Let Fq2 be the finite field with q2 elements. Most of the known Fq2-maximal curves arise as quotient curves of the Fq2-maximal Hermitian curve Hq. After a seminal paper by Garcia, Stichtenoth and Xing, many papers have provided genera of quotients of Hq, but their complete determination is a challenging open problem. In this paper we determine completely the spectrum of genera of quotients of Hq for any q≡1(mod4).

We give a decomposition, as a direct sum of indecomposable modules, of several types of Specht modules in characteristic 2. These include the Specht modules labelled by hooks, whose decomposability was considered by Murphy, [15]. Since the main arguments are essentially no more difficult for Hecke algebras at parameter q=−1, we proceed in this generality.

In his theory of unipotent characters of finite groups of Lie type, Lusztig constructed modular categories from two-sided cells in Weyl groups. Broué, Malle and Michel have extended parts of Lusztig's theory to complex reflection groups. This includes generalizations of the corresponding fusion algebras, although the presence of negative structure constants prevents them from arising from modular categories. We give here the first construction of braided pivotal monoidal categories associated with non-real reflection groups (later reinterpreted by Lacabanne as super modular categories). They are associated with cyclic groups, and their fusion algebras are those constructed by Malle.

Fix a field k of characteristic zero. If a1,…,an (n≥3) are positive integers, the integral domain Ba1,…,an=k[X1,…,Xn]/〈X1a1+⋯+Xnan〉 is called a Pham-Brieskorn ring. It is conjectured that if ai≥2 for all i and ai=2 for at most one i, then Ba1,…,an is rigid. (A ring B is said to be rigid if the only locally nilpotent derivation D:B→B is the zero derivation.) The conjecture is known to be true when n=3, and in certain special cases when n≥4. This article settles several cases not covered by previous results. For instance, we show that if a≥n≥4 then Ba,…,a︸n is rigid, and that if ∑i=1n1ai≤1n−2 then Ba1,…,an is stably rigid.

Let G(q) be a finite group of Lie type over a field with q elements, where q is a prime power. The Green functions of G(q), as defined by Deligne and Lusztig, are known in almost all cases by work of Beynon–Spaltenstein, Lusztig und Shoji. Open cases exist for groups of exceptional type E62, E7, E8 in small characteristics. We propose a general method for dealing with these cases, which proceeds by a reduction to the case where q is a prime and then uses computer algebra techniques. In this way, all open cases in type E62, E7 are solved, as well as at least one particular open case in type E8.

We show that depth(I)≤1 for each ideal I of C(X). This gives a positive answer to a conjecture in [F.Azarpanah, D. Esmaeilvandi and A.R. Salehi, Depth of ideals of C(X), J. Algebra 528 (2019) 474-496.]. The present article is in fact an attempt to complete the aforementioned paper. Using the above fact, the depth of some ideals of C(X) such as principal ideals, the ideals Op, p∈βX, and prime ideals are determined in the sense that when they are 0 and when they are 1. We have generalized Proposition 2.9 in that paper and we have shown that depth(C(X)(f)) is at most 1, for each principal ideal (f) in C(X), and it is exactly 1 if and only if intXZ(f) contains at least one non-almost P-point. Also, we prove that for each non-essential principal ideal (f), depth(C(X)(f))=1 or equivalently depth(Ann(f))=1 if and only if the set of non-almost P-points of X is dense in X. Finally, it has been shown that depth(Op)=1, for each p∈βX, if and only if X contains at least two non-almost P-points, and topological spaces X are characterized for which depth(I)=1, for each essential ideal of C(X).

The (full) extended plus closure was developed as a replacement for tight closure in mixed characteristic rings. Here it is shown by adapting André's perfectoid algebra techniques that, for complete local rings that have F-finite residue fields, this closure has the colon-capturing property. In fact, more generally, if R is a (possibly ramified) complete regular local ring of mixed characteristic that has an F-finite residue field, I and J are ideals of R, and the local domain S is a finite R-module, then (IS:J)⊆(I:J)Sepf. A consequence is that all ideals in regular local rings are closed, a fact which implies the validity of the direct summand conjecture and the Briançon–Skoda theorem in mixed characteristic.

