Mod 2 indecomposable orthogonal invariants

Over an algebraically closed base field k of characteristic 2, the ring R^G of invariants is studied, G being the orthogonal group O(n) or the special orthogonal group SO(n) and acting naturally on the coordinate ring R of the m-fold direct sum kⁿ⊕?⊕kⁿ of the standard vector representation. It is proved for O(n) (n?2) and for SO(n) (n?3) that there exist m-linear invariants with m arbitrarily large that are indecomposable (i.e., not expressible as polynomials in invariants of lower degree). In fact, they are explicitly constructed for all possible values of m. Indecomposability of corresponding invariants over immediately follows. The constructions rely on analysing the Pfaffian of the skew-symmetric matrix whose entries above the diagonal are the scalar products of the vector variables.     

Prefilter spaces and a precompletion of preuniform convergence spaces related to some well-known completions

In non-symmetric Convenient Topology the notion of pre-Cauchy filter is introduced and the construction of a precompletion of a preuniform convergence space is given from which Wyler's completion of a separated uniform limit space [O. Wyler, Ein Komplettierungsfunktor für uniforme Limesräume, Math. Nachr. 46 (1970) 1-12] as well as Weil's Hausdorff completion of a separated uniform space [A. Weil, Sur les Espaces à Structures Uniformes et sur la Topologie Générale, Hermann, Paris, 1937] can be derived (up to isomorphism). By the way, the construct PFil of prefilter spaces, i.e. of those preuniform convergence space which are ‘generated’ by their pre-Cauchy filters, is a strong topological universe filling in a gap in the theory of preuniform convergence spaces.     

Linear spaces with many projective planes

We assume that in a linear space there is a non-empty set M of points with the property that every plane containing a point of M is a projective plane. In section 3 an example is given that in general is not a projective space. But if M can be completed by two points to a generating set of P, then is a projective space.     

Homogeneous multiplicative polynomial laws are determinants

Let k be a commutative ring. Let R,B be k-algebras with B commutative. Let p:R→B be a homogeneous multiplicative polynomial law of degree n. We show that p is obtained by left and right composing a determinant with some homomorphisms of k-algebras.     

Hermitian matrices with a bounded number of eigenvalues

Conjugation covariants of matrices are applied to study the real algebraic variety consisting of complex Hermitian matrices with a bounded number of distinct eigenvalues. A minimal generating system of the vanishing ideal of degenerate three by three Hermitian matrices is given, and the structure of the corresponding coordinate ring as a module over the special unitary group is determined. The method applies also for degenerate real symmetric three by three matrices. For arbitrary n   partial information on the minimal degree component of the vanishing ideal of the variety of n×n$n\times n$ Hermitian matrices with a bounded number of eigenvalues is obtained, and some known results on sum of squares presentations of subdiscriminants of real symmetric matrices are extended to the case of complex Hermitian matrices.     

Interlacing eigenvalues on some operations of graphs

In this paper, we focus on some operations of graphs and give a kind of eigenvalue interlacing in terms of the adjacency matrix, standard Laplacian, and normalized Laplacian. Also, we explore some applications of this interlacing.     

Exceptional collections of line bundles on projective homogeneous varieties

We construct new examples of exceptional collections of line bundles on the variety of Borel subgroups of a split semisimple linear algebraic group G$G$ of rank 2 over a field. We exhibit exceptional collections of the expected length for types _A2$A_2$ and _B2=_C2$B_2=C_2$ and prove that no such collection exists for type _G2$G_2$. This settles the question of the existence of full exceptional collections of line bundles on projective homogeneous G$G$-varieties for split linear algebraic groups G$G$ of rank at most 2.     

The ring of upper triangular invariants as a module over the Dickson invariants

This research is supported in part by the Natural Sciences and Engineering Research Council of Canada     

The work of Gian-Carlo rota on invariant theory

The purpose of this paper is twofold: first, to explain Gian-Carlo Rotas work on invariant theory; second, to place this work in a broad historical and mathematical context. Rotas work falls under three specific cases: vector invariants, the invariants of binary forms, and the invariants of skew-symmetric tensors. We discuss each of these cases and show how determinants and straightening play central roles. In fact, determinants constitute all invariants in the vector case; for binary forms and skew-symmetric tensors, they constitute all invariants when invariants are represented symbolically. Consequently, we explain the symbolic method both for binary forms and for skew-symmetric tensors, where Rota developed generalizations of the usual notion of a determinant. We also discuss the Grassmann algebra, with its two operations of meet and join, which was a theme which ran through Rotas work on invariant theory almost from the very beginning.To the memory of Gian-Carlo Rota     

Set-theoretic defining equations of the tangential variety of the Segre variety

We prove a set-theoretic version of the Landsberg-Weyman Conjecture on the defining equations of the tangential variety of a Segre product of projective spaces. We introduce and study the concept of exclusive rank. For the proof of this conjecture, we use a connection to the author’s previous work and re-express the tangential variety as the variety of principal minors of symmetric matrices that have exclusive rank no more than 1. We discuss applications to semiseparable matrices, tensor rank versus border rank, context-specific independence models and factor analysis models.     

