In this paper by using some tools from graph theory, mainly the theorem on ear decomposition of factor-critical graphs given by L. Lovász and the canonical decomposition of a graph given by Edmonds and Gallai, we can describe each irreducible component of powers of edge ideals of a graph. As an application we use our results to persistence of irreducible components, to bound dstab and to count the

In this paper we study the automorphism group of the quotient Nk• of a free group Γ of finite rank n by the (k+1)-term Γk+1• of the Zassenhaus (•=Z) and the Stallings (•=S) mod-p central series, with p a prime number. Just like in the already known case for the usual terms of the lower central series, these automorphism groups fit naturally into a short exact sequence containing the automorphism groups

We introduce two notions of algebraic entropy for actions of cancellative right amenable semigroups S on discrete abelian groups A by endomorphisms; these extend the classical algebraic entropy for endomorphisms of abelian groups, corresponding to the case S=N. We investigate the fundamental properties of the algebraic entropy and compute it in several examples, paying special attention to the case

We give new improved bounds for the dominant dimension of Nakayama algebras and use those bounds to give a classification of Nakayama algebras with n simple modules that are higher Auslander algebras with global dimension at least n. The classification is then used to extend the results on the inequality for the global dimension of Nakayama algebras obtained in [18].

In this paper we first state a conjecture on the lower bound of the maximal height of characters in a p-block of a finite group. Then we show that our conjecture holds for all blocks of covering groups of a sporadic simple group, for all blocks of a quasi-simple group G with G/Z(G) isomorphic to A6,A7 or a simple group of Lie type with an exceptional covering group, for all blocks of a symmetric group

We study actions of bosonizations of quantum linear spaces on quantum algebras. Under mild conditions, we classify actions on quantum affine spaces and quantum matrix algebras. In the former case, it is shown that all actions of generalized Taft algebras are trivial extensions of actions on quantum planes. In both cases we achieve bounds on the rank of the bosonization acting on the algebra.

In this paper, we introduce the notion of a noncommutative Poisson bialgebra, and establish the equivalence between matched pairs, Manin triples and noncommutative Poisson bialgebras. Using quasi-representations and the corresponding cohomology theory of noncommutative Poisson algebras, we study coboundary noncommutative Poisson bialgebras which leads to the introduction of the Poisson Yang-Baxter

In this article, we study the weak and strong Lefschetz properties, and the related notion of almost revlex ideal, in the non-Artinian case, proving that several results known in the Artinian case hold also in this more general setting. We then apply the obtained results to the study of the Jacobian algebra of hyperplane arrangements.

For any reductive group G and a parabolic subgroup P with its Levi subgroup L, the first author in [5] introduced a ring homomorphism ξλP:Repλ−polyC(L)→H⁎(G/P,C), where Repλ−polyC(L) is a certain subring of the complexified representation ring of L (depending upon the choice of an irreducible representation V(λ) of G with highest weight λ). In this paper we study this homomorphism for G=Sp(2n) and

The absolute Galois group of the cyclotomic field K=Q(ζp) acts on the étale homology of the Fermat curve X of exponent p. We study a Galois cohomology group which is valuable for measuring an obstruction for K-rational points on X. We analyze a 2-nilpotent extension of K which contains the information needed for measuring this obstruction. We determine a large subquotient of this Galois cohomology

In this paper, we prove certain algebraic identities, which correspond to differentiation of the shuffle relation, the stuffle relation, and the relations which arise from Möbius transformations of iterated integrals. These formulas provide fundamental and useful tools in the study of iterated integrals on a punctured projective line.

