Journal of Pure and Applied Algebra最新文献

New presentations of Specht modules of symmetric groups over fields of characteristic zero have been obtained by Brauner, Friedmann, Hanlon, Stanley and Wachs. These involve generators that are column tabloids and relations that are Garnir relations with maximal number of exchanges between consecutive columns or symmetrization of Garnir relations with minimal number of exchanges between consecutive columns. In this paper, we examine Garnir relations and their symmetrization with any number of exchanges. In both cases, we provide sufficient arithmetic conditions so that the corresponding quotient is a Specht module. In particular, in the first case this yields new presentations of Specht modules if the parts of the conjugate partition that correspond to maximal number of exchanges greater than 1 are distinct. These results generalize the presentations mentioned above and offer an answer to a question of Friedmann, Hanlon and Wachs. Our approach is via representations of the general linear group.

We show that if a countably generated Lie algebra H does not contain isomorphic copies of certain finite-dimensional nilpotent Lie algebras A and B (satisfying some mild conditions), then H embeds into a quotient of $A⁎B$ that is at the same time hopfian and cohopfian. This is a Lie algebraic version of an embedding theorem proved by C. Miller and P. Schupp for groups. We also prove that any finitely presentable Lie algebra is the quotient of a finitely presented, centerless, residually nilpotent and SQ-universal Lie algebra of cohomological dimension at most 2 by an ideal that can be generated by two elements as a Lie subalgebra. This is reminiscent of the Rips construction in group theory. In both results we use the theory of Gröbner-Shirshov bases.

This article studies the reduction maps in the higher K-theory of the ring of integers in a number field arising from the canonical reduction maps at nonzero prime ideals. It proves an explicit density estimate for the subset of primes where the images of a fixed collection of elements vanish. Our result applies to a collection of elements possibly having different degrees and suggests that the linearly independent elements of global K-theory exhibit mutually independent reduction patterns. We also relate the reduction map in K-theory to the reduction map in stable cohomology of general linear groups. This connection allows us to examine the pullback of Quillen's e-classes in the cohomology of the stable general linear group over a finite field. During the proof of the main result, we construct the smallest Galois extension which trivializes a Galois cohomology class of degree one, and show that the linear independence of classes results in disjointness of corresponding field extensions.

In this paper, we give a characterization of locally nilpotent derivations on $A^2$-fibrations having kernels isomorphic to $A^1$-fibrations over Noetherian normal domains containing $Q$.

Recently, Wang and the second author constructed a bar involution and canonical basis for a quasi-permutation module of the Hecke algebra associated to a type B Weyl group W, where the basis is parameterized by left cosets of a quasi-parabolic reflection subgroup in W. In this paper we provide an alternative approach to these constructions, and then generalize these constructions to Coxeter groups which contain a product of type B Weyl groups as a parabolic subgroup.

We prove that for every prime p algebraically clean graphs of groups are virtually residually p-finite and cohomologically p-complete. We also prove that they are cohomologically good. We apply this to certain 2-dimensional Artin groups.

The use of persistent homology in applications is justified by the validity of certain stability results. At the core of such results is a notion of distance between the invariants that one associates with data sets. Here we introduce a general framework to compare distances and invariants in multiparameter persistence, where there is no natural choice of invariants and distances between them. We define amplitudes, monotone, and subadditive invariants that arise from assigning a non-negative real number to objects of an abelian category. We then present different ways to associate distances to such invariants, and we provide a classification of classes of amplitudes relevant to topological data analysis. In addition, we study the relationships as well as the discriminative power of such amplitude distances arising in topological data analysis scenarios.

The Chermak–Delgado lattice of a finite group is a modular, self-dual sublattice of the lattice of subgroups. We prove that the Chermak–Delgado lattice of a central product contains the product of the Chermak–Delgado lattices of the relevant central factors. Furthermore, we obtain information about heights of elements in the Chermak–Delgado lattice relative to their heights in the Chermak–Delgado lattices of central factors. We also explore how the central product can be used as a tool in investigating Chermak–Delgado lattices.

We establish an embedding from the Hecke algebra associated with the edge contraction of a Coxeter system along an edge to the Hecke algebra associated with the original Coxeter system.

In the representation-theoretic study of finite dimensional associative algebras over an algebraically closed field, Brenner introduced certain partially ordered sets known as hammocks to encode factorizations of maps between indecomposable finitely generated modules. In the context of domestic string algebras, Schröer introduced a simpler version of hammocks in his doctoral thesis that are bounded discrete linear orders. In this paper, we characterize the class of order types(=order isomorphism classes) of hammock linear orders for domestic string algebras as the bounded discrete ones amongst the class $L{O}_{\mathrm{fp}}$ of finitely presented linear orders–the smallest class of linear orders containing finite linear orders as well as ω, and that is closed under isomorphisms, order reversal, finite order sums and antilexicographic products.

In fact, we provide a multi-step algorithm to compute the order type of any closed interval in the hammock, and prove the correctness of this algorithm. A major step of this algorithm is the construction of a variation, which we call the arch bridge quiver, of a finite combinatorial gadget called the bridge quiver introduced by Schröer. He utilised the graph-theoretic properties of the bridge quiver for the computation of some representation-theoretic numerical invariants of domestic string algebras. The vertices of the bridge quiver are (representatives of cyclic permutations of) bands and its arrows are certain band-free strings. There is a natural but ill-behaved partial binary operation, ∘, on a superset of the set of bridges consisting of weak bridges such that bridges are precisely the ∘-irreducibles. We equip an even larger yet finite set of weak arch bridges with another partial binary operation, ${\circ }_H$, to obtain a finite category. The binary operation ${\circ }_H$ uses isomorphisms between hammocks and explicitly relies on the description of the domestic string algebra as a bound quiver algebra. Each weak arch bridge admits a unique ${\circ }_H$-factorization into arch bridges, i.e., the ${\circ }_H$-irreducibles.