Showing papers in "Algebra & Number Theory in 2009"
TL;DR: In this paper, it was shown that the center Z(C) of a fusion category is equivalent to a G-equivariantization of the relative center ZD(C).
Abstract: Let C be a fusion category faithfully graded by a finite group G and let D be the trivial component of this grading. The center Z(C) of C is shown to be canonically equivalent to a G-equivariantization of the relative center ZD(C). We use this result to obtain a criterion for C to be group-theoretical and apply it to Tambara–Yamagami fusion categories. We also find several new series of modular categories by analyzing the centers of Tambara–Yamagami categories. Finally, we prove a general result about the existence of zeroes in S-matrices of weakly integral modular categories.
138 citations
TL;DR: In this article, a theory of jeu de taquin for increasing tableaux was introduced, extending fundamental work of Schutzenberger (1977) for standard Young tableaux, and applied to give a new combinatorial rule for the K-theory Schubert calculus of Grassmannians via k-theoretic JDT, providing an alternative to the rules of Buch and others.
Abstract: We introduce a theory of jeu de taquin for increasing tableaux, extending fundamental work of Schutzenberger (1977) for standard Young tableaux. We apply this to give a new combinatorial rule for the K-theory Schubert calculus of Grassmannians via K-theoretic jeu de taquin, providing an alternative to the rules of Buch and others. This rule naturally generalizes to give a conjectural root-system uniform rule for any minuscule flag variety G/P, extending recent work of Thomas and Yong. We also present analogues of results of Fomin, Haiman, Schensted and Schutzenberger.
86 citations
TL;DR: In this article, it was shown that the Hilbert scheme Hnd of n points in Ad contains an irreducible component R84⊂H84 which can be defined by a single explicit equation and serve as a criterion for deciding whether a given ideal is a limit of distinct points.
Abstract: The Hilbert scheme Hnd of n points in Ad contains an irreducible component Rnd which generically represents n distinct points in Ad. We show that when n is at most 8, the Hilbert scheme Hnd is reducible if and only if n=8 and d≥4. In the simplest case of reducibility, the component R84⊂H84 is defined by a single explicit equation, which serves as a criterion for deciding whether a given ideal is a limit of distinct points. To understand the components of the Hilbert scheme, we study the closed subschemes of Hnd which parametrize those ideals which are homogeneous and have a fixed Hilbert function. These subschemes are a special case of multigraded Hilbert schemes, and we describe their components when the colength is at most 8. In particular, we show that the scheme corresponding to the Hilbert function (1,3,2,1) is the minimal reducible example.
83 citations
TL;DR: In this paper, the authors classify standard braided vector spaces with finite-dimensional Nichols algebra and give a PBW-basis, a closed formula of the dimension and a presentation by generators and relations of the associated Nichols algebra.
Abstract: The class of standard braided vector spaces, introduced by Andruskiewitsch and the author in \texttt{arXiv:math/0703924v2} to understand the proof of a theorem of Heckenberger \cite{H2}, is slightly more general than the class of braided vector spaces of Cartan type. In the present paper, we classify standard braided vector spaces with finite-dimensional Nichols algebra. For any such braided vector space, we give a PBW-basis, a closed formula of the dimension and a presentation by generators and relations of the associated Nichols algebra.
53 citations
TL;DR: In this article, the authors studied the conditions under which an algebraic curve can be modelled by a Laurent polynomial that is nondegenerate with respect to its Newton polytope.