Let k(x)=k(x1,…,xn) be the rational function field, and k⊂L⊂k(x) an intermediate field. Hilbert's fourteenth problem asks whether the k-algebra L∩k[x1,…,xn] is finitely generated. Various counterexamples to this problem were already given, but there are still many open cases. In this paper, we study the problem in terms of the field-theoretic properties of L. Our result implies that, for any S⊂k[x1

Let k be an algebraically closed field, char(k)=p≥2 and Er be a elementary abelian p-group of rank r≥2. Let (c,d)∈N2. We show that there exists an indecomposable module of constant Jordan type [1]c[2]d and Loewy length 2 if and only if qΓr(d,d+c)≤1 and c≥r−1, where qΓr(x,y):=x2+y2−rxy denotes the Tits form of the generalized Kronecker quiver Γr. Since p>2 and constant Jordan type [1]c[2]d imply Loewy

We characterize Harbater-Katz-Gabber curves in terms of a family of cohomology classes satisfying a compatibility condition. Our construction is applied to the description of finite subgroups of the Nottingham Group.

Every word in a free group F induces a probability measure on every finite group in a natural manner. It is an open problem whether two words that induce the same measure on every finite group, necessarily belong to the same orbit of AutF. A special case of this problem, when one of the words is the primitive word x, was settled positively by the third author and Parzanchevski [PP15]. Here we extend

The main result of this paper states that fully inert subgroups of torsion-complete abelian p-groups are commensurable with fully invariant subgroups, which have a satisfactory characterization by a classical result by Kaplansky. As the proof of this fact relies on the analogous result for direct sums of cyclic p-groups, we provide revisited and simplified proofs of the fact that fully inert subgroups

We prove that there exists an integer-valued function f on positive integers such that if a finite group G has at most k real-valued irreducible characters, then |G/Sol(G)|≤f(k), where Sol(G) denotes the largest solvable normal subgroup of G. In the case k=5, we further classify G/Sol(G). This partly answers a question of Iwasaki [15] on the relationship between the structure of a finite group and

We introduce the notion of essential support of a simple Gelfand-Tsetlin gln-module as an attempt to understand the character formula of such module. This support detects the weights having maximal possible Gelfand-Tsetlin multiplicities. Using combinatorial tools we describe the essential supports of the simple socles of the universal tableaux modules. We also prove that every simple Verma module

We study finitely generated nilpotent groups G given by full rank finite presentations 〈A|R〉Nc in the variety Nc of nilpotent groups of class at most c, where c≥2. We prove that if the deficiency |A|−|R| is at least 2 then the group G is virtually free nilpotent, it is quasi finitely axiomatizable (in particular, first-order rigid), and it is almost (up to finite factors) directly indecomposable. One

A projective hypersurface X⊆Pn has defect if hi(X)≠hi(Pn) for some i∈{n,…,2n−2} in a suitable cohomology theory. This occurs for example when X⊆P4 is not Q-factorial. We show that hypersurfaces with defect tend to be very singular: In characteristic 0, we present a lower bound on the Tjurina number, where X is allowed to have arbitrary isolated singularities. For X with mild singularities, we prove

We introduce decomposition algebras as a natural generalization of axial algebras, Majorana algebras and the Griess algebra. They remedy three limitations of axial algebras: (1) They separate fusion laws from specific values in a field, thereby allowing repetition of eigenvalues; (2) They allow for decompositions that do not arise from multiplication by idempotents; (3) They admit a natural notion

We show that an elementary class of algebras is closed under the taking of homomorphic images and direct products if and only if the class consists of all algebras that satisfy a set of (generally simultaneous) equations. For classes of regular semigroups this allows an interpretation of a universal algebraic nature that is formulated entirely in terms of the associative binary operation of the semigroup

We show in this paper that representations of a finite product of categories satisfying certain combinatorial conditions have finite Castelnuovo-Mumford regularity if and only if they are presented in finite degrees, and hence the category consisting of them is abelian. These results apply to examples such as the categories FIm and FIGm.

