Self-complementary totally symmetric plane partitions

A totally symmetric plane partition of size n is a plane partition whose three-dimensional Ferrers graph is contained in the box X_n = [1, n] × [1, n] × [1, n] and which is mapped to itself under all permutations of the coordinate axes. The complement of the Ferrers graph of such a plane partition (that is, the set of lattice points in the box X_n that do not belong to the Ferrers graph) is again totally symmetric when viewed from the vantage point of the vertex (n + 1, n + 1, n + 1). A totally symmetric plane partition is self-complementary if it is congruent (in the geometrical sense) to its complement. This cannot occur unless n = 2m is even. In this paper we give several conjectures and a few theorems concerning self-complementary totally symmetric plane partitions. In particular we describe evidence which indicates a close relationship with m by m alternating sign matrices. In an earlier paper we described the close connection between m by m alternating sign matrices and descending plane partitions with no parts exceeding m. We are thus left with three classes of objects which are all apparently interrelated. There remain many unsolved problems, the simplest of which is to prove that any two of the objects have the same cardinality.     

Embedding partial totally symmetric quasigroups

This paper concerns the embedding problem for partial totally symmetric quasigroups. For all n?9, it is shown that any partial totally symmetric quasigroup of order n can be embedded in a totally symmetric quasigroup of order v if v is even and v?2n+4, and this is the best possible such inequality.     

Totally geodesic submanifolds of the complex quadric

In this article, relations between the root space decomposition of a Riemannian symmetric space of compact type and the root space decompositions of its totally geodesic submanifolds (symmetric subspaces) are described. These relations provide an approach to the classification of totally geodesic submanifolds in Riemannian symmetric spaces; this is exemplified by the classification of the totally geodesic submanifolds in the complex quadric Qm:=SO(m+2)/(SO(2)×SO(m)) obtained in the second part of the article. The classification shows that the earlier classification of totally geodesic submanifolds of Qm by Chen and Nagano (see [B.-Y. Chen, T. Nagano, Totally geodesic submanifolds of symmetric spaces, I, Duke Math. J. 44 (1977) 745-755]) is incomplete. More specifically, two types of totally geodesic submanifolds of Qm are missing from [B.-Y. Chen, T. Nagano, Totally geodesic submanifolds of symmetric spaces, I, Duke Math. J. 44 (1977) 745-755]: The first type is constituted by manifolds isometric to CP¹×RP¹; their existence follows from the fact that Q² is (via the Segre embedding) holomorphically isometric to CP¹×CP¹. The second type consists of 2-spheres of radius which are neither complex nor totally real in Qm.     

Units in totally complex S3 fields

Let B be a totally complex number field, Galois over the rational field Q, with Galois group S₃, the symmetric group on three elements. The group of units of B has torsion free rank 2. In this paper, we determine the various inequivalent representations that occur of S₃ acting on the group of units and determine arithmetic criteria for deciding which representation occurs for a particular field. As a result, we can give a relatively simple computational procedure for determining a pair of fundamental units of B given a fundamental unit in a cubic subfield.     

On extrinsic symmetric spaces with zero mean curvature in Minkowski space-time

For an extrinsic symmetric space M in Minkowski space-time, we prove that if M is spacelike with zero mean curvature, then it is totally geodesic and if M is timelike with zero mean curvature, then it is totally geodesic or it is a flat hypersurface.     

Explicitation of characters of the symmetric group

We give a totally explicit expression for the normalized characters of the symmetric group. To cite this article: M. Lassalle, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Nonisomorphic Hadamard designs

A block b of a Hadamard design is called a good block if the symmetric difference b + b₁ is also a block for all nonparallel blocks b₁. The isomorphism classes of such designs having a good block are shown to be related to a double coset decomposition of a symmetric group. As an example, over one million mutually nonisomorphic 3-(32, 16, 7) designs of a certain type are constructed.Equivalence of Hadamard matrices is described in terms of designs and it is shown that nonisomorphic designs may arise from the same matrix.     

Submanifolds Of Maximal Nullity In Symmetric Spaces

A non-totally-geodesic submanifold of relative nullity n — 1 in a symmetric space M is a cylinder over one of the following submanifolds: a surface F ² of nullity 1 in a totally geodesic submanifold N³ ? M locally isometric to S ²(c) × ? or H ²(c) × ?; a submanifold F ^k+1 spanned by a totally geodesic submanifold F ^k(c) of constant curvature moving by a special curve in the isometry group of M; a submanifold F ^k+l of nullity k in a flat totally geodesic submanifold of M; a curve.     

