The Green polynomials via vertex operators

An iterative formula for the Green polynomial is given using the vertex operator realization of the Hall-Littlewood function. Based on this, (1) a general combinatorial formula of the Green polynomial is given; (2) several compact formulas are given for Green's polynomials associated with upper partitions of length ≤3 and the diagonal lengths ≤3; (3) a Murnaghan-Nakayama type formula for the Green polynomial is obtained; and (4) an iterative formula is derived for the bitrace of the finite general linear group G and the Iwahori-Hecke algebra of type A on the permutation module of G by its Borel subgroup.     

Multiplicities of discriminants

We compute the multiplicity of the discriminant of a line bundle ￡ over a nonsingular varietyS at a given sectionX, in terms of the Chern classes of ￡ and of the cotangent bundle ofS, and the Segre classes of the jacobian scheme ofX inS. ForS a surface, we obtain a precise formula that expresses the multiplicity as a sum of a term due to the non-reduced components of the section, and a term that depends on the Milnor numbers of the singularities ofX _red. Also, under certain hypotheses, we provide formulas for the “higher discriminants” that parametrize sections with a singular point of prescribed multiplicity. As an application, we obtain criteria for the various discriminants to be “small”. Supported in part by the Max-Planck-Institut für Mathematik     

Enveloping ringoids

We present notions of module over a universal algebra, and linear representation of a universal algebra, which have gained currency with categorical algebraists, we give several intriguing examples of these objects, showing that important aspects of universal algebras can be studied in this context, and we describe the theory ofenveloping ringoids, which have categories of left modules equivalent to corresponding categories of modules and linear representations. Algebras in many of the familiar varieties of algebras, which have underlying groups, turn out to have enveloping ringoids that are equivalent to familiar rings. Nonempty algebras in an abelian varietyV have an enveloping ringoid equivalent toR(V), the so-calledring of the variety V.Dedicated to the memory of Alan DayPresented by J. Sichler.     

A mass formula for self-dual permutation codes

For a module V over a finite semisimple algebra A we give the total number of self-dual codes in V. This enables us to obtain a mass formula for self-dual codes in permutation representations of finite groups over finite fields of coprime characteristic.     

Sum formulas for reductive algebraic groups

Let V be a Weyl module either for a reductive algebraic group G or for the corresponding quantum group Uq. If G is defined over a field of positive characteristic p, respectively if q is a primitive lth root of unity (in an arbitrary field) then V has a Jantzen filtration V=V⁰⊃V¹⊃?⊃Vr=0. The sum of the positive terms in this filtration satisfies a well-known sum formula.If T denotes a tilting module either for G or Uq then we can similarly filter the space HomG(V,T), respectively HomUq(V,T) and there is a sum formula for the positive terms here as well.We give an easy and unified proof of these two (equivalent) sum formulas. Our approach is based on an Euler type identity which we show holds without any restrictions on p or l. In particular, we get rid of previous such restrictions in the tilting module case.     

Determinant formula and a realization for the Lie algebra <Emphasis Type="Italic">W</Emphasis> (2, 2)

In this paper, an explicit determinant formula is given for the Verma modules over the Lie algebra W(2, 2). We construct a natural realization of certain vaccum module for the algebra W(2, 2) via theWeyl vertex algebra. We also describe several results including the irreducibility, characters and the descending filtrations of submodules for the Verma module over the algebra W(2, 2).     

Clifford theory for graded fusion categories

We develop a categorical analogue of Clifford theory for strongly graded rings over graded fusion categories. We describe module categories over a fusion category graded by a group G as induced from module categories over fusion subcategories associated with the subgroups of G. We define invariant C _e -module categories and extensions of C _e -module categories. The construction of module categories over C is reduced to determining invariant module categories for subgroups of G and the indecomposable extensions of these module categories. We associate a G-crossed product fusion category to each G-invariant C _e -module category and give a criterion for a graded fusion category to be a group-theoretical fusion category. We give necessary and sufficient conditions for an indecomposable module category to be extendable.     

Boundary slopes and the logarithmic limit set

The A-polynomial of a manifold whose boundary consists of a single torus is generalised to an eigenvalue variety of a manifold whose boundary consists of a finite number of tori, and the set of strongly detected boundary curves is determined by Bergman's logarithmic limit set, which describes the exponential behaviour of the eigenvalue variety at infinity. This enables one to read off the detected boundary curves of a multi-cusped manifold in a similar way to the 1-cusped case, where the slopes are encoded in the Newton polygon of the A-polynomial.     

The picard group of the variety of plane cuspidal cubics of ℙ3

We give a compactification of the varietyU of non-degenerate plane cuspidal cubics of ?³. We construct this compactification by means of the projective bundleX of a suitable vector bundleE. We describe the intersection ring ofX and, as a consequence, we obtain the intersection numbers ofU that satisfy 10 conditions of the following kinds:ρ, that the plane determined by the cuspidal cubic go through a point;c, that the cusp be on a plane;q, that the cuspidal tangent intersect a line;μ, that the cuspidal cubic intersect a line. Moreover, we prove that the Picard group of the varietyU is a product of two infinite cyclic groups generated byρ andc?q.     