The notion of 2-almost Gorenstein local ring (2-AGL ring for short) is a generalization of the notion of almost Gorenstein local ring from the point of view of Sally modules of canonical ideals. In this paper, for further developments of the theory, we discuss three different topics on 2-AGL rings. The first one is to clarify the structure of minimal presentations of canonical ideals, and the second

We study the Cantor–Bendixson rank of the space of subgroups for members of a general class of finitely generated self-replicating branch groups. In particular, we show for G either the Grigorchuk group or the Gupta–Sidki 3 group, the Cantor–Bendixson rank of Sub(G) is ω. For each natural number n, we additionally characterize the subgroups of rank n and give a description of subgroups in the perfect

Let ℓ be a commutative ring with unit. Garkusha constructed a functor from the category of ℓ-algebras into a triangulated category D, that is a universal excisive and homotopy invariant homology theory. Later on, he provided different descriptions of D, as an application of his motivic homotopy theory of algebras. Using these, it can be shown that D is triangulated equivalent to a category, denote

In this paper we construct a new “pro-p-complete” topological Kac-Moody group and compare it to various known topological Kac-Moody groups. We come across this group by investigating the process of completion of groups with BN-pairs. We would like to know whether the completion of such a group admits a BN-pair. We give explicit criteria for this to happen.

We determine a sharp lower bound for the Hilbert function in degree d of a monomial algebra failing the weak Lefschetz property over a polynomial ring with n variables and generated in degree d, for any d≥2 and n≥3. We consider artinian ideals in the polynomial ring with n variables generated by homogeneous polynomials of degree d invariant under an action of the cyclic group Z/dZ, for any n≥3 and

We study and classify the 3-dimensional Hom-Lie algebras over C. We provide first a complete set of representatives for the isomorphism classes of skew-symmetric bilinear products defined on a 3-dimensional complex vector space g. The well known Lie brackets for the 3-dimensional Lie algebras are included into appropriate isomorphism classes of such products representatives. For each product representative

Let Λ be a 1-Auslander-Gorenstein algebra. We give a necessary and sufficient condition for Λ to be a tilted algebra.

We define Θ-Hilbertianity which generalizes Hilbertianity and show that the absolute Galois group of a countable Θ-Hilbertian PAC field is an appropriate analogue of a free profinite group, thus generalizing a result of Fried-Völklein-Pop.

This paper contributes to the study of the core of ideals in Noetherian rings. We use techniques and objects from multiplicative ideal theory to develop explicit formulas for the core in various classes of one-dimensional Noetherian domains. All results are illustrated with original examples, where we explicitly compute the core.

Let C⊂Pr be a general curve of genus g embedded via a general linear series of degree d. The Maximal Rank Conjecture asserts that the restriction maps H0(OPr(m))→H0(OC(m)) are of maximal rank; this determines the Hilbert function of C. In this paper, we prove an analogous statement for the union of hyperplane sections of general curves. More specifically, if H⊂Pr is a general hyperplane, and C1,C2

Nearly 60 years ago, László Fuchs posed the problem of determining which groups can be realized as the group of units of a commutative ring. To date, the question remains open, although significant progress has been made. Along this line, one could also ask the more general question as to which finite groups can be realized as the group of units of a finite ring. In this paper, we consider the question

In this article we further develop the theory of valuation independence and study its relation with classical notions in valuation theory such as immediate and defectless extensions. We use this general theory to settle two open questions regarding vector space defectless extensions of valued fields. Additionally, we provide a characterization of such extensions within various classes of valued fields

In this paper, we investigate the problem of the simultaneous generation of r-jets of the bundle ωX⊗L⊗m, where L is an ample and globally generated line bundle on X. The results work in arbitrary characteristics. A similar problem is studied when X admits singularities.