Abstract: We study the conditions under which an algebraic curve can be modelled by a Laurent polynomial that is nondegenerate with respect to its Newton polytope. We prove that every curve of genus g ≤ 4 over an algebraically closed field is nondegenerate in the above sense. More generally, let M g be the locus of nondegenerate curves inside the moduli space of curves of genus g ≥ 2. Then we show that dimM g = min(2g +1, 3g − 3), except for g = 7 where dimM 7 = 16; thus, a generic curve of genus g is nondegenerate if and only if g ≤ 4. Subject classification: 14M25, 14H10 Let k be a perfect field with algebraic closure k. Let f ∈ k[x, y] be an irreducible Laurent polynomial, and write f = ∑ (i,j)∈Z2 cijx y . We denote by supp(f) = {(i, j) ∈ Z : cij 6= 0} the support of f , and we associate to f its Newton polytope ∆ = ∆(f), the convex hull of supp(f) in R. We assume throughout that ∆ is 2-dimensional. For a face τ ⊂ ∆, let f |τ = ∑ (i,j)∈τ cijx y . We say that f is nondegenerate if, for every face τ ⊂ ∆ (of any dimension), the system of equations (1) f |τ = x ∂f |τ ∂x = y ∂f |τ ∂y = 0 has no solutions in k ∗2 . From the perspective of toric varieties, the condition of nondegeneracy can be rephrased as follows. The Laurent polynomial f defines a curve U(f) in the torus Tk = Spec k[x , y], and Tk embeds canonically in the projective toric surface X(∆)k associated to ∆ over k. Let V (f) be the Zariski closure of the curve U(f) inside X(∆)k. Then f is nondegenerate if and only if for every face τ ⊂ ∆, we have that V (f)∩Tτ is smooth of codimension 1 in Tτ , where Tτ is the toric component of X(∆)k associated to τ . (See Proposition 1.2 for alternative characterizations.) Nondegenerate polynomials have become popular objects in explicit algebraic geometry, owing to their connection with toric geometry [4]: a wealth of geometric information about V (f) is contained in the combinatorics of the Newton polytope ∆(f). The notion was initially employed by Kouchnirenko [22], who studied nondegenerate polynomials in the context of singularity theory. Nondenegerate polynomials emerge naturally in the theory of sparse resultants [14] and admit a linear effective Nullstellensatz [8, Section 2.3]. They make an appearance in the study of real algebraic curves in maximal position [26] and in the problem of enumerating curves through a set of prescribed points [27]. In the case where k is a finite field, they arise in the construction of curves with many points [6, 23], in the p-adic cohomology theory of Adolphson and Sperber [2], and in explicit methods for computing zeta functions of varieties over k [8]. Despite their utility and seeming Date: 26 December 2008.
44 citations
TL;DR: The missing case of Batyrev and Popov's conjecture that Cox(X) is a quadratic algebra was shown in this article, where they used a complex of vector spaces whose homology determines part of the structure of the minimal free Pic(X)-graded resolution of Cox (X) over a polynomial ring.
Abstract: Let X be a del Pezzo surface of degree one over an algebraically closed field, and let Cox(X) be its total coordinate ring. We prove the missing case of a conjecture of Batyrev and Popov, which states that Cox(X) is a quadratic algebra. We use a complex of vector spaces whose homology determines part of the structure of the minimal free Pic(X)-graded resolution of Cox(X) over a polynomial ring. We show that sufficiently many Betti numbers of this minimal free resolution vanish to establish the conjecture.
41 citations
TL;DR: In this article, the exact value of the essential dimension of the maximal torus in the projective linear group PGLn for every n ≥ 1 has been derived, where p is a prime, k is a field of characteristic 6 = p and N is the normalizer of the torus.
Abstract: Let p be a prime, k be a field of characteristic 6= p and N be the normalizer of the maximal torus in the projective linear group PGLn. We compute the exact value of the essential dimension edk(N ; p) of N at p for every n ≥ 1.
41 citations
TL;DR: In this article, the authors adapt the classic Chabauty-Coleman method to study the K-rational points of a smooth projective absolutely irreducible curve of genus g >= 2 over a number field K, and give an explicit and practical criterion for showing that a given subset L subset of C(d) (K) is in fact equal to C-(d)(K).
