Combinatorics of the \widehat{\mathfrak{s}\mathfrak{l}}_2 Coinvariants: Dual Functional Realization and Recursion

We give the fermionic character formulas for the spaces of coinvariants obtained from level k integrable representations of . We establish the functional realization of the spaces dual to the coinvariant spaces. We parameterize functions in the dual spaces by rigged partitions, and prove the recursion relations for the sets of rigged partitions.     

A characterization of two classes of strongly regular graphs

The half dual polar graphsD _4,4(q) and the alternating forms graphsAlt(4,q) are characterized among strongly regular graphs with classical parameters via the geometric structures of polar spaces and affine polar spaces of rank 4, respectively.     

Star partitions and the graph isomorphism problem

A canonical basis of Rⁿ associated with a graph G on n vertices has been defined in [15] in connection with eigenspaces and star partitions of G. The canonical star basis together with eigenvalues of G determines G to an isomorphism. We study algorithms for finding the canonical basis and some of its variations. The emphasis is on the following three special cases; graphs with distinct eigenvalues, graphs with bounded eigenvalue multiplicities and strongly regular graphs. We show that the procedure is reduced in some parts to special cases of some well known combinatorial optimization problems, such as the maximal matching problem. the minimal cut problem, the maximal clique problem etc. This technique provides another proof of a result of L. Babai et al. [2] that isomorphism testing for graphs with bounded eigenvalue multiplicities can be performend in a polynomial time. We show that the canonical basis in strongly regular graphs is related to the graph decomposition into two strongly regular induced subgraphs. Examples of distinguishing between cospectral strongly regular graphs by means of the canonical basis are provided. The behaviour of star partitions of regular graphs under operations of complementation and switching is studied.     

A class of amply regular graphs related to the subconstituents of a dual polar graph

The connected components of the induced graphs on each subconstituent of the dual polar graph of the odd dimensional orthogonal spaces over a finite field are shown to be amply regular. The connected components of the graphs on the second and third subconstituents are shown to be distance-regular by elementary methods.     

Multi-way dual Cheeger constants and spectral bounds of graphs

We introduce a set of multi-way dual Cheeger constants and prove universal higher-order dual Cheeger inequalities for eigenvalues of normalized Laplace operators on weighted finite graphs. Our proof proposes a new spectral clustering phenomenon deduced from metrics on real projective spaces. We further extend those results to a general reversible Markov operator and find applications in characterizing its essential spectrum.     

Group duality with the topology of precompact convergence

The Pontryagin-van Kampen (P-vK) duality, defined for topological Abelian groups, is given in terms of the compact-open topology. Polar reflexive spaces, introduced by Köthe, are those locally convex spaces satisfying duality when the dual space is equipped with the precompact-open topology. It is known that the additive groups of polar reflexive spaces satisfy P-vK duality. In this note we consider the duality of topological Abelian groups when the topology of the dual is the precompact-open topology. We characterize the precompact reflexive groups, i.e., topological groups satisfying the group duality defined in terms of the precompact-open topology. As a consequence, we obtain a new characterization of polar reflexive spaces. We also present an example of a space which satisfies P-vK duality and is not polar reflexive. Some of our results respond to questions appearing in the literature.     

Valuations and hyperplanes of dual polar spaces

Valuations were introduced in De Bruyn and Vandecasteele (Valuations of near polygons,preprint, 2004) as a very important tool for classifying near polygons. In the present paper we study valuations of dual polar spaces. We will introduce the class of the SDPS-valuations and characterize these valuations. We will show that a valuation of a finite thick dual polar space is the extension of an SDPS-valuation if and only if no induced hex valuation is ovoidal or semi-classical. Each SDPS-valuation will also give rise to a geometric hyperplane of the dual polar space.     

Bounded distance preserving surjections in the projective geometry of matrices

We examine surjective maps which preserve a fixed bounded distance in both directions on some classical dual polar spaces.     

Projective embeddings of dual polar spaces arising from a class of alternative division rings

We discuss some recent results of us regarding a class of polar spaces which includes the nonembeddable polar spaces introduced by Tits [Tits, J., “Buildings of spherical type and finite BN-pairs,” Lecture Notes in Mathematics 386, Springer-Verlag, Berlin-New York, 1974]. These results include an elementary construction of the polar space, a construction of a polarized embedding of the corresponding dual polar space and the determination whether this projective embedding is universal and unique (as a polarized embedding).     

