On the decomposition of the Foulkes module

The Foulkes module ${H^{(a^b)}}$ is the permutation module for the symmetric group S _ab given by the action of S _ab on the collection of set partitions of a set of size ab into b sets each of size a. The main result of this paper is a sufficient condition for a simple ${\mathbb{C} S_{ab}}$ -module to have zero multiplicity in ${H^{(a^b)}}$ . A special case of this result implies that no Specht module labelled by a hook partition (ab ? r, 1 ^r ) with r ≥ 1 appears in ${H^{(a^b)}}$ .     

Characterization of the automorphism group of quaternary linear Hadamard codes

A quaternary linear Hadamard code ${\mathcal{C}}$ is a code over ${\mathbb{Z}_4}$ such that, under the Gray map, gives a binary Hadamard code. The permutation automorphism group of a quaternary linear code ${\mathcal{C}}$ of length n is defined as ${{\rm PAut}(\mathcal{C}) = \{\sigma \in S_{n} : \sigma(\mathcal{C}) = \mathcal{C}\}}$ . In this paper, the order of the permutation automorphism group of a family of quaternary linear Hadamard codes is established. Moreover, these groups are completely characterized by computing the orbits of the action of ${{\rm PAut}(\mathcal{C})}$ on ${\mathcal{C}}$ and by giving the generators of the group. Since the dual of a Hadamard code is an extended 1-perfect code in the quaternary sense, the permutation automorphism group of these codes is also computed.     

Stanley depth of weakly polymatroidal ideals

Let \({\mathbb{K}}\) be a field and \({S = \mathbb{K}[x_1,\ldots,x_n]}\) be the polynomial ring in n variables over the field \({\mathbb{K}}\) . In this paper, it is shown that Stanley’s conjecture holds for S/I if I is a weakly polymatroidal ideal.     

Stable moduli spaces of high-dimensional manifolds

We prove an analogue of the Madsen–Weiss theorem for high-dimensional manifolds. In particular, we explicitly describe the ring of characteristic classes of smooth fibre bundles whose fibres are connected sums of g copies of S ⁿ ×S ⁿ , in the limit ${g \to \infty}$ . Rationally it is a polynomial ring in certain explicit generators, giving a high-dimensional analogue of Mumford’s conjecture. More generally, we study a moduli space ${\mathcal{N}(P)}$ of those null-bordisms of a fixed (2n–1)-dimensional manifold P which are (n–1)-connected relative to P. We determine the homology of ${\mathcal{N}(P)}$ after stabilisation using certain self-bordisms of P. The stable homology is identified with that of an infinite loop space.     

On the Stanley depth of weakly polymatroidal ideals

Let ${\mathbb{K}}$ be a field and ${S = \mathbb{K}[x_1,\dots,x_n]}$ be the polynomial ring in n variables over the field ${\mathbb{K}}$ . In this paper, it is shown that Stanley’s conjecture holds for I and S/I if I is a product of monomial prime ideals or I is a high enough power of a polymatroidal or a stable ideal generated in a single degree.     

Random and algebraic permutations’ statistics

An algebraic permutation $\hat{A}\in S(N=n^{m})$ is the permutation of the N points of the finite torus ? _n ^m , realized by a linear operator A∈SL(m,? _n ). The statistical properties of algebraic permutations are quite different from those of random permutations of N points. For instance, the period length T(A) grows superexponentially with N for some (random) permutations A of N elements, whereas $T(\hat{A})$ is bounded by a power of N for algebraic permutations  $\hat{A}$ . The paper also contains a strange mean asymptotics formula for the number of points of the finite projective line P¹(? _n ) in terms of the zeta function.     