We prove several characterizations of Gorenstein rings in terms of vanishings of derived functors of certain modules or complexes whose scalars are restricted via contracting endomorphisms. These results can be viewed as analogues of results of Kunz (in the case of the Frobenius) and Avramov–Hochster–Iyengar–Yao (in the case of general contracting endomorphisms).

In this paper, we study the F-rationality of the Rees algebra and the extended Rees algebra of m-primary ideals in excellent local rings (R,m) of prime characteristic. We partially answer some conjectures and questions raised by Hara et al. (2002) [9].

Let S be a deeply embedded, equicharacteristic, Artinian Gorenstein local ring. We prove that if R is a non-Gorenstein quotient of S of small colength, then every totally reflexive R-module is free. Indeed, the second syzygy of the canonical module of R has a direct summand T which is a test module for freeness over R in the sense that if Tor+R(T,N)=0, for some finitely generated R-module N, then N is free.

We show that the Koszul homology algebra of the second Veronese subalgebra of a polynomial ring over a field of characteristic zero is generated, as an algebra, by the homology classes corresponding to the syzygies of the lowest linear strand.

We generalize a result of Eisenbud, Huneke and Ulrich on the maximal graded shifts of a module with prescribed annihilator and prove a linear regularity bound for ideals in a polynomial ring depending only on the first p−c steps in the resolution, where p=pd(S/I) and c=codim(I).

Free extensions of graded Artinian algebras were introduced by T. Harima and J. Watanabe, and were shown to preserve the strong Lefschetz property. The Jordan type of a multiplication map m by a nilpotent element of an Artinian algebra is the partition determining the sizes of the blocks in a Jordan matrix for m. We show that a free extension C of the Artinian algebra A with fiber B is a deformation of the usual tensor product. This has consequences for the generic Jordan types of A,B and C: we show that the Jordan type of C is at least that of the usual tensor product in the dominance order (Theorem 2.5). In particular this gives a different proof of the T. Harima and J. Watanabe result concerning the strong Lefschetz property of a free extension. Examples illustrate that a non-strong-Lefschetz graded Gorenstein algebra A with non-unimodal Hilbert function may nevertheless have a non-homogeneous element with strong Lefschetz Jordan type, and may have an A-free extension that is strong Lefschetz. We apply these results to algebras of relative coinvariants of linear group actions on a polynomial ring.

In this paper we prove that the unitary groups SUn(q2) are (2,3)-generated for any prime power q and any integer n≥8. By previous results this implies that, if n≥3, the groups SUn(q2) and PSUn(q2) are (2,3)-generated, except when (n,q)∈{(3,2),(3,3),(3,5),(4,2),(4,3),(5,2)}.

Given a selfinjective artin algebra Λ, we consider the category Sinj(Λ) of all embeddings of a left Λ-module in a finitely generated injective left Λ-module. We show that Sinj(Λ) is a Frobenius category with Auslander-Reiten sequences such that the categories Λ-mod and Sinj(Λ) are stably equivalent and Sinj(Λ) has twice as many indecomposable injective objects as Λ-mod.

In this paper the authors study the class of locally graded groups all of whose subgroups are permutable or are soluble and satisfy a certain rank condition. The rank conditions in question include groups of finite abelian section rank, minimax groups and polycyclic groups. In each case necessary and sufficient conditions are given for a locally graded group to have all subgroups permutable or soluble with the given rank condition.

Throughout this paper, G always denotes a group and L(G) is the lattice of all subgroups of G. If K⊴H≤G, then H/K is called a section of G; such a section is called normal if K,H⊴G. We call any set Σ of normal sections of G a stratification of G provided: (i) Σ is G-closed, that is, H/K∈Σ whenever H/K≃GT/L∈Σ, and (ii) L/K,H/L∈Σ for each triple K

A rational map ϕ:Pkm⇢Pkn is defined by homogeneous polynomials of a common degree d. We establish a linear bound in terms of d for the number of (m−1)-dimensional fibers of ϕ, by using ideals of minors of the Jacobian matrix. In particular, we answer affirmatively Question 11 in [9].

