of Σ _r the method presented in this paper is also interesting. As an application of this result, it is proved that if E = ℝ ⁿ and F = ℝ ^m , then Σ _r is a smooth and path connected submanifold of B(ℝ ⁿ , ℝ ^m ) and its dimension is dimΣ _r = (m+n)r−r ² for each r, 0 <- r < min {n,m}. Supported by the National Science Foundation of China (Grant No.10671049 and 10771101).     

Maps on spaces of symmetric matrices preserving idempotence

Suppose F is an arbitrary field. Let |F| be the number of the elements of F. Let M_n(F) be the space of all n×n matrices over F, and let S_n(F) be the subset of M_n(F) consisting of all symmetric matrices. Let V{S_n(F),M_n(F)}, a map Φ:V→V is said to preserve idempotence if A-λB is idempotent if and only if Φ(A)-λΦ(B) is idempotent for any A,BV and λF. It is shown that: when the characteristic of F is not 2, |F|>3 and n4, Φ:S_n(F)→S_n(F) is a map preserving idempotence if and only if there exists an invertible matrix PM_n(F) such that Φ(A)=PAP^-1 for every AS_n(F) and P^tP=aI_n for some nonzero scalar a in F.     

A degree condition for the existence of regular factors in K1, n-free graphs

A graph is called K_{1, n}-free if it contains no K_{1, n} as an induced subgraph. Let n(≥3), r be integers (if r is odd, r ≥ n − 1). We prove that every K_{1, n}-free connected graph G with r|V(G)| even has an r-factor if its minimum degree is at least. $ \left(n+{{n-1}\over{r}}\right) \left\lceil {n\over{2(n-1)}}r \right\rceil - {{n-1}\over{r}}\left(\left\lceil {n\over{2(n-1)}}r \right\rceil \right)^2+n-3. $ This degree condition is sharp. © 1996 John Wiley & Sons, Inc.     

Codes for Identification in the King Lattice

Let G=(V, E) be an undirected graph and C a subset of vertices. If the sets B _r (F)C are distinct for all subsets FV with at most k elements, then C is called an (r,k)-identifying code in G. Here B _r (F) denotes the set of all vertices that are within distance r from at least one vertex in F. We consider the two-dimensional square lattice with diagonals (the king lattice). We show that the optimal density of an (r,2)-identifying code is 1/4 for all r3. For r=1,2, we give constructions with densities 3/7 and 3/10, and we prove the lower bounds 9/22 and 31/120, respectively. The research of the authors was supported by the Academy of Finland under grants 44002 and 46186     

The centralizer of a semi-linear transformation

Let V be a finite-dimensional vector space over a division ring D, where D is finite-dimensional over its center F. Suppose T is a semi-linear transformation on V with associated automorphism σ of D. The centralizer of T is the ring C(T) of all linear transformations on V which commute with T. If σ^r is the identity on D for some r ? 1 and no smaller positive power of σ is an inner automorphism, then the center of C(T) is computed to be polynomials in T^r with coefficients from F₀, where F₀ is the subfield of F left elementwise fixed by σ. A matrix version of this theorem is also given.     

Integral formulae on hypersurfaces in Riemannian manifolds

The coefficients of the complete set of n fundamental forms of a hypersuface V_n−1 imbedded in an n-dimensional Riemannian space V_n, as recently introduced[(5)], are used to construct certain tensor fields over V_n−1 which display some remarkable features. In particular, the divergences of these tensor fields can be expressed very simply in terms of polynomials involving the curvature tensor of V_n, the coefficients of the n fundamental forms, and the r^th curvatures of V_n−1. As the result of an application of the generalized divergence theorem of Gauss to these relations a set of integral formulae on V_n−1 is obtained. The integrands of these integral formulae can be expressed very simply in terms of the n fundamental forms of V_n−1. By successive specialization it is indicated how known integral theorems([2], [3], [6], [7], [8]) can be derived as particular cases, which is possible partly as a result of the fact that the polynomial referred to above vanishes identically whenever V_n is a space of constant curvature. This research was supported by the National Research Council of Canada. Entrata in Redazione il 21 agosto 1970.     

Vectors with a prescribed minimal polynomial

Let V be a finite dimensional vector space over the field Fand φ (x)?F[x].Letx V → V be a linear operator. Let S_φ be the set consisting of the vectors whose minimal polynomial φ(x)together with the zero vector We give necessary and sufficieni condition for S _φ to be a subspace.     

Minimal underlying division rings of sets of points of a projective space

Let V be a vector space over a division ring K. Let P be a spanning set of points in Σ:=PG(V). Denote by K(P) the family of sub-division rings F of K having the property that there exists a basis B_F of V such that all points of P are represented as F-linear combinations of B_F. We prove that when K is commutative, then K(P) admits a least element. When K is not commutative, then, in general, K(P) does not admit a minimal element. However we prove that under certain very mild conditions on P, any two minimal elements of K(P) are conjugate in K, and if K is a quaternion division algebra then K(P) admits a minimal element.     

The Vector Space Kinna‐Wagner Principle is Equivalent to the Axiom of Choice

We show that the axiom of choice AC is equivalent to the Vector Space Kinna‐Wagner Principle, i.e., the assertion: “For every family 𝒱= {V_i : i ∈ k} of non trivial vector spaces there is a family ℱ = {F_i : i ∈ k} such that for each i ∈ k, F_i is a non empty independent subset of V_i”. We also show that the statement “every vector space over ℚ has a basis” implies that every infinite well ordered set of pairs has an infinite subset with a choice set, a fact which is known not to be a consequence of the axiom of multiple choice MC.