The notions of almost centralizer and almost commutator are introduced and basic properties are established, such as the three subgroup lemma. These notions are used to study M˜c-groups introduced here, that is, groups for which, in every definable sections, every descending chain of centralizers each having infinite index in its predecessor stabilizes after finitely many steps. The Fitting subgroup

We study properties of Engel elements in weakly branch groups, lying in the group of automorphisms of a spherically homogeneous rooted tree. More precisely, we prove that the set of bounded left Engel elements is always trivial in weakly branch groups. In the case of branch groups, the existence of non-trivial left Engel elements implies that these are all p-elements and that the group is virtually

The notions of stable and Morse subgroups of finitely generated groups generalize the concept of a quasiconvex subgroup of a word-hyperbolic group. For a word-hyperbolic group G, Kapovich [31] provided a partial algorithm which, on input a finite set S of G, halts if S generates a quasiconvex subgroup of G and runs forever otherwise. In this paper, we give various detection and decidability algorithms

Lawrence-Krammer representations are an important family of linear representations of Artin-Tits groups of small type, which are known, under some assumptions on the parameters, to be faithful when the type is spherical (or more generally when they are restricted to the Artin-Tits monoid) and irreducible when the type is connected. Here, we investigate an analogue of these representations — introduced

The glider Brauer-Severi variety GBS(A) of a central simple algebra A over a field K is introduced as the set of all irreducible left glider ideals in A for some filtration FA. For fields we deduce that GBS(K) equals R(K)×Z, the product of the Riemann surface of K and Z. For a csa A over K it turns out that GBS(A)=BS(A)×GBS(K), where BS(A) denotes the classical Brauer-Severi variety of A.

We completely characterize the bigraded resolutions of Ulrich bundles of arbitrary rank on the Segre fourfold P2×P2. We characterize the Ulrich bundles V of arbitrary rank on P2×P2 with h1(V⊗Ω⊠Ω)=0 or with h1(V⊗Ω(−1)⊠Ω(−1))=0 or obtained as pullback from P2 and we construct more complicated examples.

We show that an automaton group or semigroup is infinite if and only if it admits an ω-word (i.e. a right-infinite word) with an infinite orbit, which solves an open problem communicated to us by Ievgen V. Bondarenko. In fact, we prove a generalization of this result, which can be applied to show that finitely generated subgroups and subsemigroups as well as principal left ideals of automaton semigroups

If a Nakayama algebra is not cyclic, it has finite global dimension. For a cyclic Nakayama algebra, there are many characterizations of when it has finite global dimension. In [She17], Shen gave such a characterization using Ringel's resolution quiver. In [IZ92], the second author, with Zacharia, gave a cyclic homology characterization for when a monomial relation algebra has finite global dimension

We present a Las Vegas polynomial-time algorithm that takes as input a subgroup of GL(d,Fq) and, subject to the existence of certain oracles, determines its composition factors, provided that none of those factors is isomorphic to one of B22(22k+1), F42(22k+1), D43(2k), or G22(32k+1), for any k.

We studied both the double cross product and the bicrossproduct constructions for the Hom-Hopf algebras of general (α,β)-type. This allows us to consider the universal enveloping Hom-Hopf algebras of Hom-Lie algebras, which are of (α,Id)-type. We show that the universal enveloping Hom-Hopf algebras of a matched pair of Hom-Lie algebras form a matched pair of Hom-Hopf algebras. We observe also that

We show that the number of lines in an m–homogeneous supersolvable line arrangement is upper bounded by 3m−3 and we classify the m–homogeneous supersolvable line arrangements with two modular points up-to lattice-isotopy. We also prove the nonexistence of unexpected curves for supersolvable line arrangements obtained as cones over generic line arrangements, or cones over arbitrary line arrangements

Given a quasi-compact and quasi-separated (qcqs) scheme X of finite type over a Noetherian ring R having an ample family of line bundles, we construct a closed embedding of X into a smooth (qcqs) scheme over R having an ample family of line bundles. Such a smooth scheme arises as an open subscheme of the multihomogeneous Proj of a Zn-graded ring.