This paper is a sequel to [8] where we introduced an invariant, called canonical degree, of Cohen–Macaulay local rings that admit a canonical ideal. Here to each such ring R with a canonical ideal, we attach a different invariant, called bi-canonical degree, which in dimension 1 appears also in [12] as the residue of R. The minimal values of these functions characterize specific classes of Cohen–Macaulay rings. We give a uniform presentation of such degrees and discuss some computational opportunities offered by the bi-canonical degree.

Let R be a commutative Noetherian local ring and M,N be finitely generated R-modules. We prove a number of results of the form: if HomR(M,N) has some nice properties and ExtR1≤i≤n(M,N)=0 for some n, then M (and sometimes N) must be close to free. Our methods are quite elementary, yet they suffice to give a unified treatment, simplify, and sometimes extend a number of results in the literature.

A construction due to Knörrer shows that if N is a maximal Cohen–Macaulay module over a hypersurface defined by f+y2, then the first syzygy of N/yN decomposes as the direct sum of N and its own first syzygy. This was extended by Herzog–Popescu to hypersurfaces f+yn, replacing N/yN by N/yn−1N. We show, in the same setting as Herzog–Popescu, that the first syzygy of N/ykN is always an extension of N by its first syzygy, and moreover that this extension has useful approximation properties. We give two applications. First, we construct a ring Λ over which every finitely generated module has an eventually 2-periodic projective resolution, prompting us to call it a “non-commutative hypersurface ring”. Second, we give upper bounds on the dimension of the stable module category (a.k.a. the singularity category) of a hypersurface defined by a polynomial of the form x1a1+…+xdad.

We prove an “nth moment =1” result for irreducible Weil representations of degree (qn+1)/(q+1) of special unitary groups SUn(q) for any odd n≥3 and any prime power q.

We establish a duality of the non-periodic Gaudin model associated with superalgebra glm|n and the non-periodic Gaudin model associated with algebra glk. The Hamiltonians of the Gaudin models are given by expansions of a Berezinian of an (m+n)×(m+n) matrix in the case of glm|n and of a column determinant of a k×k matrix in the case of glk. We obtain our results by proving Capelli type identities for both cases and comparing the results.

Let k be a field of positive characteristic. Building on the work of the second named author, we define a new class of k-algebras, called diagonally F-regular algebras, for which the so-called Uniform Symbolic Topology Property (USTP) holds effectively. We show that this class contains all essentially smooth k-algebras. We also show that this class contains certain singular algebras, such as the affine cone over Pkr×Pks, when k is perfect. By reduction to positive characteristic, it follows that USTP holds effectively for the affine cone over PCr×PCs and more generally for complex varieties of diagonal F-regular type.

The formalism of injective stabilization of additive functors is used to define a new notion of the torsion submodule of a module. It applies to arbitrary modules over arbitrary rings. For arbitrary modules over commutative domains it coincides with the classical torsion, and for finitely presented modules over arbitrary rings it coincides with the Bass torsion. A formally dual approach – based on projective stabilization – gives rise to a new concept: the cotorsion quotient module of a module. This is done in complete generality – the new concept is defined for any module over any ring. Unlike torsion, cotorsion does not have classical prototypes. General properties of these constructs are established. It is shown that the Auslander-Gruson-Jensen functor applied to the cotorsion functor returns the torsion functor. As a consequence, a ring is one-sided absolutely pure if and only if each pure injective on the other side is cotorsion-free. If the injective envelope of the ring is finitely presented, then the right adjoint of the Auslander-Gruson-Jensen functor applied to the torsion functor returns the cotorsion functor. This correspondence establishes a duality between torsion and cotorsion over such rings. In particular, this duality applies to artin algebras. It is also shown that, over any ring, the character module of the torsion of a module is isomorphic to the cotorsion of the character module of the module. Under various finiteness conditions on the injective envelope of the ring, the derived functors of torsion and cotorsion are computed.