On the chromatic numbers of spheres in ℝ n

Let χ(S _r ^n?1 )) be the minimum number of colours needed to colour the points of a sphere S _r ^n?1 of radius $r \geqslant \tfrac{1} {2}$ in ? ⁿ so that any two points at the distance 1 apart receive different colours. In 1981 P. Erd?s conjectured that χ(S _r ^n?1 )→∞ for all $r \geqslant \tfrac{1} {2}$ . This conjecture was proved in 1983 by L. Lovász who showed in [11] that χ(S _r ^n?1 ) ≥ n. In the same paper, Lovász claimed that if $r < \sqrt {\frac{n} {{2n + 2}}}$ , then χ(S _r ^n?1 ) ≤ n+1, and he conjectured that χ(S _r ^n?1 ) grows exponentially, provided $r \geqslant \sqrt {\frac{n} {{2n + 2}}}$ . In this paper, we show that Lovász’ claim is wrong and his conjecture is true: actually we prove that the quantity χ(S _r ^n?1 ) grows exponentially for any $r > \tfrac{1} {2}$ .     

Matrix Representations of the Low Order Real Clifford Algebras

In this paper we construct the matrix subalgebras ${L_{r,s}(\mathbb{R})}$ of the real matrix algebra ${M_{2^{r+s}} (\mathbb{R})}$ when 2 ≤ r + s ≤ 3 and we show that each ${L_{r,s}(\mathbb{R})}$ is isomorphic to the real Clifford algebra ${\mathcal{C} \ell_{r,s}}$ . In particular, we prove that the algebras ${L_{r,s}(\mathbb{R})}$ can be induced from ${L_{0,n}(\mathbb{R})}$ when 2 ≤ r + s = n ≤ 3 by deforming vector generators of ${L_{0,n}(\mathbb{R})}$ to multiply the specific diagonal matrices. Also, we construct two subalgebras ${T_4(\mathbb{C})}$ and ${T_2(\mathbb{H})}$ of matrix algebras ${M_4(\mathbb{C})}$ and ${M_2(\mathbb{H})}$ , respectively, which are both isomorphic to the Clifford algebra ${\mathcal{C} \ell_{0,3}}$ , and apply them to obtain the properties related to the Clifford group Γ_0,3.     

d-strong Edge Colorings of Graphs

For a proper edge coloring of a graph G the palette S(v) of a vertex v is the set of the colors of the incident edges. If S(u) ≠ S(v) then the two vertices u and v of G are distinguished by the coloring. A d-strong edge coloring of G is a proper edge coloring that distinguishes all pairs of vertices u and v with distance 1 ≤ d (u, v) ≤ d. The d-strong chromatic index ${\chi_{d}^{\prime}(G)}$ of G is the minimum number of colors of a d-strong edge coloring of G. Such colorings generalize strong edge colorings and adjacent strong edge colorings as well. We prove some general bounds for ${\chi_{d}^{\prime}(G)}$ , determine ${\chi_{d}^{\prime}(G)}$ completely for paths and give exact values for cycles disproving a general conjecture of Zhang et al. (Acta Math Sinica Chin Ser 49:703–708 2006)).     

Arc-transitive Dihedrants of Odd Prime-power Order

Let G be a finite group with identity element 1, and S be a subset of G such that ${1 \notin S}$ and S = S ^?1. The Cayley graph Cay(G, S) has vertex set G, and x, y in G are adjacent if and only if ${xy^{-1} \in S}$ . In this paper we classify the connected, arc-transitive Cayley graphs ${{\rm Cay}(D_{2p^n}, S),}$ where ${D_{2p^n}}$ is the dihedral group of order 2p ⁿ , p is an odd prime.     

Estimate for index of hypersurfaces in spheres with null higher order mean curvature

In a recent paper (Barros, Sousa in: Kodai Math. J. 2009) the authors proved that closed oriented non-totally geodesic minimal hypersurfaces of the Euclidean unit sphere have index of stability greater than or equal to n + 3 with equality occurring at only Clifford tori provided their second fundamental forms A satisfy the pinching: |A|² ≥ n. The natural generalization for this pinching is ?(r + 2)S _r+2 ≥ (n ? r)S _r  > 0. Under this condition we shall extend such result for closed oriented hypersurface Σ ⁿ of the Euclidean unit sphere ${\mathbb{S}^{n+1}}$ with null S _r+1 mean curvature by showing that the index of r-stability, ${Ind_{\Sigma^n}^{r}}$ , also satisfies ${Ind_{\Sigma^n}^{r}\ge n+3}$ . Instead of the previous hypothesis if we consider ${\frac{S_{r+2}}{{S_r}}}$ constant we have the same conclusion. Moreover, we shall prove that, up to Clifford tori, closed oriented hypersurfaces ${\Sigma^{n}\subset \mathbb{S}^{n+1}}$ with S _r+1 = 0 and S _r+2 < 0 have index of r-stability greater than or equal to 2n + 5.     

