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In this paper the authors consider four questions of primary interest for the representation theory of reductive algebraic groups: (i) Donkin's Tilting Module Conjecture, (ii) the Humphreys-Verma Question, (iii) whether St r ⊗ L (λ) is a tilting module for L (λ) an irreducible representation of p r -restricted highest weight, and (iv) whether Ext G 1 1 (L (λ) , L (μ)) (− 1) is a tilting module where L (λ) and L (μ) have p -restricted highest weight. The authors establish affirmative answers to each of these questions with a new uniform bound, namely p ≥ 2 h − 4 where h is the Coxeter number. Notably, this verifies these statements for infinitely many more cases. Later in the paper, questions (i)-(iv) are considered for rank two groups where there are counterexamples (for small primes) to these questions. [ABSTRACT FROM AUTHOR]

The G -representation variety R G (Σ g) parametrizes the representations of the fundamental group π 1 (Σ g) of a closed surface into an algebraic group G. For G the groups of n × n upper triangular matrices or unipotent upper triangular matrices, we compare two methods for computing algebraic invariants of R G (Σ g). Using the geometric method initiated by González-Prieto, Logares and Muñoz, based on a Topological Quantum Field Theory (TQFT), we compute the virtual classes of R G (Σ g) in the Grothendieck ring of varieties for n = 1 , ... , 5. Introducing the notion of algebraic representatives we are able to efficiently compute the TQFT. Using the arithmetic method initiated by Hausel and Rodriguez-Villegas, we compute the E -polynomials of R G (Σ g) for n = 1 , ... , 10. For both methods, we describe how the computations can be performed algorithmically. Furthermore, we discuss how the representation varieties for the groups of unipotent upper triangular matrices are related to Higman's conjecture. The computations of this paper can be seen as positive evidence towards a generalized motivic version of the conjecture. [ABSTRACT FROM AUTHOR]

An LR-structure is a tetravalent vertex-transitive graph together with a special type of a decomposition of its edge-set into cycles. LR-structures were introduced in a paper by P. Potočnik and S. Wilson (2014) [12] , as a tool to study tetravalent semisymmetric graphs of girth 4. In this paper, we use the methods of group amalgams to resolve some problems left open in the above-mentioned paper. [ABSTRACT FROM AUTHOR]

In our previous work with Grifo and Hà, we showed the stable Harbourne-Huneke containment and Chudnovsky's conjecture for the defining ideal of sufficiently many general points in P N. In this paper, we establish the conjectures for all remaining cases, and hence, give the affirmative answer to Harbourne-Huneke containment and Chudnovsky's conjecture for the general points in P N for all N. Our new technique is to develop the Cremona reduction process that provides effective lower bounds for the Waldschmidt constant of the defining ideals of generic points in projective spaces. [ABSTRACT FROM AUTHOR]

Given a finite group G and an irreducible complex character χ of G , the codegree of χ is defined as the integer cod (χ) = | G : ker (χ) | / χ (1). It was conjectured by G. Qian in [16] that, for every element g of G , there exists an irreducible character χ of G such that cod (χ) is a multiple of the order of g ; the conjecture has been verified under the assumption that G is solvable ([16]) or almost-simple ([13]). In this paper, we prove that Qian's conjecture is true for every finite group whose Fitting subgroup is trivial, and we show that the analysis of the full conjecture can be reduced to groups having a solvable socle. [ABSTRACT FROM AUTHOR]

The aim of this paper is to prove a conjecture made by T. Fukaya in 2008. This conjecture concerns the exact value of the Z 2 -cup-length of the Grassmann manifold G ˜ n , 3 of oriented 3-planes in R n. Along the way, we calculate the heights of the Stiefel–Whitney classes of the canonical vector bundle over G ˜ n , 3. [ABSTRACT FROM AUTHOR]