Trivial module for ortho-symplectic Lie superalgebras and Littlewood’s formula

We show that the denominator identity for ortho-symplectic Lie superalgebras osp(k|2n) is equivalent to the Littlewood’s formula. Such an equivalence also implies the relation between the trivial module and generalized Verma modules for osp(k|2n). Furthermore, we discuss the harmonic representative elements of the Kostant’s u-cohomology with trivial coefficients.     

On the support of complete intersection zero-cycles

On an algebraic varietyY ? ? _N we will call complete intersection a 0-cycle when it is the intersection of Y with a codimension n complete intersection of ? _N . We consider the following problem: Let E?Y be given. Does E contain the support of a complete intersection 0-cycle? The two main theorems shown in this article give the answers in some cases: first, a negative answer for E some “big” subset of a singular irreducible algebraic variety; secondly, a positive answer for some “small” subset, on any algebraic variety.     

Weakly Koszul-like modules

The Koszul-like property for any finitely generated graded modules over a Koszul-like algebra is investigated and the notion of weakly Koszul-like module is introduced. We show that a finitely generated graded module M is a weakly Koszul-like module if and only if it can be approximated by Koszul-like graded submodules, which is equivalent to the fact that G(M) is a Koszul-like module, where G(M) denotes the associated graded module of M. As applications, the relationships between minimal graded projective resolutions of M and G(M), and Koszul-like submodules are established. Moreover, the Koszul dual of a weakly Koszul-like module is proved to be generated in degree 0 as a graded E(A)-module.     

An affirmative answer to a question of Ciliberto

We give an example of a nondegeneraten-dimensional smooth projective varietyX inP ²ⁿ⁺¹ with the canonical bundle ample a varietyX whose tangent variety TanX has dimension less than 2n over an algebraically closed field of any characteristic whenn≥9. This varietyX is not ruled by lines and the embedded tangent space at a general point ofX intersectsX at some other points, so that this yields an affirmative answer to a question of Ciliberto.     

Weyl modules, Demazure modules, KR-modules, crystals, fusion products and limit constructions

We study finite-dimensional representations of current algebras, loop algebras and their quantized versions. For the current algebra of a simple Lie algebra of type ADE, we show that Kirillov-Reshetikhin modules and Weyl modules are in fact all Demazure modules. As a consequence one obtains an elementary proof of the dimension formula for Weyl modules for the current and the loop algebra. Further, we show that the crystals of the Weyl and the Demazure module are the same up to some additional label zero arrows for the Weyl module.For the current algebra Cg of an arbitrary simple Lie algebra, the fusion product of Demazure modules of the same level turns out to be again a Demazure module. As an application we construct the Cg-module structure of the Kac-Moody algebra -module V(?Λ₀) as a semi-infinite fusion product of finite-dimensional Cg-modules.     

Monoid varieties defined byx n+1 =x are local

We offer a new proof of the theorem in the title. In fact, we prove that for any varietyH of groups of finite exponent, the varietyCR(H) of all completely regular monoids with subgroups fromH, is local. The analogous result holds for pseudovarieties. A previously published proof of the theorem in the title has been found deficient.     

On the zeta function associated with module classes of a number field

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The goal of this note is to generalize a formula of Datskovsky and Wright on the zeta function associated with integral binary cubic forms. We show that for a fixed number field K of degree d, the zeta function associated with decomposable forms belonging to K in d−1 variables can be factored into a product of Riemann and Dedekind zeta functions in a similar fashion. We establish a one-to-one correspondence between the pure module classes of rank d−1 of K and the integral ideals of width <d−1. This reduces the problem to counting integral ideals of a special type, which can be solved using a tailored Moebius inversion argument. As a by-product, we obtain a characterization of the conductor ideals for orders of number fields.

Video

For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=RePyaF8vDnE.     

Boolean-like algebras

Using Vaggione’s concept of central element in a double-pointed algebra, we introduce the notion of Boolean-like variety as a generalisation of Boolean algebras to an arbitrary similarity type. Appropriately relaxing the requirement that every element be central in any member of the variety, we obtain the more general class of semi-Boolean-like varieties, which still retain many of the pleasing properties of Boolean algebras. We prove that a double-pointed variety is discriminator if and only if it is semi-Boolean-like, idempotent, and 0-regular. This theorem yields a new Maltsev-style characterisation of double-pointed discriminator varieties.     

Factorization of transformations over a local ring

Let V be a finitely generated free module over a local ring R, and π an invertible linear transformation for V. Then π is a product of simple mappings. In addition, the determinant of each simple factor but one can be chosen to be any given unit in R. The smallest number of factors needed is not greater than r+1, where r is the codimension of the vector space that is associated with the module of vectors fixed under π.     

On the second successive minimum of the Jacobian of a Riemann surface

To a compact Riemann surface of genus g can be assigned a principally polarized abelian variety (PPAV) of dimension g, the Jacobian of the Riemann surface. The Schottky problem is to discern the Jacobians among the PPAVs. Buser and Sarnak showed that the square of the first successive minimum, the squared norm of the shortest non-zero vector in the lattice of a Jacobian of a Riemann surface of genus g is bounded from above by log(4g), whereas it can be of order g for the lattice of a PPAV of dimension g. We show that in the case of a hyperelliptic surface this geometric invariant is bounded from above by a constant and that for any surface of genus g the square of the second successive minimum is equally of order log(g). We obtain improved bounds for the kth successive minimum of the Jacobian, if the surface contains small simple closed geodesics.