Sharing Set and Normal Families of Entire Functions

Let ${\mathcal{F}}$ be a family of holomorphic functions defined in a domain ${\mathcal{D}}$ , let k( ≥ 2) be a positive integer, and let S = {a, b}, where a and b are two distinct finite complex numbers. If for each ${f \in \mathcal{F}}$ , all zeros of f(z) are of multiplicity at least k, and f and f ^(k) share the set S in ${\mathcal{D}}$ , then ${\mathcal{F}}$ is normal in ${\mathcal{D}}$ . As an application, we prove a uniqueness theorem.     

An upper bound for the height for regular affine automorphisms of {{{\mathbb{A}}^ n}}

Kawaguchi (Math. Ann. 335(2):285–310, 359–374, 2006) proved a height inequality for ${h\bigl(f(P)\bigr)}$ when f is a regular affine automorphism of ${{\mathbb{A}}^2}$ , and he conjectured that a similar estimate is also true for regular affine automorphisms of ${{\mathbb{A}}^n}$ for n ≥ 3. In this paper we prove Kawaguchi’s conjecture. This implies that Kawaguchi’s theory of canonical heights for regular affine automorphisms of projective space is true in all dimensions.     

A Note on Toeplitz’ Conjecture

In 1911, Toeplitz made a conjecture asserting that every Jordan curve in $\mathbb{R}^{2}$ contains four points forming the corners of a square. Here Conjecture C is presented, which states that the side length of the largest square on a closed curve that consists of edges of an n×n grid is at least $1/\sqrt{2}$ times the side length of the largest axis-aligned square contained inside the curve. Conjecture C implies Toeplitz’ conjecture and is verified computationally for n≤13.     

On cyclic fixed points of spectra

For a finite $p$ -group $G$ and a bounded below $G$ -spectrum $X$ of finite type mod  $p$ , the $G$ -equivariant Segal conjecture for $X$ asserts that the canonical map $X^G \rightarrow X^{hG}$ , from $G$ -fixed points to $G$ -homotopy fixed points, is a $p$ -adic equivalence. Let $C_{p^n}$ be the cyclic group of order  $p^n$ . We show that if the $C_p$ -equivariant Segal conjecture holds for a $C_{p^n}$ -spectrum $X$ , as well as for each of its geometric fixed point spectra $\varPhi ^{C_{p^e}}(X)$ for $0 < e < n$ , then the $C_{p^n}$ -equivariant Segal conjecture holds for  $X$ . Similar results also hold for weaker forms of the Segal conjecture, asking only that the canonical map induces an equivalence in sufficiently high degrees, on homotopy groups with suitable finite coefficients.     

A Spanning Tree with High Degree Vertices

Let G be a connected graph, let ${X \subset V(G)}$ and let f be a mapping from X to {2, 3, . . .}. Kaneko and Yoshimoto (Inf Process Lett 73:163–165, 2000) conjectured that if |N _G (S) ? X| ≥ f (S) ? 2|S| + ω _G (S) + 1 for any subset ${S \subset X}$ , then there exists a spanning tree T such that d _T (x) ≥ f (x) for all ${x \in X}$ . In this paper, we show a result with a stronger assumption than this conjecture; if |N _G (S) ? X| ≥ f (S) ? 2|S| + α(S) + 1 for any subset ${S \subset X}$ , then there exists a spanning tree T such that d _T (x) ≥ f (x) for all ${x \in X}$ .     