In this paper we show that the Poisson analogue of the Noether's Problem has a positive solution for essentially all symplectic reflection groups — the analogue of complex reflection groups in the symplectic world. Our proofs are constructive, and generalize and refine previously known results. The results of this paper can be thought as analogues of the Noncommutative Noether Problem and the Gelfand-Kirillov Conjecture for rational Cherednik algebras in the quasi-classical limit. An abstract framework to understand these results is introduced. As a consequence for complex reflection groups, we obtain the Poisson rationality of the Calogero-Moser spaces associated to any of them, and we verify the Gelfand-Kirillov Conjecture for trigonometric Cherednik algebras and the Poisson rationality of their corresponding Calogero-Moser spaces. [ABSTRACT FROM AUTHOR]

We prove the conjecture that exotic and block-exotic fusion systems coincide holds for all fusion systems on exceptional p -groups of maximal nilpotency class, where p ≥ 5. This is done by considering a family of exotic fusion systems discovered by Parker and Stroth. Together with a previous result by the author, which we also generalise in this paper, and a result by Grazian and Parker this implies the conjecture for fusion systems on such groups. Considering small primes, there are no exotic fusion systems on 2-groups of maximal class and for p = 3 , we prove block-exoticity of two exotic fusion systems described by Diaz–Ruiz–Viruel. [ABSTRACT FROM AUTHOR]

A combinatorial property that characterizes Cohen-Macaulay binomial edge ideals has long been elusive. A recent conjecture ties the Cohen-Macaulayness of a binomial edge ideal J G to special disconnecting sets of vertices of its underlying graph G , called cut sets. More precisely, the conjecture states that J G is Cohen-Macaulay if and only if J G is unmixed and the collection of the cut sets of G is an accessible set system. In this paper we prove the conjecture theoretically for all graphs with up to 12 vertices and develop an algorithm that allows to computationally check the conjecture for all graphs with up to 15 vertices and all blocks with whiskers where the block has at most 11 vertices. This significantly extends previous computational results. [ABSTRACT FROM AUTHOR]

In the paper, we give an affirmative answer to the conjecture in [1]. We prove that a Shamsuddin derivation D is simple if and only if Aut (K [ x , y 1 , ... , y n ]) D = { i d }. In addition, we calculate the isotropy groups of the Shamsuddin derivations D = ∂ x + ∑ j = 1 r (a (x) y j + b j (x)) ∂ j of K [ x , y 1 , ... , y r ]. We also prove that Im D is a Mathieu-Zhao subspace if and only if a (x) ∈ K. [ABSTRACT FROM AUTHOR]

Berenstein and Zelevinsky introduced quantum cluster algebras [3] and the triangular bases [4]. The support conjecture in [12] asserts that the support of a triangular basis element for a rank-2 cluster algebra is bounded by an explicitly described region that is possibly concave. In this paper, we prove the support conjecture for all skew-symmetric rank-2 cluster algebras. [ABSTRACT FROM AUTHOR]

A conjecture raised by Cossey in 2007 asserts that if G is a finite p -solvable group and φ is an irreducible p -Brauer character of G with vertex Q , then the number of lifts of φ is at most | Q : Q ′ |. This conjecture is now known to be true in several situations for p odd, but there has been little progress for p even. The main obstacle appeared in characteristic two is that all the vertex pairs of a lift might be neither linear nor conjugate. In this paper we show that if χ is a lift of an irreducible 2-Brauer character in a solvable group, then χ has a linear Navarro vertex if and only if all the vertex pairs of χ are linear, and in that case all of the twisted vertices of χ are conjugate. Our result can also be used to study other lifting problems of Brauer characters in characteristic two. As an application, we prove a weaker form of Cossey's conjecture for p = 2 "one vertex at a time". [ABSTRACT FROM AUTHOR]

In this article we study semi-abelian analogues of Schanuel conjecture. As showed by the first author, Schanuel conjecture is equivalent to the Generalized Period conjecture applied to 1-motives without abelian part. Extending her methods, the second, the third and the fourth authors have introduced the abelian analogue of Schanuel conjecture as the Generalized Period conjecture applied to 1-motives without toric part. As a first result of this paper, we define the semi-abelian analogue of Schanuel conjecture as the Generalized Period conjecture applied to 1-motives. C. Cheng et al. proved that Schanuel conjecture implies the algebraic independence of the values of the iterated exponential and the values of the iterated logarithm, answering a question of M. Waldschmidt. The second, the third and the fourth authors have investigated a similar question in the setup of abelian varieties: the Weak Abelian Schanuel conjecture implies the algebraic independence of the values of the iterated abelian exponential and the values of an iterated generalized abelian logarithm. The main result of this paper is that a Relative Semi-abelian conjecture implies the algebraic independence of the values of the iterated semi-abelian exponential and the values of an iterated generalized semi-abelian logarithm. [ABSTRACT FROM AUTHOR]

