On the Maximally Clustered Elements of Coxeter Groups

We continue the study of the maximally clustered elements for simply laced Coxeter groups which were recently introduced by Losonczy. Such elements include as a special case the freely braided elements introduced by Losonczy and the author, which in turn constitute a superset of the i ji-avoiding elements of Fan. Our main result is to classify the MC-finite Coxeter groups, namely, those Coxeter groups having finitely many maximally clustered elements. Remarkably, any simply laced Coxeter group having finitely many i ji-avoiding elements also turns out to be MC-finite.     

Schubert Varieties and Free Braidedness

We give a simple necessary and sufficient condition for a Schubert variety X_w to be smooth when w is a freely braided element of a simply laced Weyl group; such elements were introduced by the authors in a previous work. This generalizes in one direction a result of Fan concerning varieties indexed by short-braid avoiding elements. We also derive generating functions for the freely braided elements that index smooth Schubert varieties. All results are stated and proved only for the simply laced case.     

Smooth and palindromic Schubert varieties in affine Grassmannians

We completely determine the smooth and palindromic Schubert varieties in affine Grassmannians, in all Lie types. We show that an affine Schubert variety is smooth if and only if it is a closed parabolic orbit. In particular, there are only finitely many smooth affine Schubert varieties in a given Lie type. An affine Schubert variety is palindromic if and only if it is a closed parabolic orbit, a chain, one of an infinite family of “spiral” varieties in type A, or a certain 9-dimensional singular variety in type B ₃. In particular, except in type A there are only finitely many palindromic affine Schubert varieties in a fixed Lie type. Moreover, in types D and E an affine Schubert variety is smooth if and only if it is palindromic; in all other types there are singular palindromics. The proofs are for the most part combinatorial. The main tool is a variant of Mozes’ numbers game, which we use to analyze the Bruhat order on the coroot lattice. In the proof of the smoothness theorem we also use Chevalley’s cup product formula.     

INTERSECTION COHOMOLOGY AND LIE ALGEBRA HOMOLOGY

For a semisimple algebraic group G over C, we try to make a comparative study between intersection cohomology of Schubert varieties and Lie algebra homology of certain nilpotent Lie algebras. We prove that when all simple factors of G are simply laced, these two are the same as vector spaces over C at the first homology level. We give counter-examples in the general case and state a conjecture as a possible direction for generalisation.     

Cohomology of line bundles on Schubert varieties: The rank two case

In this paper we describe vanishing and non-vanishing of cohomology of “most” line bundles over Schubert subvarieties of flag varieties for rank 2 semisimple algebraic groups.     

A partial Horn recursion in the cohomology of flag varieties

Horn recursion is a term used to describe when non-vanishing products of Schubert classes in the cohomology of complex flag varieties are characterized by inequalities parameterized by similar non-vanishing products in the cohomology of “smaller” flag varieties. We consider the type A partial flag variety and find that its cohomology exhibits a Horn recursion on a certain deformation of the cup product defined by Belkale and Kumar (Invent. Math. 166:185–228, 2006). We also show that if a product of Schubert classes is non-vanishing on this deformation, then the associated structure constant can be written in terms of structure constants coming from induced Grassmannians.     

On the automorphic theta representation for simply laced groups

We construct an automorphic realization of the minimal representation of a split, simply laced groupG, over a number field. The realization is by a residue, at a certain point, of an Eisenstein series, induced from the Borel subgroup. This residue representation is square integrable and defines the automorphic theta representation. It has “very few” Fourier coefficients, which turn out to have some extra invariance properties. This research was supported by the Basic Research Foundation administered by the Israel Academy of Sciences and Humanities.     

Torus quotients of homogeneous spaces — minimal dimensional Schubert varieties admitting semi-stable points

In this paper, for any simple, simply connected algebraic group G of type B,C or D and for any maximal parabolic subgroup P of G, we describe all minimal dimensional Schubert varieties in G/P admitting semistable points for the action of a maximal torus T with respect to an ample line bundle on G/P. We also describe, for any semi-simple simply connected algebraic group G and for any Borel subgroup B of G, all Coxeter elements τ for which the Schubert variety X(τ) admits a semistable point for the action of the torus T with respect to a non-trivial line bundle on G/B.     