Let H be a finite-dimensional unimodular pivotal quasi-Hopf algebra over a field k, and let H-mod be the pivotal tensor category of finite-dimensional H-modules. We give a bijection between left (resp. right) modified traces on the tensor ideal H-pmod of projective modules and left (resp. right) cointegrals for H. The non-zero left/right modified traces are non-degenerate, and we show that non-degenerate left/right modified traces can only exist for unimodular H. This generalises results of Beliakova, Blanchet, and Gainutdinov [1] from Hopf algebras to quasi-Hopf algebras. As an example we compute cointegrals and modified traces for the family of symplectic fermion quasi-Hopf algebras.

Breda, Nedela and Širáň (2005) classified the regular maps on surfaces of Euler characteristic −p for every prime p. This classification relies on three key theorems, each proved using the highly non-trivial characterisation of finite groups with dihedral Sylow 2-subgroups, due to D. Gorenstein and J.H. Walter (1965). Here we give new proofs of those three facts (and hence the entire classification) using somewhat more elementary group theory, using without referring to the Gorenstein-Walter theorem.

Building on prior work of Bogomolov, Garibaldi, Guralnick, Igusa, Kordonskiĭ, Merkurjev and others, we show that the Noether Problem for Spinn has a positive solution for every n≤14 over an arbitrary field of characteristic ≠2.

We describe all finite-dimensional pointed Hopf algebras whose infinitesimal braiding is a fixed Yetter-Drinfeld module decomposed as the sum of two simple objects: a point and the one of transpositions of the symmetric group in three letters. We give a presentation by generators and relations of the corresponding Nichols algebra and show that Andruskiewitsch-Schneider Conjecture holds for this kind of pointed Hopf algebras.

We use ring-theoretic methods and methods from the theory of skew braces to produce set-theoretic solutions to the reflection equation. We also use set-theoretic solutions to construct solutions of the parameter dependent reflection equation.

Given an additive category C and an integer n⩾2. We form a new additive category C[ϵ]n consisting of objects X in C equipped with an endomorphism ϵX satisfying ϵXn=0. First, using the descriptions of projective and injective objects in C[ϵ]n, we not only establish a connection between Gorenstein flat modules over a ring R and R[t]/(tn), but also prove that an Artinian algebra R satisfies some homological conjectures if and only if so does R[t]/(tn). Then we show that the corresponding homotopy category K(C[ϵ]n) is a triangulated category when C is an idempotent complete exact category. Moreover, under some conditions for an abelian category A, the natural quotient functor Q from K(A[ϵ]n) to the derived category D(A[ϵ]n) produces a recollement of triangulated categories. Finally, we prove that if A is an Ab4-category with a compact projective generator, then D(A[ϵ]n) is a compactly generated triangulated category.

Given a finite group G and two unitary G-representations V and W, possible restrictions on topological degrees of equivariant maps between representation spheres S(V) and S(W) are usually expressed in a form of congruences modulo the greatest common divisor of lengths of orbits in S(V) (denoted by α(V)). Effective applications of these congruences is limited by answers to the following questions: (i) under which conditions, is α(V)>1? and (ii) does there exist an equivariant map with the degree easy to calculate? In the present paper, we address both questions. We show that α(V)>1 for each irreducible non-trivial C[G]-module if and only if G is solvable. This provides a new solvability criterion for finite groups. For non-solvable groups, we use 2-transitive actions to construct complex representations with non-trivial α-characteristic. Regarding the second question, we suggest a class of Norton algebras without 2-nilpotents giving rise to equivariant quadratic maps, which admit an explicit formula for the degree.

We classify simple associative and Lie algebras inside the Deligne categories Rep(St), answering a question posed by Etingof.

