Orbit semigroups and the representation type of quivers

We show that a finite, connected quiver Q without oriented cycles is a Dynkin or Euclidean quiver if and only if all orbit semigroups of representations of Q are saturated.     

General presentations of algebras

For any finite dimensional basic associative algebra, we study the presentation spaces and their relation with the representation spaces. We prove two theorems about a general presentation, one on its subrepresentations and the other on its canonical decomposition. As a special case, we consider rigid presentations. We show how to complete a rigid presentation and study the number of nonisomorphic direct summands and different complements. Based on that, we construct a simplicial complex governing the canonical decompositions of rigid presentations and provide some examples.     

The number of faces of balanced Cohen-Macaulay complexes and a generalized Macaulay theorem

A Cohen-Macaulay complex is said to be balanced of typea=(a ₁,a ₂, ...,a _s ) if its vertices can be colored usings colors so that every maximal face gets exactlya _i vertices of thei:th color. Forb=(b ₁,b ₂, ...,b _s ), 0≦b≦a, letf _b denote the number of faces havingb _i vertices of thei:th color. Our main result gives a characterization of thef-vectorsf=(f _b )_0≦b≦a′ or equivalently theh-vectors, which can arise in this way from balanced Cohen-Macaulay complexes. As part of the proof we establish a generalization of Macaulay’s compression theorem to colored multicomplexes. Finally, a combinatorial shifting technique for multicomplexes is used to give a new simple proof of the original Macaulay theorem and another closely related result. First and third authors partially supported by the National Science Foundation.     

The genus of nearly complete graphs-case 6

The genus of a complete graph equals the least integer greater than or equal to (E-3V+6)/6, whereE andV are the numbers of edges and vertices of the graph. This paper extends the class of graphs known to have this property, concentrating on graphs whose number of vertices is congruent to 6 modulo 12.This research was initiated with partial support from NSF grant GJ-28403 at Columbia University and continued while the author was an IBM Postdoctoral Fellow in the Mathematical Sciences Department at the Thomas J. Watson Research Center, Yorktown Heights, N.Y. The author is also an Alfred P. Sloan Fellow.     

Subword complexes in Coxeter groups

Let (Π,Σ) be a Coxeter system. An ordered list of elements in Σ and an element in Π determine a subword complex, as introduced in Knutson and Miller (Ann. of Math. (2) (2003), to appear). Subword complexes are demonstrated here to be homeomorphic to balls or spheres, and their Hilbert series are shown to reflect combinatorial properties of reduced expressions in Coxeter groups. Two formulae for double Grothendieck polynomials, one of which appeared in Fomin and Kirillov (Proceedings of the Sixth Conference in Formal Power Series and Algebraic Combinatorics, DIMACS, 1994, pp. 183-190), are recovered in the context of simplicial topology for subword complexes. Some open questions related to subword complexes are presented.     

Finite Generation of Units in Alternative Loop Rings

Let L be an RA loop, that is, a loop whose loop ring in any characteristic is an alternative, but not associative, ring. Suppose that L is finite and that any noncommutative division algebra appearing as a simple component in the Wedderburn decomposition of Q L is the classical Cayley–Dickson algebra over Q. Then the unit loop of the alternative loop ring Z L of L over the ring of rational integers is finitely generated.     

On associated quasi connections

The purpose of this paper is to give a necessary and sufficient condition on the existence of associated splittings (defined in this paper) and to consider some applications to associated quasi-connections on fibred manifolds and vector bundles, using the idea and extending Theorem 1 from [2]. In Section 1, a general condition on the existence of associated splittings is given. In Section 2, the basic constructions concerning q.c.s. used in the next Section are briefly described following [7]; they extend the q.c.s. of Wang [8, 1, 2]. In Section 3 there are proved two theorems on associated q.c.s. using essentially the main theorem from Section 1.     

