In this paper we introduce the notion of G-pre-weighted homogeneous map germ, (G is one of Mather’s groups A or K) and show that any G-pre-weight ed homogeneous map germ is G-finitely determined. We also give an explicit “order”, based on the Newton polyhedron of a pre-weighted homogeneous germ of function, such that the topological structure is preserved after perturbations by terms of higher order.

Let $(W,S)$ be a finite Coxeter group. Kazhdan and Lusztig introduced the concept of $W$-graphs, and Gyoja proved that every irreducible representation of the Iwahori–Hecke algebra $H(W,S)$ can be realized as a $W$-graph. Gyoja defined an auxiliary algebra for this purpose which—to the best of the author’s knowledge—was never explicitly mentioned again in the literature after Gyoja’s proof (although the underlying ideas were reused). The purpose of this paper is to resurrect this $W$-graph algebra, and to study its structure and its modules. A new explicit description of it as a quotient of a certain path algebra is given. A general conjecture is proposed which would imply strong restrictions on the structure of $W$-graphs. This conjecture is then proven for Coxeter groups of type $I_{2}(m)$, $B_{3}$ and $A_{1}$–$A_{4}$.

We give a presentation of cyclotomic q-Schur algebras by generators and defining relations. As an application, we give an algorithm for computing decomposition numbers of cyclotomic q-Schur algebras.

Let f: E→ P′(C) be an elliptic fibration with non-constant j-invariant, and possessing a section σ0. The group of sections of f is then naturally identified with the group of points defined over a function field C(u) of the generic fibre of f.

Let be a solvable group of order g. Let p be a prime and let g = pa g′ with (p,g′) = 1. In [4] we have tried to find sufficient conditions for to possess an irreducible character of p-defect 0, that is, a character whose degree is divisible by pa.

Quasi-socle ideals, that is, ideals of the form I = Q: mq (q ≥ 2), with Q parameter ideals in a Buchsbaum local ring (A,m), are explored in connection to the question of when I is integral over Q and when the associated graded ring G(I) ⊕ n≥0In/In+1 of I is Buchsbaum. The assertions obtained by Wang in the Cohen-Macaulay case hold true after necessary modifications of the conditions on parameter ideals Q and integers q. Examples are explored.

In this paper, we investigate several classes of groups, among which the most general is defined as follows:

Definition 1.1. A finite group G is a SR-group if it contains a subgroup P1 of order 3 satisfying:

• (a) A/S3-subgroup P2 of NGP1 is elementary of order 9;

• (b) NG(P2)/P2 acts semi-regularly by conjugation on the conjugates of P1 contained in P2.

We show that if is a frame and {ψＱ}Ｑ∈Q ∈ ∩ Mα(ℝn) is its dual frame (for the definition of Mα(ℝn), see Definition 2.1), where Q is the collection of dyadic cubes, then for any f ∈ S′(ℝn), there exists a sequence of polynomials, PL,L′,L″, such that

(0.1)

in the topology of S′(ℝn), where δ(i) = max(2i, 1). We prove this result by explicitly constructing the polynomials PL,L′,L″. Furthermore, using the above result, we assert that the linear span of the one-dimensional wavelet system is dense in a function space if and only if the dual space of this function space has an trivial intersection with the set of polynomials. This is proved by using the annihilator of the one-dimensional wavelet system.

If U=U(0), U(i) is a maximal subgroup of U(i + 1) and U(n) = V, then we term the U(i) a densest chain connecting U and V; and n is the length of this chain. The subgroup U of V will furthermore be termed an n-uniserial subgroup of V if there exists one and only one densest chain connecting U and V and if its length is n. The principal aim of this note is to give characterizations of dedekind groups [ = groups all of whose subgroups are normal] in terms of the normality of uniserial subgroups. We quote one of our results:

D. Mumford has shown an excelent algebralization of theory of theta constants and theta functions in his papers: On the equations defining abelian varieties I, II, III (Invent. Math. 1. 237-354 (1966), 3. 75-135 (1967), 3. 215-244) (1967). Our starting point and idea, however, are something different from those of Mumford; we begin our study at characterizing abelian addition formulae among all the possible addition formulae, and we want to give expressions to everything in words of matric notations.

We consider the moduli space of genus 4 curves endowed with a $g^1_3$ (which maps with degree 2 onto the moduli space of genus 4 curves). We prove that it defines a degree $\frac {1}{2}(3^{10}-1)$ cover of the nine-dimensional Deligne–Mostow ball quotient such that the natural divisors that live on that moduli space become totally geodesic (their normalizations are eight-dimensional ball quotients). This isomorphism differs from the one considered by S. Kondō, and its construction is perhaps more elementary, as it does not involve K3 surfaces and their Torelli theorem: the Deligne–Mostow ball quotient parametrizes certain cyclic covers of degree 6 of a projective line and we show how a level structure on such a cover produces a degree 3 cover of that line with the same discriminant, yielding a genus 4 curve endowed with a $g^1_3$.

We describe a general method for expanding a truncated $G$-iterative Hasse–Schmidt derivation, where $G$ is an algebraic group. We give examples of algebraic groups for which our method works.

Building on the embedding of an n-abelian category $\mathscr {M}$ into an abelian category $\mathcal {A}$ as an n-cluster-tilting subcategory of $\mathcal {A}$ , in this paper, we relate the n-torsion classes of $\mathscr {M}$ with the torsion classes of $\mathcal {A}$ . Indeed, we show that every n-torsion class in $\mathscr {M}$ is given by the intersection of a torsion class in $\mathcal {A}$ with $\mathscr {M}$ . Moreover, we show that every chain of n-torsion classes in the n-abelian category $\mathscr {M}$ induces a Harder–Narasimhan filtration for every object of $\mathscr {M}$ . We use the relation between $\mathscr {M}$ and $\mathcal {A}$ to show that every Harder–Narasimhan filtration induced by a chain of n-torsion classes in $\mathscr {M}$ can be induced by a chain of torsion classes in $\mathcal {A}$ . Furthermore, we show that n-torsion classes are preserved by Galois covering functors, thus we provide a way to systematically construct new (chains of) n-torsion classes.

In this paper we introduce and study Miyashita action in the context of monoidal categories aiming by this to provide a common framework of previous studies in the literature. We make a special emphasis of this action on Azumaya monoids. To this end, we develop the theory of invertible bimodules over different monoids (a sort of Morita contexts) in general monoidal categories as well as their corresponding Miyashita action. Roughly speaking, a Miyashita action is a homomorphism of groups from the group of all isomorphic classes of invertible subobjects of a given monoid to its group of automorphisms. In the symmetric case, we show that for certain Azumaya monoids, which are abundant in practice, the corresponding Miyashita action is always an isomorphism of groups. This generalizes Miyashita’s classical result and sheds light on other applications of geometric nature which cannot be treated using the classical theory. In order to illustrate our methods, we give a concrete application to the category of comodules over commutative (flat) Hopf algebroids. This obviously includes the special cases of split Hopf algebroids (action groupoids), which for instance cover the situation of the action of an affine algebraic group on an affine algebraic variety.

Let G be a finite group and ℒ(G) the lattice consisting of all subgroups of G. It is well known that ℒ(G) is distributive if and only if G is cyclic (cf. [2, p. 173]). Moreover, the classical result of Iwasawa [8] says that ℒ(G) is pure if and only if G is supersolvable. Here, a finite lattice is called pure if all of maximal chains in it have same length and a finite group G is called supersolvable if ℒ(G) has a maximal chain which consists of normal subgroups of G.