This paper presents an E -derivation analogue of a result on derivations due to van den Essen, Wright and Zhao. We prove that the image of a locally finite K - E -derivation of polynomial algebras in two variables over a field K of characteristic zero is a Mathieu subspace. This result together with that of van den Essen, Wright and Zhao confirms the LFED conjecture in the case of polynomial algebras in two variables. [ABSTRACT FROM AUTHOR]

In this paper, we present some partial results for the geometrically m -step solvable Grothendieck conjecture in anabelian geometry. Among other things, we prove the geometrically 3-step solvable Grothendieck conjecture for genus 0 curves over fields finitely generated over the prime field of arbitrary characteristic. [ABSTRACT FROM AUTHOR]

In this paper, we consider the homogeneous coordinate rings A (Y n , d) ≅ K [ Ω n , d ] of monomial projections Y n , d of Veronese varieties parameterized by subsets Ω n , d of monomials of degree d in n + 1 variables where: (1) Ω n , d contains all monomials supported in at most s variables and, (2) Ω n , d is a set of monomial invariants of a finite diagonal abelian group G ⊂ GL (n + 1 , K) of order d. Our goal is to study when K [ Ω n , d ] is a quadratic algebra and, if so, when K [ Ω n , d ] is Koszul or G-quadratic. For the family (1), we prove that K [ Ω n , d ] is quadratic when s ≥ ⌈ n + 2 2 ⌉. For the family (2), we completely characterize when K [ Ω 2 , d ] is quadratic in terms of the group G ⊂ GL (3 , K) , and we prove that K [ Ω 2 , d ] is quadratic if and only if it is Koszul. We also provide large families of examples where K [ Ω n , d ] is G-quadratic. [ABSTRACT FROM AUTHOR]

Blickle, Mustaţă and Smith proposed two conjectures on the limits of F -pure thresholds. One conjecture asks whether or not the limit of a sequence of F -pure thresholds of principal ideals on regular local rings of fixed dimension can be written as an F -pure thresshold in lower dimension. Another conjecture predicts that any F -pure threshold of a formal power series can be written as the F -pure threshold of a polynomial. In this paper, we prove that the first conjecture has a counterexample but a weaker statement still holds. We also give a partial affirmative answer to the second conjecture. [ABSTRACT FROM AUTHOR]

Let Γ be a connected (X , 2) -arc-transitive bipartite graph with bi-parts Δ 0 and Δ 1 , where X ≤ Aut (Γ). Let X + be the subgroup of X fixing Δ 0 setwise. In this paper, we first prove that if X + is primitive and faithful on Δ 0 and Δ 1 , then the actions of X + on Δ 0 and Δ 1 are primitive of the same type X with X ∈ { HA , SA , PA }. This is then used to prove that if X + is quasiprimitive on Δ 0 of type HA or TW, then either soc (X +) ⊴ Aut (Γ) , or Γ ≅ K q , q (q a prime power) or Γ ≅ K 2 r , 2 r − 2 r K 2 (r ≥ 2). This confirms a conjecture of C. H. Li from 2008. [ABSTRACT FROM AUTHOR]

Let G = O p ′ ( G ¯ F) be a finite simple group of Lie type defined over a field of characteristic p , where F is a Steinberg endomorphism of the ambient simple algebraic group G ¯. Let T ¯ be an F -stable maximal torus of G ¯ and set N = N G (T ¯). A conjecture due to Vdovin asserts that if G ≇ L 3 (2) then N ∩ N x is a p -group for some x ∈ G. In this paper, we use a combination of probabilistic and computational methods to calculate the base size for the natural action of G on G / N , which allows us to prove a stronger, and suitably modified, version of Vdovin's conjecture. [ABSTRACT FROM AUTHOR]