We study the class of virtually uniserial modules and rings as a nontrivial generalization of uniserial modules and rings. An R-module M is virtually uniserial if for every finitely generated submodule 0≠K⊆M, K/Rad(K) is virtually simple. Also, an R-module M is called virtually serial if it is a direct sum of virtually uniserial modules and a left virtually uniserial (resp., left virtually serial) ring is a ring which is virtually uniserial (resp., serial) as a left R-module. We give some useful properties of virtually (uni)serial modules and rings. In particular, it is shown that every left virtually uniserial module is uniform and Bézout. Also, we show that if R is a left virtually serial ring, then R/J(R)≅∏i=1tMni(Di) where t,n1,…,nt∈N and each Di is a principal left ideal domain. As a consequence, we obtain that a ring R is left virtually serial with J(R)=0 if and only if R≅∏i=1tMni(Di) where t,n1,…,nt∈N and each Di is a principal left ideal domain with J(Di)=0. Also, several classes of rings for which every virtually uniserial module (resp., ring) is uniserial are given. Noetherian left virtually uniserial rings are characterized. Finally, we obtain some structure theorems for (commutative) rings over which every (finitely generated) module is virtually serial.

We aim in this paper to investigate the connections between some properties of blocks and of their dominating blocks. We find conditions to verify that a block is inertial if and only if its dominating block is inertial. In some situations the equality of the factor fusion systems associated with a block and with its Brauer correspondent block give information about the hyperfocal subgroups.

Let G be a finite group. The ring RK(G) of virtual characters of G over the field K is a λ-ring; as such, it is equipped with the so-called Γ-filtration, first defined by Grothendieck. We explore the properties of the associated graded ring RK⁎(G), and present a set of tools to compute it through detailed examples. In particular, we use the functoriality of RK⁎(−), and the topological properties of the Γ-filtration, to explicitly determine the graded character ring over the complex numbers of every group of order at most 8, as well as that of dihedral groups of order 2p for p prime.

Let (L,N) be a pair of Lie algebras, in which N is an ideal of L. We prove if dim⁡(N)=n and dim⁡(L/N)=m, then the dimension of the second relative homology of (L,N) is equal to 12n(n+2m−1)−t, for some non-negative integer t. Also, we characterize all pairs of nilpotent Lie algebras for which t=0,1,2,3.

Supermanifolds are known to admit both differential forms and integral forms, thus any appropriate super analogue of the de Rham theory should take both types of forms into account. However, the cohomology of Lie superalgebras studied so far in the literature involves only differential forms when interpreted as a de Rham theory for Lie supergroups. Thus a new cohomology theory of Lie superalgebras is needed to fully incorporate differential-integral forms, and we investigate such a theory here. This new cohomology is defined by a BRST complex of Lie superalgebra modules, and includes the standard Lie superalgebra cohomology as a special case. General properties expected of a cohomology theory are established for the new cohomology, and examples of the new cohomology groups are computed.

We construct families of principally polarized abelian varieties whose theta divisor is irreducible and contains an abelian subvariety. These families are used to construct examples when the Gauss map of the theta divisor is only generically finite and not finite. That is, the Gauss map in these cases has at least one positive-dimensional fiber. We also obtain lower-bounds on the dimension of Andreotti-Mayer loci.

From the viewpoint of higher homological algebra, we introduce pure semisimple n-abelian categories, which are analogs of pure semisimple abelian categories. Let Λ be an Artin algebra and M be an n-cluster tilting subcategory of Mod-Λ. We show that M is pure semisimple if and only if each module in M is a direct sum of finitely generated modules. Let m be an n-cluster tilting subcategory of mod-Λ. We show that Add(m) is an n-cluster tilting subcategory of Mod-Λ if and only if m has an additive generator if and only if Mod(m) is locally finite. This generalizes Auslander's classical results on pure semisimplicity of Artin algebras.

We introduce topological rewriting systems as a generalisation of abstract rewriting systems, where we replace the set of terms by a topological space. Abstract rewriting systems correspond to topological rewriting systems for the discrete topology. We introduce the topological confluence property as an approximation of the confluence property. Using a representation of linear topological rewriting systems with continuous reduction operators, we show that the topological confluence property is characterised by lattice operations. Using this characterisation, we show that standard bases induce topologically confluent rewriting systems on formal power series. Finally, we investigate duality for reduction operators that we relate to series representations and syntactic algebras. In particular, we use duality for proving that an algebra is syntactic or not.

