Noncommutative symmetric functions and an amazing matrix

We present a simple way to derive the results of Diaconis and Fulman [P. Diaconis, J. Fulman, Foulkes characters, Eulerian idempotents, and an amazing matrix, arXiv:1102.5159] in terms of noncommutative symmetric functions.     

Brill-Gordan loci, transvectants and an analogue of the Foulkes conjecture

The hypersurfaces of degree d in the projective space Pn correspond to points of PN, where . Now assume d=2e is even, and let X(n,d)⊆PN denote the subvariety of two e-fold hyperplanes. We exhibit an upper bound on the Castelnuovo regularity of the ideal of X(n,d), and show that this variety is r-normal for r?2. The latter result is representation-theoretic, and says that a certain GLn+1-equivariant morphism

Sr(S2e(Cn+1))→S₂(Sre(Cn+1))     

When is a matrix a difference of two idempotents

Necessary and sufficient conditions are given for a matrix A to be a sum or a difference of two idempotents. These conditions are expressed in terms of (i) the block structure of A under similarity, as well as (ii) the elementary divisor structure of A.     

Logarithmic residues, generalized idempotents, and sums of idempotents in Banach algebras

In a commutative Banach algebraB the set of logarithmic residues (i.e., the elements that can be written as a contour integral of the logarithmic derivative of an analyticB-valued function), the set of generalized idempotents (i.e., the elements that are annihilated by a polynomial with non-negative integer simple zeros), and the set of sums of idempotents are all the same. Also, these (coinciding) sets consist of isolated points only and are closed under the operations of addition and multiplication. Counterexamples show that the commutativity condition onB is essential. The results extend to logarithmic residues of meromorphicB-valued functions.     

Eulerian* -polynomial identities on matrix algebras

In this paper we considerably strengthen a result of Ribenboim on noetherian generalised power series rings. While Ribenboim proves his result under the restrictive assumption that the monoid occuring in the definition of the geralised power series ring in cancellative we prove a corresponding result for arbitrary ordered monoids.     

Exchange rings,units and idempotents

An associative ring R with identity is semiperfect if and only if every element of R is a sum of a unit and an idempotent, and R contains no infinite set of orthogonal idempotents. A ring which contains no infinite set of orthogonal idempotents is an exchange ring if and only if every element is a sum of a unit and an idempo-tent     

On decomposing any matrix as a linear combination of three idempotents

In a recent article, we gave a full characterization of matrices that can be decomposed as linear combinations of two idempotents with prescribed coefficients. In this one, we use those results to improve on a recent theorem of Rabanovich: we establish that every square matrix is a linear combination of three idempotents (for an arbitrary coefficient field rather than just one of characteristic 0).     

Spectral shift function, amazing and multifaceted

Several formula representations for the I. M. Lifshits — M. G. Kreîn spectral shift function (SSF) are discussed and intercompared. It is pointed out that the equivalence of these representations is not apparent, and different properties of the SSF are revealed by different formulas. The presentation is informal and contains no proofs.To the memory of the great mathematician Mark Grigor'evich Kreîn     

Special idempotents and projections

Semigroup Forum - We define, for any special matching of a finite graded poset, an idempotent, regressive and order preserving function. We consider the monoid generated by such functions. We call...     

Bisimple inverse semigroups whose idempotents are an interval

Bisimple locally compact topological inverse semigroups whose maximal subgroups are compact and whose set of idempotents is a real interval are studied in this paper. The structure of such semigroups is shown to be entirely determined by their maximal subgroups in case the semigroups come from a certain subclass which includes the connected ones. The factor semigroups moduloH are completely determined.     

On generalized idempotents,lattice-dilations and contractions inf-rings

This paper falls into two related parts, concerned respectively with generalized idempotents (associated to superunities and subunities) and with lattice-dilations and contractions in anf-ringA. We characterize the mappings F∶:A→A satisfying ¦F(x) — F(y)¦=u ¦x — y¦ (u-dilations) as the mappings of the formF(x)=x(u — 2e) +b, withe being a generalized idempotent andb depending only onF, and obtain an analogous result for lattice-contractions. The structure of the sets of all the homogeneous ones (both cases) is studied and is related with the algebraic and lattice-theoretic properties of the ring.     

Eulerian pairs and Eulerian recurrence systems

In this paper, we introduce the definitions of Eulerian pair and Hermite-Biehler pair. We also characterize a duality relation between Eulerian recurrences and Eulerian recurrence systems. This generalizes and unifies Hermite-Biehler decompositions of several enumerative polynomials, including up-down run polynomials for symmetric groups, alternating run polynomials for hyperoctahedral groups, flag descent polynomials for hyperoctahedral groups and flag ascent-plateau polynomials for Stirling permutations. We derive some properties of associated polynomials. In particular, we prove the alternatingly increasing property and the interlacing property of the ascent-plateau and left ascent-plateau polynomials for Stirling permutations.     

Schur idempotents and hyperreflexivity

We show that the set of Schur idempotents with hyperreflexive range is a Boolean lattice which contains all contractions. We establish a preservation result for sums which implies that the weak* closed span of a hyperreflexive and a ternary masa-bimodule is hyperreflexive, and prove that the weak* closed span of finitely many tensor products of a hyperreflexive space and a hyperreflexive range of a Schur idempotent (respectively, a ternary masa-bimodule) is hyperreflexive.     

Involutions and characters of upper triangular matrix groups

We study the realizability over of representations of the group of upper-triangular matrices over . We prove that all the representations of are realizable over if , but that if , has representations not realizable over . This theorem is a variation on a result that can be obtained by combining work of J. Arregi and A. Vera-López and of the authors, but the proof of the theorem in this paper is much more natural.

     

The asymptotic numbers of regular tournaments,Eulerian digraphs and Eulerian oriented graphs

LetRT(n), ED(n) andEOG(n) be the number of labelled regular tournaments, labelled loop-free simple Eulerian digraphs, and labelled Eulerian oriented simple graphs, respectively, onn vertices. Then, asn,, for any>0. The last two families of graphs are also enumerated by their numbers of edges. The proofs use the saddle point method applied to appropriaten-dimensional integrals.     

Generalized Eulerian numbers and the topology of the Hessenberg variety of a matrix

Let Agl(n, C) and let p be a positive integer. The Hessenberg variety of degree p for A is the subvariety Hess(p, A) of the complete flag manifold consisting of those flags S ₁ S _n–1 in ⁿ which satisfy the condition AS _i S _i+p ,for all i. We show that if A has distinct eigenvalues, then Hess(p, A) is smooth and connected. The odd Betti numbers of Hess(p, A) vanish, while the even Betti numbers are given by a natural generalization of the Eulerian numbers. In the case where the eigenvalues of A have distinct moduli, |₁|<<|₁|, these results are applied to determine the dimension and topology of the submanifold of U(n) consisting of those unitary matrices P for which A ₀=P ^-1 AP is in Hessenberg form and for which the diagonal entries of the QR-iteration initialized at A ₀ converge to a given permutation of ₁,_n.Research partially supported by the National Science Foundation under Grant ECS-8696108.