Asymptotic resurgences for ideals of positive dimensional subschemes of projective space

Recent work of Ein–Lazarsfeld–Smith and Hochster–Huneke raised the containment problem of determining which symbolic powers of an ideal are contained in a given ordinary power of the ideal. Bocci–Harbourne defined a quantity called the resurgence to address this problem for homogeneous ideals in polynomial rings, with a focus on zero-dimensional subschemes of projective space. Here we take the first steps toward extending this work to higher dimensional subschemes. We introduce new asymptotic versions of the resurgence and obtain upper and lower bounds on them for ideals I$I$ of smooth subschemes, generalizing what is done in Bocci and Harbourne (2010)  [5]. We apply these bounds to ideals of unions of general lines in P^N${\mathbb{P}}^N$. We also pose a Nagata type conjecture for symbolic powers of ideals of lines in P³${\mathbb{P}}^3$.     

Almost equal group multiplications

In a recent paper entitled “A commutative analogue of the group ring” we introduced, for each finite group (G,⋅), a commutative graded Z-algebra R_(G,⋅) which has a close connection with the cohomology of (G,⋅). The algebra R_(G,⋅) is the quotient of a polynomial algebra by a certain ideal I_(G,⋅) and it remains a fundamental open problem whether or not the group multiplication ⋅ on G can always be recovered uniquely from the ideal I_(G,⋅).Suppose now that (G,×) is another group with the same underlying set G and identity element e∈G such that I_(G,⋅)=I_(G,×). Then we show here that the multiplications ⋅ and × are at least “almost equal” in a precise sense which renders them indistinguishable in terms of most of the standard group theory constructions. In particular in many cases (for example if (G,⋅) is Abelian or simple) this implies that the two multiplications are actually equal as was claimed in the previously cited paper.     

A commutative analogue of the group ring

Throughout, all rings R will be commutative with identity element. In this paper we introduce, for each finite group G, a commutative graded Z-algebra R_G. This classifies the G-invariant commutative R-algebra multiplications on the group algebra R[G] which are cocycles (in fact coboundaries) with respect to the standard “direct sum” multiplication and have the same identity element.In the case when G is an elementary Abelian p-group it turns out that R_G is closely related to the symmetric algebra over F_p of the dual of G. We intend in subsequent papers to explore the close relationship between G and R_G in the case of a general (possibly non-Abelian) group G.Here we show that the Krull dimension of R_G is the maximal rank r of an elementary Abelian subgroup E of G unless either E is cyclic or for some such E its normalizer in G contains a non-trivial cyclic group which acts faithfully on E via “scalar multiplication” in which case it is r+1.     

Criteria of unitary equivalence of Hermitian operators with a degenerate spectrum

Nonimprovable, in general, estimates of the number of necessary and sufficient conditions for two Hermitian operators to be unitarily equaivalent in a unitary space are obtained when the multiplicities of eigenvalues of operators can be more than 1. The explicit form of these conditions is given. In the Appendix the concept of conditionally functionally independent functions is given and the corresponding necessary and sufficient conditions are presented.     

A tight bound on the projective dimension of four quadrics

Motivated by Stillman's question, we show that the projective dimension of an ideal generated by four quadric forms in a polynomial ring is at most 6; moreover, this bound is tight. We achieve this bound, in part, by giving a characterization of the low degree generators of ideals primary to height three primes of multiplicities one and two.     

Injective spaces via adjunction

The work of the present author and his coauthors over the past years gives evidence that it may be useful to regard each topological space as a kind of enriched category, by interpreting the convergence relation x→x between ultrafilters and points of a topological space X as arrows in X. Naturally, this point of view opens the door to the use of concepts and ideas from enriched Category Theory for the investigation of topological spaces. Topological theories introduced by the author provide a convenient general setting for appropriately transferring these concepts and ideas to the world of topological spaces and some other geometric objects such as approach spaces. Using tools like adjunction and the Yoneda lemma, we show that the cocomplete spaces are precisely the injective spaces, and they are algebras for a suitable monad on . This way we obtain enriched versions of known results about injective topological spaces and continuous lattices.     

Borel sets and sectional matrices

Following the path trodden by several authors along the border between Algebraic Geometry and Algebraic Combinatorics, we present some new results on the combinatorial structure of Borel ideals. This enables us to prove theorems on the shape of thesectional matrix of a homogeneous ideal, which is a new invariant stronger than the Hilbert function. The authors were partially supported by the Consiglio Nazionale delle Ricerche (CNR).