We show that for any solvable group G and a Drinfel'd twist J, kGJ is solvable in the sense of the intrinsic definition of solvability given in [2]. More generally, if a Hopf algebra H has a normal solvable series so does HJ. Furthermore, while solvable groups are defined as having certain commutative quotients, quasitriangular normally solvable Hopf algebras have appropriate quantum commutative quotients. We end with a detailed example.

We correct three issues in the paper “The Zariski topology on sets of semistar operations without finite-type assumptions”.

Let k be a field of arbitrary characteristic, A a Noetherian k-algebra and consider the polynomial ring A[x]=A[x0,…,xn]. We consider graded submodules of A[x]m having a special set of generators, a marked basis over a quasi-stable module. Such a marked basis shares many interesting properties with a Gröbner basis, including the existence of a Noetherian reduction relation. The set of submodules of A[x]m having a marked basis over a given quasi-stable module possesses a natural affine scheme structure which we will exhibit. Furthermore, the syzygies of a module with such a marked basis are generated by a marked basis, too (over a suitable quasi-stable module in ⊕i=1m′A[x](−di)). We apply marked bases and related properties to the investigation of Quot functors (and schemes). More precisely, for a given Hilbert polynomial, we explicitly construct (up to the action of a general linear group) an open cover of the corresponding Quot functor, made up of open subfunctors represented by affine schemes. This provides a new proof that the Quot functor is the functor of points of a scheme. We also exhibit a procedure to obtain the equations defining a given Quot scheme as a subscheme of a suitable Grassmannian. Thanks to the good behaviour of marked bases with respect to Castelnuovo-Mumford regularity, we can adapt our methods in order to study the locus of the Quot scheme given by an upper bound on the regularity of its points.

We study Betti numbers of graded finitely generated modules over a quadratic complete intersection. In the case of codimension 1, we give a natural class of quadratic forms Q whose Clifford algebras are division rings, and deduce the possible Betti numbers of modules over the hypersurfaces Q=0. Our approach leads to a new version of the Betti degree Conjecture. In higher codimensions, we obtain formulas for the Betti numbers in terms of the ranks of certain free modules in a higher matrix factorization.

A compact Riemann surface is called pseudo-real if it admits anti-conformal (orientation-reversing) automorphisms, but no anti-conformal automorphism of order 2, or equivalently, if the surface is reflexible but not definable over the reals. It is known that there exist pseudo-real surfaces of genus g for every integer g≥2, and the number of automorphisms of any such surface is bounded above by 12(g−1). In this paper, we extend the concepts of symmetric genus, strong symmetric genus and symmetric cross-cap genus of a group by defining and investigating two new parameters, as follows: (1) the pseudo-real genus ψ(G) of a finite group G is the smallest genus of those pseudo-real surfaces on which G acts faithfully as a group of automorphisms, some of which might reverse orientation, and (2) the strong pseudo-real genus ψ⁎(G) of G is the smallest genus of those pseudo-real surfaces on which G acts faithfully as a group of automorphisms, some of which do reverse orientation, when there exists such a surface for G. Our main theorem is that for every integer g≥2, there exists a finite group G for which ψ(G)=ψ⁎(G)=g, and hence that the range of each of the functions ψ and ψ⁎ is the set of all integers g≥2. We also give an example of a group G for which ψ⁎(G) is defined but ψ(G)<ψ⁎(G).

Motivated by the theory of Riemann surfaces and specifically the significance of Weierstrass points, we study finite simple groups from a permutation action point of view. We classify all possibilities for finite simple groups acting faithfully on a compact Riemann surface of genus at least 2 in such a way that all non-trivial elements have at most two fixed points on each non-regular orbit and at most four fixed points in total.

We give a short combinatorial proof of Asai's decomposition formula for Lusztig induction of unipotent characters in groups of classical type, relying solely on the Mackey formula.