In this paper, we introduce new invariants to a singularity (V , 0) , i.e., the derivation Lie algebras L k (V) of the higher Nash blow-up local algebra M k (V). A new conjecture about the non-existence of negative weighted derivations of L k (V) for weighted homogeneous isolated hypersurface singularities is proposed. We verify this conjecture partially. Moreover, we compute the Lie algebra L 2 (V) for binomial isolated singularities. We also formulate a sharp upper estimate conjecture for the dimension of L k (V) for weighted homogeneous isolated hypersurface singularities and verify this conjecture for a large class of singularities. [ABSTRACT FROM AUTHOR]

This paper is a continuation of our previous paper in Li and Zhang (2018) [13]. We prove that if (n , q − 1) = 1 , PSL n (q) satisfies the inductive blockwise Alperin weight condition for any prime ℓ different from the defining characteristic. [ABSTRACT FROM AUTHOR]

This paper proves Lee-Lee's conjecture that establishes a coincidence between the set of associated roots of non-self-intersecting curves in an n -punctured disc and the set of real Schur roots of acyclic (valued) quivers with n vertices. [ABSTRACT FROM AUTHOR]

By the reduction theorems of Navarro–Tiep and Späth, a way to prove the Alperin weight conjecture and its blockwise version is to verify the so-called inductive Alperin weight condition and inductive blockwise Alperin weight condition for all finite simple groups respectively. In this paper, we establish the inductive blockwise Alperin weight condition for simple groups of type B and odd primes, using a criterion given by Brough and Späth recently. [ABSTRACT FROM AUTHOR]

In this paper, we study higher dimensional anabelian geometry. We prove the Grothendieck conjecture for certain types of hyperbolic polycurves (i.e., algebraic varieties in the form of successive fibrations by hyperbolic curves) over a sub- p -adic field (i.e., a field isomorphic to a subfield of a field finitely generated over the p -adic field Q p) for some prime number p. Among other things, we prove the Grothendieck conjecture for hyperbolic polycurves obtained as certain types of finite étale covers of the moduli space of curves of genus two with ordered r marked points over a sub- p -adic field. [ABSTRACT FROM AUTHOR]

Let G be a finite group. A subgroup Q of G is defined, by Glauberman, to be a centrally large subgroup , or CL- subgroup , of G if | Q | ⋅ | Z (Q) | ⩾ | Q ⁎ | ⋅ | Z (Q ⁎) | for every subgroup Q ⁎ of G. A CL-subgroup of G that is minimal under inclusion is called a minimal CL -subgroup of G. Glauberman (2006) [10] studied some properties of CL-subgroups, especially minimal CL-subgroups. He showed that if Q 1 and Q 2 are a pair of minimal CL-subgroups, then (⁎) Q i = (Q 1 ∩ Q 2) Z (Q i) for i = 1 , 2. We say that two CL-subgroups Q 1 and Q 2 (not necessarily minimal) are twisted CL -subgroups if they satisfy the above property (⁎). In this paper, we extend Glauberman's work to prove some further properties of CL-subgroups, especially twisted CL-subgroups. In addition, we give a negative answer to a conjecture proposed by Scoppola et al. (2018) [18]. [ABSTRACT FROM AUTHOR]

Abstract Let G be an infinite group and let X be a finite generating set for G such that the growth series of G with respect to X is a rational function; in this case G is said to have rational growth with respect to X. In this paper a result on sizes of spheres (or balls) in the Cayley graph Γ (G , X) is obtained: namely, the size of the sphere of radius n is bounded above and below by positive constant multiples of n α λ n for some integer α ≥ 0 and some λ ≥ 1. As an application of this result, a calculation of degree of commutativity (d. c.) is provided: for a finite group F , its d. c. is defined as the probability that two randomly chosen elements in F commute, and Antolín, Martino and Ventura have recently generalised this concept to all finitely generated groups. It has been conjectured that the d. c. of a group G of exponential growth is zero. This paper verifies the conjecture (for certain generating sets) when G is a right-angled Artin group or, more generally, a graph product of groups of rational growth in which centralisers of non-trivial elements are "uniformly small". [ABSTRACT FROM AUTHOR]