Eigenvalues of the derangement graph

We consider the Cayley graph on the symmetric group Sn generated by derangements. It is well known that the eigenvalues of this graph are indexed by partitions of n. We investigate how these eigenvalues are determined by the shape of their corresponding partitions. In particular, we show that the sign of an eigenvalue is the parity of the number of cells below the first row of the corresponding Ferrers diagram. We also provide some lower and upper bounds for the absolute values of these eigenvalues.     

On the simple connectedness of hyperplane complements in dual polar spaces

Let Δ be a dual polar space of rank n≥4, H be a hyperplane of Δ and Γ?Δ?H be the complement of H in Δ. We shall prove that, if all lines of Δ have more than 3 points, then Γ is simply connected. Then we show how this theorem can be exploited to prove that certain families of hyperplanes of dual polar spaces, or all hyperplanes of certain dual polar spaces, arise from embeddings.     

The Eigenspaces of the Bose-Mesner-Algebras of the Association Schemes Corresponding to Projective Spaces and Polar Spaces

In an earlier paper 7, some properties of the eigenspaces of the Bose-Mesner-algebras of association schemes are figured out, leaving open the problem of determining the eigenspaces. In the present paper, these eigenspaces and the eigenvalues are determined for projective spaces and for polar spaces. This allows characterizations of certain sets of subspaces of these geometries.     

Antidesigns and regularity of partial spreads in dual polar graphs

We give several examples of designs and antidesigns in classical finite polar spaces. These types of subsets of maximal totally isotropic subspaces generalize the dualization of the concepts of m ‐ovoids and tight sets of points in generalized quadrangles. We also consider regularity of partial spreads and spreads. The techniques that we apply were developed by Delsarte. In some polar spaces of small rank, some of these subsets turn out to be completely regular codes. © 2010 Wiley Periodicals, Inc. J Combin Designs 19: 202‐216, 2011     

Embeddings of flag-transitive classical locally polar geometries of rank 3

The group-admissible embeddings of flag-transitive classical locally polar geometries of rank 3 are determined, as well as those of truncations of the related dual polar spaces.     

Suborbits of a point stabilizer in the orthogonal group on the last subconstituent of orthogonal dual polar graphs

As one of the serial papers on suborbits of point stabilizers in classical groups on the last subconstituent of dual polar graphs, the corresponding problem for orthogonal dual polar graphs over a finite field of odd characteristic is discussed in this paper. We determine all the suborbits of a point-stabilizer in the orthogonal group on the last subconstituent, and calculate the length of each suborbit. Moreover, we discuss the quasi-strongly regular graphs and the association schemes based on the last subconstituent, respectively.     

Characters of Fundamental Representations of Quantum Affine Algebras

We give closed formulae for the q-characters of the fundamental representations of the quantum loop algebra of a classical Lie algebra, in terms of a family of partitions satisfying some simple properties. We also give the multiplicities of the eigenvalues of the imaginary subalgebra in terms of these partitions.     

Kollineationen von Translationsstrukturen

Translationstructures are generalized affine spaces. They can be described algebraically by partitions of groups. For desarguesian affine spaces the group is a vectorspace and the partition is the set of all onedimensional subspaces. In this case each collineation fixing 0 is a regular semilinear mapping, i.e. an automorphism of the vectorspace. In the general case it is a mapping called equivalence. Each equivalence of a partition is an automorphism iff the set of translations of the group is a normal subgroup of the collineationgroup. The translations form a normal subgroup, if the group is finite or abelian. We prove some theorems for the infinite non abelian case.     

A Lower Bound for the Dimension of Trivariate Spline Spaces

We consider trivariate C^r spline spaces of degree d defined on arbitrary tetrahedral partitions. A lower bound for the dimension of trivariate spline spaces over arbitrary tetrahedral partitions for d > r is computed. This is the first general lower bound known.     

Circular trichotomy of the spectrum of regular matrix pencils

We introduce a method for approximating the right and left deflating subspaces of a regular matrix pencil corresponding to the eigenvalues inside, on and outside the unit circle. The method extends the iteration used in the context of spectral dichotomy, where the assumption on the absence of eigenvalues on the unit circle is removed. It constructs two matrix sequences whose null spaces and the null space of their sum lead to approximations of the deflating subspaces corresponding to the eigenvalues of modulus less than or equal to 1, equal to 1 and larger than or equal to 1. An orthogonalization process is then used to extract the desired delating subspaces. The resulting algorithm is an inverse free, easy to implement, and sufficiently fast. The derived convergence estimates reveal the key parameters, which determine the rate of convergence. The method is tested on several numerical examples.