We analyze families of Markov chains that arise from decomposing tensor products of irreducible representations. This illuminates the Burnside-Brauer theorem for building irreducible representations, the McKay correspondence, and Pitman's 2M−X theorem. The chains are explicitly diagonalizable, and we use the eigenvalues/eigenvectors to give sharp rates of convergence for the associated random walks. For modular representations, the chains are not reversible, and the analytical details are surprisingly intricate. In the quantum group case, the chains fail to be diagonalizable, but a novel analysis using generalized eigenvectors proves successful.

In this paper, we introduce the notion of generalized Brauer morphism for FG-modules with respect to a p-subgroup of G. This generalization is based on p-indecomposable decompositions of FG-modules. The generalized Brauer morphism is compatible with the Brauer morphism introduced by M. Broué when applied to p-permutation modules. We generalized some well-known results of p-permutation modules through generalized Brauer morphism to arbitrary modules. In particular, we show that the vertices of an indecomposable module M are the maximal p-subgroups P such that the image of M under the generalized Brauer morphism with respect to P is nonzero and that the image is the Green correspondent of M. As an example, we also compute the generalized Brauer morphisms for indecomposable modules with cyclic vertices.

We study modulo p reduction of compatible systems of p-adic representations of the absolute Galois group of Q, arising from an algebraic automorphic representation. In particular, we prove that there is a field extension of Q with the Galois group G2(p), ramified at 5 and p only, for a set of primes p of density one.

We prove several conjectures about the cohomology of Deligne–Lusztig varieties: invariance under conjugation in the braid group, behaviour with respect to translation by the full twist, parity vanishing of the cohomology for the variety associated with the full twist. In the case of split groups of type A, and using previous results of the second author, this implies Broué–Michel's conjecture on the disjointness of the cohomology for the variety associated to any good regular element. That conjecture was inspired by Broué's abelian defect group conjecture and the specific form Broué conjectured for finite groups of Lie type [4, Rêves 1 et 2].

We make some conjectures about tensor products of modular representations for a finite p-group in characteristic p, for p=2 and p=3. The easiest to state is that if M is a module of odd dimension for a 2-group over an algebraically closed field of characteristic two, then k is the only direct summand of M⊗M⁎ of odd dimension, and the remaining indecomposable summands have dimension divisible by four. As a consequence, the odd dimensional modules form an abelian group, where the composition law is to take the unique indecomposable summand of odd dimension of the tensor product, and where the inverse of an odd dimensional module is its dual. In characteristic three the conjectures are more complicated, and in larger characteristics it is not at all clear what the corresponding conjectures should be. We prove a number of theorems relating to these conjectures, and describe examples illustrating them. The examples were computed using the computer algebra package Magma.

We show that the group of bounded sequences of elements of Z[2] is an example of an abelian group with several well known, and not so well known, pathological properties. It appears to be simpler than all previously known examples for some of these properties, and at least simpler to describe for others.

The Brauer-Chen algebra is a generalization of the algebra of Brauer diagrams to arbitrary complex reflection groups, that admits a natural monodromic deformation. We determine the generic representation theory of the first non trivial quotient of this algebra. We also define natural extensions of this algebra and prove that they similarly admit natural monodromic deformations.

We study crossed S-matrices for braided G-crossed categories and reduce their computation to a submatrix of the de-equivariantization. We study the more general case of a category containing the symmetric category Rep(A,z) with A a finite cyclic group and z∈A such that z2=1. We give two examples of such categories, which enable us to recover the Fourier matrix associated with the big family of unipotent characters of the dihedral groups with automorphism as well as the Fourier matrix of the big family of unipotent characters of the Ree group of type F42.

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

We prove that the cohomology group of a Deligne–Lusztig variety defines a Morita equivalence in a case which is not covered by the argument in [2], specifically we consider the situation for semisimple elements in type D whose centralizer has non-cyclic component group. Some arguments use considerations already present in an unpublished note by Bonnafé, Dat and Rouquier.

Let H be an abelian subgroup of a finite group G and π the set of prime divisors of |H|. We prove that |HOπ(G)/Oπ(G)| is bounded above by the largest character degree of G. A similar result is obtained when H is nilpotent.

We prove a determinant formula for the standard integral form of a lattice vertex operator algebra.