Possible spectra of totally positive matrices

For any given set S of n distinct positive numbers, we construct a symmetric n-by-n (strictly) totally positive matrix whose spectrum is S. Thus, in order to be the spectrum of an n-by-n totally positive matrix, it is necessary and sufficient that n numbers be positive and distinct.     

Dérivée en s=1 de la fonction L p-adique du carré symétrique dʼune courbe elliptique sur un corps totalement réel

We prove a formula for the derivative of the p-adic L-function associated with the symmetric square representation of an elliptic curve over a totally real field in which p is inert, under certain assumptions on the conductor. In particular, this proves a conjecture of Greenberg on trivial zeros. The method is to generalize unpublished calculations of Greenberg and Tilouine.     

Lightlike Hypersurfaces in Indefinite Trans-Sasakian Manifolds

This paper deals with lightlike hypersurfaces of indefinite trans-Sasakian manifolds of type (α, β), tangent to the structure vector field. Characterization Theorems on parallel vector fields, integrable distributions, minimal distributions, Ricci-semi symmetric, geodesibility of lightlike hypersurfaces are obtained. The geometric configuration of lightlike hypersurfaces is established. We prove, under some conditions, that there are no parallel and totally contact umbilical lightlike hypersurfaces of trans-Sasakian space forms, tangent to the structure vector field. We show that there exists a totally umbilical distribution in an Einstein parallel lightlike hypersurface which does not contain the structure vector field. We characterize the normal bundle along any totally contact umbilical leaf of an integrable screen distribution. We finally prove that the geometry of any leaf of an integrable distribution is closely related to the geometry of a normal bundle and its image under ${\overline{\phi}}$ .     

A covering theorem for quasi-groups

It is shown that if Q is a quasi-group of order n and k is moderately large, there exists a subset A of Q of size k such that if t is the least number of left translates of A needed to cover Q, then $\text{t >}\text{c(n}\text{log}\text{n)}\text{k}$.     

Factoring symmetric totally nonpositive matrices and inverses with a diagonal pivoting method

In this paper, we consider how to factor symmetric totally nonpositive matrices and their inverses by taking advantage of the symmetric property. It is well-known that the Bunch-Kaufman algorithm is the most commonly used pivoting strategy which can, however, produce arbitrarily large entries in the lower triangular factor for such matrices as illustrated by our example. Therefore, it is interesting to show that when the Bunch-Parlett algorithm is simplified for these matrices, it only requires O(n ²) comparisons with the growth factor being nicely bounded by 4. These facts, together with a nicely bounded lower triangular factor and a pleasantly small relative backward error, show that the Bunch-Parlett algorithm is more preferable than the Bunch-Kaufman algorithm when dealing with these matrices.     

Diagonalizing matrices over C(X)

A complete characterization of those compact Hausdorff spaces is given such that for every n, each normal element in the algebra C(X)?M_n of continuous functions from X to $\text{M}$_n can be continuously diagonalized. The conditions are that X be a sub-Stonean space with dim X ? 2 and carries no nontrivial G-bundles over any closed subset, for G a symmetric group or the circle group. In particular, diagonalization is assured on every totally disconnected sub-Stonean space, but also on connected spaces of the form β(Y)/Y, where Y is a simply-connected (noncompact) graph.     

Symmetries of Null Geometry in Indefinite Kenmotsu Manifolds

Null hypersurfaces have metrics with vanishing determinants and this degeneracy of these metrics leads to several difficulties. In this paper, null hypersurfaces of indefinite Kenmotsu space forms, tangent to the structure vector field, are studied with specific attention to locally symmetric, semi-symmetric and Ricci semi-symmetric null hypersurfaces. We show that locally symmetric and semi-symmetric null hypersurfaces are totally geodesic and parallel. These also hold for Ricci semi-symmetric null hypersurfaces, under a certain condition. We prove that, in null Einstein hypersurfaces of an indefinite Kenmotsu space form, tangent to the structure vector field, the local symmetry, semisymmetry and Ricci semi-symmetry notions are equivalent. For totally contact umbilical null hypersurfaces, we show that there are η-“Weyl” connections adapted to the induced structure on the null hypersurface.     