By a φ-variety V, we mean a supervariety or a ⁎-variety generated by an associative algebra over a field F of characteristic zero. In this case, we consider its sequence of φ-codimensions cnφ(V) and say that V is minimal of polynomial growth nk if cnφ(V) grows like nk, but any proper φ-subvariety grows like nt with t

For a finite irreducible subgroup H⊂PSL(Cn) and an irreducible, H-invariant curve Z⊂P(Cn) such that C(Z)H=C(t), a standard differential operator Lst∈C(t)[ddt] is constructed. For n=2 this is essentially Klein's work. For n>2 an actual calculation of Lst is done by computing an evaluation of invariants C[X1,…,Xn]H→C(t) and applying a scalar form of a theorem of E. Compoint in a “Procedure”. Also in

We extend the computations in [AGM02], [AGM08], [AGM10] to find the cohomology in degree five of a congruence subgroup of SL4(Z) with coefficients in a field twisted by a nebentype character, along with the action of the Hecke algebra on the cohomology. This is the top cuspidal degree. For each Hecke eigenclass we find, we produce the unique Galois representation that appears to be attached to it.

We present a practical algorithm which, given a non-archimedean local field K and any two elements A,B∈SL2(K), determines after finitely many steps whether or not the subgroup 〈A,B〉≤SL2(K) is discrete and free of rank two. This makes use of the Ping Pong Lemma applied to the action of SL2(K) by isometries on its Bruhat-Tits tree. The algorithm itself can also be used for two-generated subgroups of

We prove that the word problem of the Brin-Thompson group nV over a finite generating set is coNP-complete for every n≥2. It is known that {nV:n≥1} is an infinite family of infinite, finitely presented, simple groups. We also prove that the word problem of the Thompson group V over a certain infinite set of generators, related to boolean circuits, is coNP-complete.

In this paper we investigate the image of the center Z of the distribution algebra Dist(GL(m|n)) of the general linear supergroup over a ground field of positive characteristic under the Harish-Chandra morphism h:Z→Dist(T) obtained by the restriction of the natural map Dist(GL(m|n))→Dist(T). We define supersymmetric elements in Dist(T) and show that each image h(c) for c∈Z is supersymmetric. The central

In this paper characters of the normalizer of d-split Levi subgroups in SLn(q) and SUn(q) are parametrized with a particular focus on the Clifford theory between the Levi subgroup and its normalizer. These results are applied to verify the Alperin-McKay conjecture for primes ℓ with ℓ∤6(q2−1) and the Alperin weight conjecture for ℓ-blocks of those quasi-simple groups with abelian defect. The inductive

We show that a compactly generated locally compact group of polynomial growth having no non-trivial compact normal subgroups can be embedded as a co-compact subgroup into a semidirect product of a connected, simply connected, nilpotent Lie group and a compact group. There is also a uniqueness statement for this extension.

We introduce the notion of a matching Rota-Baxter algebra motivated by the recent work on multiple pre-Lie algebras arising from the study of algebraic renormalization of regularity structures [10], [19]. This notion is also related to iterated integrals with multiple kernels and solutions of the associative polarized Yang-Baxter equation. Generalizing the natural connection of Rota-Baxter algebras

For a finite group G, the Hurwitz space Hr,gin(G) is the space of genus g covers of the Riemann sphere with r branch points and the monodromy group G. In this paper, we give a complete list of primitive genus one systems of affine type. That is, we assume that G is a primitive group of affine type. Under this assumption we determine the braid orbits on the suitable Nielsen classes, which is equivalent

We continue the project started in [8] to describe the structure of the finite groups G of characteristic p in terms of their Baumann components and the conjugacy class Baup(G). The reduction theorem proved in [8] allows to assume that G has a unique Baumann component. In this paper we use this property to determine the isomorphism type of G/Op(G) and the action of G on Ω1(Z(Op(G))). In addition, we

We define and study cyclotomic quotients of affine Hecke algebras of type D. We establish an isomorphism between (direct sums of blocks of) these cyclotomic quotients and a generalisation of cyclotomic quiver Hecke algebras which are a family of Z-graded algebras closely related to algebras introduced by Shan, Varagnolo and Vasserot. To achieve this, we first complete the study of cyclotomic quotients