On ideals with the Rees property

A homogeneous ideal I of a polynomial ring S is said to have the Rees property if, for any homogeneous ideal ${J \subset S}$ which contains I, the number of generators of J is smaller than or equal to that of I. A homogeneous ideal ${I \subset S}$ is said to be ${\mathfrak{m}}$ -full if ${\mathfrak{m}I:y=I}$ for some ${y \in \mathfrak{m}}$ , where ${\mathfrak{m}}$ is the graded maximal ideal of ${S}$ . It was proved by one of the authors that ${\mathfrak{m}}$ -full ideals have the Rees property and that the converse holds in a polynomial ring with two variables. In this note, we give examples of ideals which have the Rees property but are not ${\mathfrak{m}}$ -full in a polynomial ring with more than two variables. To prove this result, we also show that every Artinian monomial almost complete intersection in three variables has the Sperner property.     

Zhelobenko invariants, Bernstein-Gelfand-Gelfand operators and the analogue Kostant Clifford algebra conjecture

Let $ \mathfrak{g} $ be a complex simple Lie algebra and $ \mathfrak{h} $ a Cartan subalgebra. The Clifford algebra C( $ \mathfrak{g} $ ) of g admits a Harish-Chandra map. Kostant conjectured (as communicated to Bazlov in about 1997) that the value of this map on a (suitably chosen) fundamental invariant of degree 2?m?+?1 is just the zero weight vector of the simple (2?m?+?1)-dimensional module of the principal s-triple obtained from the Langlands dual $ {\mathfrak{g}^\vee } $ . Bazlov [1] settled this conjecture positively in type A. The hard part of the Kostant Clifford algebra conjecture is a question concerning the Harish-Chandra map for the enveloping algebra U( $ \mathfrak{g} $ ) composed with evaluation at the half sum ?? of the positive roots. The analogue Kostant conjecture is obtained by replacing the Harish-Chandra map by a ??generalized Harish-Chandra?? map. This map had been studied notably by Zhelobenko [15]. The proof given here involves a symmetric algebra version of the Kostant conjecture, the Zhelobenko invariants in the adjoint case, and, surprisingly, the Bernstein-Gelfand-Gelfand operators introduced in their study [3] of the cohomology of the flag variety.     

On the tolerance lattice of tolerance factors

Although the notion of a tolerance is a natural generalization of the notion of a congruence, many properties of factor lattices modulo congruences are not, in general, valid for factor lattices modulo tolerances. In this paper, for a lattice L of a finite length, we define a new partial order ? on $\operatorname{Tol}\, (L)$ such that for every ${S\in \operatorname{Tol}\, (L)}$ with T?S, a tolerance S/T is induced on the factor lattice L/T. This partial order is a particular restriction of ? and thus we can prove for tolerances some analogous results to the homomorphism theorem and the second isomorphism theorem for congruences. The poset $(\operatorname{Tol}\, (L), \sqsubseteq)$ is not always a lattice, but it can be converted into a specific commutative join-directoid. Then, for every ${T\in \operatorname{Tol}\, (L)}$ , $(\operatorname{Tol}\, (L/T),\sqsubseteq)$ constitutes a subdirectoid of the directoid based on the poset $(\operatorname{Tol}\, (L),\sqsubseteq)$ and this specific directoid structure is preserved by the direct product of lattices.     

Betti tables of p-Borel-fixed ideals

In this note we provide a counterexample to a conjecture of Pardue (Thesis (Ph.D.), Brandeis University, 1994), which asserts that if a monomial ideal is p-Borel-fixed, then its $\mathbb{N}$ -graded Betti table, after passing to any field, does not depend on the field. More precisely, we show that, for any monomial ideal I in a polynomial ring S over the ring $\mathbb{Z}$ of integers and for any prime number p, there is a p-Borel-fixed monomial S-ideal J such that a region of the multigraded Betti table of $J(S \otimes_{\mathbb{Z}}\ell)$ is in one-to-one correspondence with the multigraded Betti table of $I(S \otimes_{\mathbb{Z}}\ell)$ for all fields ? of arbitrary characteristic. There is no analogous statement for Borel-fixed ideals in characteristic zero. Additionally, the construction also shows that there are p-Borel-fixed ideals with noncellular minimal resolutions.