Abstract The aim of this paper is to present some contributions to the theory of finite transformation monoids. The dominating influence that permutation groups have on transformation monoids is used to describe and characterise transitive transformation monoids and primitive transitive transformation monoids. We develop a theory that not only includes the analogs of several important theorems of the classical theory of permutation groups but also contains substantial information about the algebraic structure of the transformation monoids. Open questions naturally arising from the substantial paper of Steinberg (2010) [11] have been answered. Our results can also be considered as a further development in the hunt for a solution of the Černý conjecture. [ABSTRACT FROM AUTHOR]

Generalised quadrangles, as a special case of generalised polygons, were introduced by Jacques Tits [14] in 1959. About twenty years later Kantor [7] showed that an elation generalised quadrangle is completely characterized by its so-called 4-gonal family or Kantor family. Motivated by the Ahrens and Szekeres Quadrangle [1] , Ghinelli [4] established a variation of the procedure of Kantor [7] , by introducing the notion of an AS -configuration of order n. Such a configuration gives rise to a skew-translation generalised quadrangle of order (n , n) , and conversely, outlined in J. Bamberg, S.P. Glasby, E. Swartz [2]. The only known groups of even order admitting an AS-configuration are elementary abelian 2-groups. Moreover, Payne [8] conjectured that there are no other examples. Indeed, he found a proof for n = 4. Moreover it is also true for n = 8 , compare J. Bamberg, S.P. Glasby, E. Swartz [2]. The purpose of this paper is to prove Theorem A group of even order admitting an AS-configuration of order n is elementary abelian, if n is not a square. In other words, the only skew-translation generalised quadrangles of order (n , n) , n ≡ 0 (mod 2) , are translation generalised quadrangles, if n is not a square. K. Thas claims in his preprint [12] that he found among other things a proof of Payne's conjecture. Unfortunately, in the main part of his proof there is a gap not easy to fill, [13]. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the properties of inner and outer projections with a view toward the Eisenbud-Goto regularity conjecture or the characterization of varieties satisfying certain extremal conditions. For example, if X is a quadratic scheme, the depth and regularity X and those of its inner projection from a smooth point are equal. In general, the above equalities do not hold for non-quadratic schemes. Therefore it is natural to investigate the algebraic invariants (e.g., depth and regularity) of X and its projected image in general. We develop a framework which provides partial answers and explains their relations using the partial elimination ideal theory. Our main theorems recover several preceding results in the literature. We also give some interesting examples and applications to illustrate our results. [ABSTRACT FROM AUTHOR]

For an irreducible complex character χ of the finite group G , let π (χ) denote the set of prime divisors of the degree χ (1) of χ. Denote then by ρ (G) the union of all the sets π (χ) and by σ (G) the largest value of | π (χ) | , as χ runs in Irr (G). The ρ-σ conjecture , formulated by Bertram Huppert in the 80's, predicts that | ρ (G) | ≤ 3 σ (G) always holds, whereas | ρ (G) | ≤ 2 σ (G) holds if G is solvable; moreover, O. Manz and T.R. Wolf proposed a "strengthened" form of the conjecture in the general case, asking whether | ρ (G) | ≤ 2 σ (G) + 1 is true for every finite group G. In this paper we study the strengthened ρ - σ conjecture for the class of finite groups having a trivial Fitting subgroup : in this context, we prove that the conjecture is true provided σ (G) ≤ 5 , but it is false in general if σ (G) ≥ 6. Instead, we establish that | ρ (G) | ≤ 3 σ (G) − 4 holds for every finite group with a trivial Fitting subgroup and with σ (G) ≥ 6 (this being the right, best possible bound). Also, we improve the up-to-date best bound for the solvable case, showing that we have | ρ (G) | ≤ 3 σ (G) whenever G belongs to one particular class including all the finite solvable groups, and we improve the up-to-date best bound obtained in [18] for the general case. [ABSTRACT FROM AUTHOR]