Direct and inverse spectral problems for generalized strings

Let the functionQ be holomorphic in he upper half plane ⁺ and such that ImQ(z 0 and ImzQ(z) 0 ifz ⁺. A basic result of M.G. Krein states that these functionsQ are the principal Titchmarsh-Weyl coefficiens of a (regular or singular) stringS[L,m] with a (non-decreasing) mass distribution functionm on some interval [0,L) with a free left endpoint 0. This string corresponds to the eigenvalue problemdf + fdm = 0; f(0–) = 0. In this note we show that the set of functionsQ which are holomorphic in ⁺ and such that the kernel has negative squares of ⁺ and ImzQ(z) 0 ifz ⁺ is the principal Titchmarsh-Weyl coefficient of a generalized string, which is described by the eigenvalue problemdf +f dm + ² fdD = 0 on [0,L),f(0–) = 0. Here is the number of pointsx whereD increases or 0 >m(x + 0) –m(x – 0) –; outside of these pointsx the functionm is locally non-decreasing and the functionD is constant.To the memory of M.G. Krein with deep gratitude and affection.This author is supported by the Fonds zur Förderung der wissenschaftlichen Forschung of Austria, Project P 09832     

Radical cube zero selfinjective algebras of finite complexity

One of our main results is a classification of all the possible quivers of selfinjective radical cube zero finite-dimensional algebras over an algebraically closed field having finite complexity. In the paper (Erdmann and Solberg, 2011) [5] we classified all weakly symmetric algebras with support varieties via Hochschild cohomology satisfying Dade’s Lemma. For a finite-dimensional algebra to have such a theory of support varieties implies that the algebra has finite complexity. Hence this paper is a partial extension of [5].     

Neeman's characterization of K(R-Proj) via Bousfield localization

Let R be an associative ring with unit and denote by $K(\mathrm{R}\text{-}\mathrm{Proj})$ the homotopy category of complexes of projective left R-modules. Neeman proved the theorem that $K(\mathrm{R}\text{-}\mathrm{Proj})$ is ${\mathrm{?}}_1$-compactly generated, with the category $K^{+}(\mathrm{R}\text{-}\mathrm{proj})$ of left bounded complexes of finitely generated projective R-modules providing an essentially small class of such generators. Another proof of Neeman's theorem is explained, using recent ideas of Christensen and Holm, and Emmanouil. The strategy of the proof is to show that every complex in $K(\mathrm{R}\text{-}\mathrm{Proj})$ vanishes in the Bousfield localization $K(\mathrm{R}\text{-}\mathrm{Flat})/〈K^{+}(\mathrm{R}\text{-}\mathrm{proj})〉$.     

Schur indices of association schemes

In this paper we introduce and investigate the theory of Schur indices arising from simple components of the rational adjacency algebras of association schemes and investigate methods for computing these indices.     

Direct products of simple modules over Dedekind domains

Let R be a (commutative) Dedekind domain and let the R-module M be a direct product of simple R-modules. Then any homomorphism from a closed submodule K of M to M can be lifted to M. Received: 9 December 2002     

On the equivariant formality of Kähler manifolds with torus group actions

We consider simply connected compact Kähler manifolds which have a holomorphic action of a torus group. We use the existing equivariant models for rational homotopy to show that these spaces satisfy an equivariant formality condition over the complex numbers.     

Free groups and involutions in the unit group of a group algebra

We study the existence of free groups of rank 2 in the group generated by the involutions in the unit group of a group algebra over a non-absolute field.Received: 25 March 2004     

Constructing cospectral graphs

Some new constructions for families of cospectral graphs are derived, and some old ones are considerably generalized. One of our new constructions is sufficiently powerful to produce an estimated 72% of the 51039 graphs on 9 vertices which do not have unique spectrum. In fact, the number of graphs of ordern without unique spectrum is believed to be at leastn ³ g _–1 for some>0, whereg _n is the number of graphs of ordern andn 7.     

Multiplicities in the Trace Cocharacter Sequence of Two 4 × 4 Matrices

We find explicitly the generating functions of the multiplicities in the pure and mixed trace cocharacter sequences of two 4 × 4 matrices over a field of characteristic 0. We determine the asymptotic behavior of the multiplicities and show that they behave as polynomials of 14th degree.Partially supported by Grant MM-1106/2001 of the Bulgarian Foundation for Scientific Research.     

Topology of matching,chessboard, and general bounded degree graph complexes

We survey results and techniques in the topological study of simplicial complexes of (di-, multi-, hyper-)graphs whose node degrees are bounded from above. These complexes have arisen in a variety of contexts in the literature. The most well-known examples are the matching complex and the chessboard complex. The topics covered here include computation of Betti numbers, representations of the symmetric group on rational homology, torsion in integral homology, homotopy properties, and connections with other fields.In memory of Gian-Carlo Rota