Rates of convex approximation in non-hilbert spaces

This paper deals with sparse approximations by means of convex combinations of elements from a predetermined “basis” subsetS of a function space. Specifically, the focus is on therate at which the lowest achievable error can be reduced as larger subsets ofS are allowed when constructing an approximant. The new results extend those given for Hilbert spaces by Jones and Barron, including, in particular, a computationally attractive incremental approximation scheme. Bounds are derived for broad classes of Banach spaces; in particular, forL _p spaces with 1<p<∞, theO (n ^−1/2) bounds of Barron and Jones are recovered whenp=2. One motivation for the questions studied here arises from the area of “artificial neural networks,” where the problem can be stated in terms of the growth in the number of “neurons” (the elements ofS) needed in order to achieve a desired error rate. The focus on non-Hilbert spaces is due to the desire to understand approximation in the more “robust” (resistant to exemplar noise)L _p, 1 ≤p<2, norms. The techniques used borrow from results regarding moduli of smoothness in functional analysis as well as from the theory of stochastic processes on function spaces.     

Regularp-groups. II

We investigate minimal irregularp-groups, and derive several criteria for regularity as a consequence. For example, ap-group is regular if all its subgroups have “few” generators. It is also shown that for varieties, regularity is equivalent to “good” power structure. We slose with some examples.     

Schubert polynomials of types A--D

Schubert polynomials of type B, C, and D have been described first by S. Billey and M. Haiman [BH] using a combinatorial method. In this paper we give a unified algebraic treatment of Schubert polynomials of types A–D in the style of the Lascoux–Schützenberger theory in type A, i.e. Schubert polynomials are generated by the application of sequences of divided difference operators to “top polynomials”. The use of the creation operators for Q-Schur and P-Schur functions allows us to give: (1) simple and natural forms of the “top polynomials”, (2) formulas for the easy computation with all divided differences, (3) recursive structures, and (4) simplified derivations of basic properties. Received: 23 July 1998     

A KAM phenomenon for singular holomorphic vector fields

Let X be a germ of holomorphic vector field at the origin of Cⁿ and vanishing there. We assume that X is a good perturbation of a “nondegenerate” singular completely integrable system. The latter is associated to a family of linear diagonal vector fields which is assumed to have nontrivial polynomial first integrals (they are generated by the so called “resonant monomials”). We show that X admits many invariant analytic subsets in a neighborhood of the origin. These are biholomorphic to the intersection of a polydisc with an analytic set of the form “resonant monomials = constants”. Such a biholomorphism conjugates the restriction of X to one of its invariant varieties to the restriction of a linear diagonal vector field to a toric variety. Moreover, we show that the set of “frequencies” defining the invariant sets is of positive measure.     

The <Emphasis Type="Italic">R</Emphasis>-observability and <Emphasis Type="Italic">R</Emphasis>-controllability of linear differential-algebraic systems

We study the R-controllability (the controllability within the attainability set) and the R-observability of time-varying linear differential-algebraic equations (DAE). We analyze DAE under assumptions guaranteeing the existence of a structural form (which is called “equivalent”) with separated “differential” and “algebraic” subsystems. We prove that the existence of this form guarantees the solvability of the corresponding conjugate system, and construct the corresponding “equivalent form” for the conjugate DAE. We obtain conditions for the R-controllability and R-observability, in particular, in terms of controllability and observability matrices. We prove theorems that establish certain connections between these properties.     

Dynkin Diagram Classification of λ-Minuscule Bruhat Lattices and of d-Complete Posets

d-Complete posets are defined to be posets which satisfy certain local structural conditions. These posets play or conjecturally play several roles in algebraic combinatorics related to the notions of shapes, shifted shapes, plane partitions, and hook length posets. They also play several roles in Lie theory and algebraic geometry related to -minuscule elements and Bruhat distributive lattices for simply laced general Weyl or Coxeter groups, and to -minuscule Schubert varieties. This paper presents a classification of d-complete posets which is indexed by Dynkin diagrams.     

Restriction varieties and geometric branching rules

This paper develops a new method for studying the cohomology of orthogonal flag varieties. Restriction varieties are subvarieties of orthogonal flag varieties defined by rank conditions with respect to (not necessarily isotropic) flags. They interpolate between Schubert varieties in orthogonal flag varieties and the restrictions of general Schubert varieties in ordinary flag varieties. We give a positive, geometric rule for calculating their cohomology classes, obtaining a branching rule for Schubert calculus for the inclusion of the orthogonal flag varieties in Type A flag varieties. Our rule, in addition to being an essential step in finding a Littlewood–Richardson rule, has applications to computing the moment polytopes of the inclusion of SO(n) in SU(n), the asymptotic of the restrictions of representations of SL(n) to SO(n) and the classes of the moduli spaces of rank two vector bundles with fixed odd determinant on hyperelliptic curves. Furthermore, for odd orthogonal flag varieties, we obtain an algorithm for expressing a Schubert cycle in terms of restrictions of Schubert cycles of Type A flag varieties, thereby giving a geometric (though not positive) algorithm for multiplying any two Schubert cycles.     

p-convergence in measure of a sequence of measurable functions and corresponding minimal elements of c 0

We characterize convergence in measure of a sequence (f_n)_n of measurable functions to a measurable function f by elements of c₀, which express the quality of convergence of (f_n)_n to f. This characterization motivates the introduction of a new notion of convergence, called “p-convergence in measure” (p > 0), which is stronger than convergence in measure. We prove the existence of “minimal” elements in c₀ which characterize the convergence in measure of (f_n)_n to f.      