In this paper we verify Navarro's refinement of the McKay conjecture for quasi-simple groups of Lie type in their defining characteristic. Navarro's refinement takes into account the action of specific Galois automorphisms on the characters presents in the McKay conjecture [12]. Our proof of this case of the conjecture relies on a character correspondence constructed by Maslowski in [11]. Building on this we verify the inductive condition for Navarro's refinement from [14] for most groups of Lie type in defining characteristic. [ABSTRACT FROM AUTHOR]

Motivated by a 2004 conjecture by the author and J. Sangroniz, Y. Yang has recently proved that if G is solvable then the index in G of the 8th term of the ascending Fitting series is bounded in terms of the largest number of zeros in a row in the character table of G. In this note, we prove this result for arbitrary finite groups and propose a stronger form of the 2004 conjecture. We conclude the paper showing some possible ways to prove this strengthened conjecture. [ABSTRACT FROM AUTHOR]

The classification of the nilpotent Jacobians with some structure has been an object of study because of its relationship with the Jacobian conjecture. In this paper we classify the polynomial maps in dimension n of the form H = (u (x , y) , u 2 (x , y , x 3) , ... , u n − 1 (x , y , x n) , h (x , y)) with JH nilpotent. In addition we prove that the maps X + H are invertible, which shows that for this kind of maps the Jacobian conjecture is verified. [ABSTRACT FROM AUTHOR]

This paper represents a step in our program towards the proof of the Pierce–Birkhoff conjecture. In the nineteen eighties J. Madden proved that the Pierce–Birkhoff conjecture for a ring A is equivalent to a statement about an arbitrary pair of points α , β ∈ Sper A and their separating ideal < α , β > ; we refer to this statement as the local Pierce–Birkhoff conjecture at α , β . In [8] we introduced a slightly stronger conjecture, also stated for a pair of points α , β ∈ Sper A and the separating ideal < α , β > , called the Connectedness conjecture . In this paper, for each pair ( α , β ) with h t ( < α , β > ) = dim ⁡ A , we define a natural number, called complexity of ( α , β ) . Complexity 0 corresponds to the case when one of the points α , β is monomial; this case was settled in all dimensions in [8] . In the present paper we introduce a new conjecture, called the Strong Connectedness conjecture , and prove that the strong connectedness conjecture in dimension n − 1 implies the connectedness conjecture in dimension n in the case when h t ( < α , β > ) ≤ n − 1 . We prove the Strong Connectedness conjecture in dimension 2, which gives the Connectedness and the Pierce–Birkhoff conjectures in any dimension in the case when h t ( < α , β > ) ≤ 2 . Finally, we prove the Connectedness (and hence also the Pierce–Birkhoff) conjecture in the case when dim ⁡ A = h t ( < α , β > ) = 3 , the pair ( α , β ) is of complexity 1 and A is excellent with residue field R . [ABSTRACT FROM AUTHOR]

Up to equivalence, this paper classifies all the irreducible unitary representations with non-zero Dirac cohomology for the following simple real exceptional Lie groups: EI = E 6 (6) , EIV = E 6 (− 26) , FI = F 4 (4) , FII = F 4 (− 20). Along the way, we find an irreducible unitary representation of F 4 (4) whose Dirac index vanishes, while its Dirac cohomology is non-zero. This disproves a conjecture raised in 2015 asserting that there should be no cancellation between the even part and the odd part of the Dirac cohomology. [ABSTRACT FROM AUTHOR]

In this paper characters of the normalizer of d -split Levi subgroups in SL n (q) and SU n (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 (q 2 − 1) and the Alperin weight conjecture for ℓ -blocks of those quasi-simple groups with abelian defect. The inductive Alperin-McKay condition and inductive Alperin weight condition by the second author are verified for certain blocks of SL n (q) and SU n (q). [ABSTRACT FROM AUTHOR]

The paper presented here focuses on the classification of trivial source Specht modules. We completely classify the trivial source Specht modules labelled by hook partitions. We also classify the trivial source Specht modules labelled by two-part partitions in the odd characteristic case. Moreover, in the even characteristic case, we prove a result for the classification of the trivial source Specht modules labelled by partitions with 2-weight 2, which justifies a conjecture of [16]. [ABSTRACT FROM AUTHOR]