Crystals and total positivity on orientable surfaces

We develop a combinatorial model of networks on orientable surfaces, and study weight and homology generating functions of paths and cycles in these networks. Network transformations preserving these generating functions are investigated. We describe in terms of our model the crystal structure and R-matrix of the affine geometric crystal of products of symmetric and dual symmetric powers of type A. Local realizations of the R-matrix and crystal actions are used to construct a double affine geometric crystal on a torus, generalizing the commutation result of Kajiwara et al. (Lett Math Phys, 60(3):211–219, 2002) and an observation of Berenstein and Kazhdan (MSJ Mem, 17:1–9, 2007). We show that our model on a cylinder gives a decomposition and parametrization of the totally non-negative part of the rational unipotent loop group of GL _n .     

The number of equivalence classes of symmetric sign patterns

This paper shows that the number of sign patterns of totally non-zero symmetric n-by-n matrices, up to conjugation by permutation and signature matrices and negation, is equal to the number of unlabelled graphs on n vertices.     

Curved flats,pluriharmonic maps and constant curvature immersions into pseudo-Riemannian space forms

We study two aspects of the loop group formulation for isometric immersions with flat normal bundle of space forms. The first aspect is to examine the loop group maps along different ranges of the loop parameter. This leads to various equivalences between global isometric immersion problems among different space forms and pseudo-Riemannian space forms. As a corollary, we obtain a non-immersibility theorem for spheres into certain pseudo-Riemannian spheres and hyperbolic spaces. The second aspect pursued is to clarify the relationship between the loop group formulation of isometric immersions of space forms and that of pluriharmonic maps into symmetric spaces. We show that the objects in the first class are, in the real analytic case, extended pluriharmonic maps into certain symmetric spaces which satisfy an extra reality condition along a totally real submanifold. We show how to construct such pluriharmonic maps for general symmetric spaces from curved flats, using a generalised DPW method.      

Cellular automata, quasigroups and symmetries

Summary. We investigate cellular automata (CA) with a local rule f: G² ? G \phi : G^2 \rightarrow G , where the local rule defines a quasigroup structure (Latin square) on the finite set G. If the quasigroup is semisymmetric or totally symmetric, some top-down equilateral triangular subsets of the CA-orbits, the so-called \triangledown \triangledown -configurations, exhibit certain symmetries. The most interesting symmetries are the rotational and the total (dihedral) symmetries, which may be considered in conjunction with certain automorphisms.¶We first explore the conditions for quasigroups to be symmetric (or for local CA-rules to allow symmetric \triangledown \triangledown -configurations), and how to construct symmetric quasigroups by prolongation, i.e., by steadily increasing the order of the quasigroup, thereby conserving the symmetry. Then we study rotationally or totally symmetric \triangledown \triangledown -configurations. We begin with the existence of symmetric \triangledown \triangledown -configurations of arbitrary size N and N o 0,1 mod 3 N \equiv 0,1\,{\rm mod}\,3 , and show that symmetric \triangledown \triangledown -configurations of size N o 2 mod 3 N \equiv 2\,{\rm mod}\,3 exist under mild conditions on J. We present explicit formulas for the number of distinct symmetric \triangledown \triangledown -configurations. By studying the combined group action of the dihedral (or rotational) group and the automorphism group of the quasigroup G on the \triangledown \triangledown -configurations of size N, we are able to classify and count the number of different equivalence classes of \triangledown \triangledown -configurations.     

Monotonicity of strata in the stratification of the cone of totally positive matrices

According to a theorem of Bjorner [5], there exists a stratified space whose strata are labeled by the elements of [u, v] for every interval [u, v] in the Bruhat order of a Coxeter group W, and each closed stratum (respectively the boundary of each stratum) has the homology of a ball (respectively of a sphere). In [6], Fomin and Shapiro suggest a natural geometric realization of these stratified spaces for a Weyl group W of a semi-simple Lie group G, and then prove its validity in the case of the symmetric group. The stratified spaces arise as links in the Bruhat decomposition of the totally non-negative part of the unipotent radical of G. In this article, we verify the topological regularity property of the strata formed as a result of Bruhat partial ordering on the elements of theWeyl group (of rank 4) of a semi-simple simply connected algebraic group G which is SL(4,?) in our case here. The Weyl group here is the Coxeter group S ₄.