Generalized low pass filters and MRA frame wavelets

A tight frame wavelet ψ is an L ²(ℝ) function such that {ψ jk(x)} = {2^j/2 ψ(2 ^j x −k), j, k ∈ ℤ},is a tight frame for L ² (ℝ).We introduce a class of “generalized low pass filters” that allows us to define (and construct) the subclass of MRA tight frame wavelets. This leads us to an associated class of “generalized scaling functions” that are not necessarily obtained from a multiresolution analysis. We study several properties of these classes of “generalized” wavelets, scaling functions and filters (such as their multipliers and their connectivity). We also compare our approach with those recently obtained by other authors.     

Four positive formulae for type A quiver polynomials

We give four positive formulae for the (equioriented type A) quiver polynomials of Buch and Fulton [BF99 ]. All four formulae are combinatorial, in the sense that they are expressed in terms of combinatorial objects of certain types: Zelevinsky permutations, lacing diagrams, Young tableaux, and pipe dreams (also known as rc-graphs). Three of our formulae are multiplicity-free and geometric, meaning that their summands have coefficient 1 and correspond bijectively to components of a torus-invariant scheme. The remaining (presently non-geometric) formula is a variant of the conjecture of Buch and Fulton in terms of factor sequences of Young tableaux [BF99 ]; our proof of it proceeds by way of a new characterization of the tableaux counted by quiver constants. All four formulae come naturally in “doubled” versions, two for double quiver polynomials, and the other two for their stable limits, the double quiver functions, where setting half the variables equal to the other half specializes to the ordinary case. Our method begins by identifying quiver polynomials as multidegrees [BB82 , Jos84 , BB85 , Ros89 ] via equivariant Chow groups [EG98 ]. Then we make use of Zelevinsky’s map from quiver loci to open subvarieties of Schubert varieties in partial flag manifolds [Zel85 ]. Interpreted in equivariant cohomology, this lets us write double quiver polynomials as ratios of double Schubert polynomials [LS82 ] associated to Zelevinsky permutations; this is our first formula. In the process, we provide a simple argument that Zelevinsky maps are scheme-theoretic isomorphisms (originally proved in [LM98 ]). Writing double Schubert polynomials in terms of pipe dreams [FK96 ] then provides another geometric formula for double quiver polynomials, via [KM05 ]. The combinatorics of pipe dreams for Zelevinsky permutations implies an expression for limits of double quiver polynomials in terms of products of Stanley symmetric functions [Sta84 ]. A degeneration of quiver loci (orbit closures of GL on quiver representations) to unions of products of matrix Schubert varieties [Ful92 , KM05 ] identifies the summands in our Stanley function formula combinatorially, as lacing diagrams that we construct based on the strands of Abeasis and Del Fra in the representation theory of quivers [AD80 ]. Finally, we apply the combinatorial theory of key polynomials to pass from our lacing diagram formula to a double Schur function formula in terms of peelable tableaux [RS95a , RS98 ], and from there to our formula of Buch–Fulton type.     

How to Play M 13?

A puzzle called “M ₁₃” J. H. Conway has described recently is explained. We report an implementation of the puzzle in the programming language Java. The program allows the human user to “play M ₁₃” interactively (and to cheat by solving it automatically). The program is an example on how to bring to life a nice piece of discrete mathematics. In this sense it presents not only a didactical way of seeing “mathematics at work”, but also displays the stabilizer chain method developed by C. Sims to solve group theoretic puzzles, the most famous of which being Rubik's cube.     

The Number of Hierarchical Orderings

An ordered set-partition (or preferential arrangement) of n labeled elements represents a single “hierarchy” these are enumerated by the ordered Bell numbers. In this note we determine the number of “hierarchical orderings” or “societies”, where the n elements are first partitioned into m ≤ n subsets and a hierarchy is specified for each subset. We also consider the unlabeled case, where the ordered Bell numbers are replaced by the composition numbers. If there is only a single hierarchy, we show that the average rank of an element is asymptotic to n/(4 log 2) in the labeled case and to n/4 in the unlabeled case. This revised version was published online in September 2006 with corrections to the Cover Date.