Let C ⊂ P r 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 H 0 (O P r (m)) → H 0 (O C (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 ⊂ P r is a general hyperplane, and C 1 , C 2 , ... , C n are general curves, we show H 0 (O H (m)) → H 0 (O (C 1 ∪ C 2 ∪ ⋯ ∪ C n) ∩ H (m)) is of maximal rank, except for some counterexamples when m = 2. As explained in [5] , this result plays a key role in the author's proof of the Maximal Rank Conjecture [7]. [ABSTRACT FROM AUTHOR]

In [10] , Conjecture 6.6, Migliore, the first author, and Nagel conjectured that, for all n ≥ 4 , the artinian ideal I = (L 0 d , ... , L 2 n + 1 d) ⊂ R = k [ x 0 , ... , x 2 n ] generated by the d -th powers of 2 n + 2 general linear forms fails to have the weak Lefschetz property if and only if d > 1. This paper is entirely devoted to prove partially this conjecture. More precisely, we prove that R / I fails to have the weak Lefschetz property, provided 4 ≤ n ≤ 8 , d ≥ 4 or d = 2 r , 1 ≤ r ≤ 8 , 4 ≤ n ≤ 2 r (r + 2) − 1. [ABSTRACT FROM AUTHOR]

The Hall–Paige conjecture asserts that a finite group has a complete mapping if and only if its Sylow subgroups are not cyclic. The conjecture is now proved, and one aim of this paper is to document the final step in the proof (for the sporadic simple group J 4). We apply this result to prove that primitive permutation groups of simple diagonal type with three or more simple factors in the socle are non-synchronizing. We also give the simpler proof that, for groups of affine type, or simple diagonal type with two socle factors, synchronization and separation are equivalent. Synchronization and separation are conditions on permutation groups which are stronger than primitivity but weaker than 2-homogeneity, the second of these being stronger than the first. Empirically it has been found that groups which are synchronizing but not separating are rather rare. It follows from our results that such groups must be primitive of almost simple type. [ABSTRACT FROM AUTHOR]

Let X be a reduced closed subscheme in P n , π : X → π (X) ⊂ P n − 1 be a projection from a point outside X and Z i (X) ⊂ π (X) be the closed subscheme defined by the i -th partial elimination ideal K i (I X) , which is supported on the (i + 1) -th multiple points of π. In this paper, motivated from projection methods to prove Eisenbud-Goto conjecture on regularity in many cases, we describe the syzygetic behaviors and Castelnuovo normality of the projection with a viewpoint of the regularity of the partial elimination ideal K i (I X) , i ≥ 1 (or that of the multiple locus Z i (X) of π). We also give some applications to the syzygies and Castelnuovo normality of successive projections, which recover and generalize some known results in [1,3,15,16]. [ABSTRACT FROM AUTHOR]

Each (equigenerated) squarefree monomial ideal in the polynomial ring S = K [ x 1 , ... , x n ] represents a family of subsets of [ n ] , called a (uniform) clutter. In this paper, we introduce a class of uniform clutters, called decomposable clutters, whose associated ideal has linear quotients and hence linear resolution over all fields. We show that chordality of these clutters guarantees the correctness of a conjecture raised by R.S. Simon [23] on extendable shellebility of d -skeletons of a simplex 〈 [ n ] 〉 , for all d. We then prove this conjecture for d ≥ n − 3. [ABSTRACT FROM AUTHOR]

Abstract Let G be a finite p -group. The breadth b (x) of an element x of a group G is defined as log p of the size of the conjugacy class of x in G. M.R. Vaughan-Lee proved that if the breadths of all elements in G are bounded by some natural number n , then | G ′ | ≤ p n (n + 1) 2 . In the present paper, we prove that if | G ′ | > p n (n + 1) 2 , then G can be generated by the elements of breadths > n , thereby confirming the hypothesis of James Wiegold from the "Kourovka notebook". [ABSTRACT FROM AUTHOR]

Abstract Let F be a non-archimedean local field and G = G (F) the group of F -rational points of a connected reductive F -group. The Langlands classification of complex irreducible admissible representations of G gives a parametrization of the set of all such representations in terms of triples (P , σ , ν) , where P ⊂ G is a standard F -parabolic subgroup, σ is an irreducible tempered representation of the standard Levi-group M P and ν ∈ R ⊗ X ⁎ (M P) is regular and positive with respect to P. In this paper we consider Langlands' L-parameters [ ϕ ] which conjecturally will serve as a system of parameters for the representations π and which are equivalence classes of L -homomorphisms ϕ of the absolute Galois group Γ = Gal (F ‾ | F) with image in Langlands' L-group G L. Our goal is to establish a classification of L-parameters [ ϕ ] in terms of triples (P , [ ϕ b ] , ν) such that (P , ν) is as before and [ ϕ b ] is a bounded L-parameter of M P. In an Appendix we also deal with the archimedean case F = R , C. This case is easier because the Weil group W R has a natural polar decomposition W R = R + × × W R 0. [ABSTRACT FROM AUTHOR]

Abstract Bouc proposed the following conjecture: a finite group G is nilpotent if and only if its largest quotient B -group β (G) is nilpotent. And Thévenaz proposed another conjecture: a finite group G is solvable if and only if its largest quotient B -group β (G) is solvable. Bouc has proven that his conjecture holds when G is solvable. In this paper, we consider some cases when G is not solvable. Let S be a nonabelian simple group except the Chevalley groups A n (q) , D n (q) , E 6 (q) , and A n 2 (q) , if the group G has exactly one composition factor isomorphic to S , then β (G) is not solvable, of course, is not nilpotent. That means we prove the conjectures in these cases. [ABSTRACT FROM AUTHOR]

It is known that the fundamental group homomorphism π 1 ( T 2 ) → π 1 ( S 3 ∖ K ) induced by the inclusion of the boundary torus into the complement of a knot K in S 3 is a complete knot invariant. Many classical invariants of knots arise from the natural (restriction) map induced by the above homomorphism on the SL 2 -character varieties of the corresponding fundamental groups. In our earlier work [3] , we proposed a conjecture that the classical restriction map admits a canonical deformation into a two-parameter family of affine cubic surfaces in C 3 . In this paper, we show that (modulo some mild technical conditions) our conjecture follows from a known conjecture of Brumfiel and Hilden [1] on the algebraic structure of the peripheral system of a knot. We then confirm the Brumfiel–Hilden conjecture for an infinite class of knots, including all torus knots, 2-bridge knots, and certain pretzel knots. We also show the class of knots for which the Brumfiel–Hilden conjecture holds is closed under taking connect sums and knot coverings. [ABSTRACT FROM AUTHOR]

In this paper, we will discuss a conjecture on the hyperfocal subalgebra of the block which is proposed by R. Kessar, M. Linckelmann and G. Navarro. We will prove this conjecture under the condition that the hyperfocal subgroup is normal and give two applications. [ABSTRACT FROM AUTHOR]

In this paper, we prove an extension of Zaks' conjecture on integral domains with semi-regular proper homomorphic images (with respect to finitely generated ideals) to arbitrary rings (i.e., possibly with zero-divisors). The main result extends and recovers Levy's related result on Noetherian rings [23, Theorem] and Matlis' related result on Prüfer domains [26, Theorem] . It also globalizes Couchot's related result on chained rings [10, Theorem 11] . New examples of rings with semi-regular proper homomorphic images stem from the main result via trivial ring extensions. [ABSTRACT FROM AUTHOR]

In [2] , Conjecture 5.5.2, Harbourne, Schenck and Seceleanu conjectured that, for r = 6 and all r ≥ 8 , the artinian ideal I = ( ℓ 1 2 , … , l r + 1 2 ) ⊂ K [ x 1 , … , x r ] generated by the square of r + 1 general linear forms ℓ i fails the Weak Lefschetz property. This paper is entirely devoted to prove this Conjecture. It is worthwhile to point out that half of the Conjecture – namely, the case when the number of variables r is even – was already proved in [5] , Theorem 6.1. [ABSTRACT FROM AUTHOR]

In [2] , Bazzoni and Glaz (2007) conjecture that the weak global dimension of a Gaussian ring is 0 , 1 or ∞. In this paper, we prove their conjecture. [ABSTRACT FROM AUTHOR]