Abstract: Let C be a smooth projective absolutely irreducible curve of genus g >= 2 over a number field K, and denote its Jacobian by J. Let d >= 1 be an integer and denote the d-th symmetric power of C by C-(d). In this paper we adapt the classic Chabauty-Coleman method to study the K-rational points of C-(d). Suppose that J(K) has Mordell-Weil rank at most g - d . We give an explicit and practical criterion for showing that a given subset L subset of C-(d) (K) is in fact equal to C-(d) (K).
40 citations
TL;DR: In this paper, T-adic exponential sums associated to a Laurent polynomial f were introduced and the Hodge bound for the Newton polygon of L-functions of Tadic exponential sum was established.
Abstract: We introduce T-adic exponential sums associated to a Laurent polynomial f. They interpolate all classical pm-power order exponential sums associated to f. We establish the Hodge bound for the Newton polygon of L-functions of T-adic exponential sums. This bound enables us to determine, for all m, the Newton polygons of L-functions of pm-power order exponential sums associated to an f that is ordinary for m = 1. We also study deeper properties of L-functions of T-adic exponential sums. Along the way, we discuss new open problems about the T-adic exponential sum itself.
34 citations
TL;DR: In this paper, a modification of the Leonard-Pellikaan algorithm is presented, which computes the integral closure of a reduced ring that is finitely generated over a finite field.
Abstract: We present an algorithm for computing the integral closure of a reduced ring that is finitely generated over a finite field. Leonard and Pellikaan [2003] devised an algorithm for computing the integral closure of weighted rings that are finitely generated over finite fields. Previous algorithms proceed by building successively larger rings between the original ring and its integral closure [de Jong 1998; Seidenberg 1970; 1975; Stolzenberg 1968; Vasconcelos 1991; 2000]; the Leonard‐Pellikaan algorithm instead starts with the first approximation being a finitely generated module that contains the integral closure, and successive steps produce submodules containing the integral closure. The weights in [Leonard and Pellikaan 2003] impose strong restrictions, and play a crucial role in various steps of their algorithm; see Remark 1.7. We present a modification of the Leonard‐Pellikaan algorithm that works in much greater generality: it computes the integral closure of a reduced ring that is finitely generated over a finite field. We discuss an implementation of the algorithm in Macaulay 2, and provide comparisons with de Jong’s algorithm [1998].
34 citations
TL;DR: In this paper, the authors consider two obstructionions to the Hasse principle on X: the Brauer-Manin obstruction applied to etale covers of X and the descent obstruction on X and prove that the first one is at least as strong as the second.
Abstract: Soit X une variete projective lisse geometriquement integre sur un corps de nombres. On considere deux obstructions au principe de Hasse sur X : l’obstruction de Brauer–Manin appliquee aux revetements etales de X et l’obstruction de descente sur X. On demontre que la premiere est plus forte que la seconde. On en deduit, grâce a un exemple recent de Poonen, que l’obstruction de descente est insuffisante pour expliquer tous les contrexemples au principe de Hasse. Let X be a smooth, projective and geometrically integral variety over a number field. We consider two obstructions to the Hasse principle on X: the Brauer–Manin obstruction applied to etale covers of X and the descent obstruction on X. We prove that the first one is at least as strong as the second. Combining this with a recent example of Poonen shows that the descent obstruction is not sufficient to explain all counterexamples to the Hasse principle.
TL;DR: In this article, a series indexed by rooted trees and with coefficients in Q(q) was defined and studied, and it was shown that it is related to a family of Lie idempotents.
Abstract: We define and study a series indexed by rooted trees and with coefficients in Q(q). We show that it is related to a family of Lie idempotents. We prove that this series is a q-deformation of a more classical series and that some of its coefficients are Carlitz q-Bernoulli numbers.
TL;DR: In this paper, weak bimonoids in braided monoidal categories are studied and shown to be in one-to-one correspondence with quantum categories with a separable Frobenius object-of-objects.
Abstract: We develop the theory of weak bimonoids in braided monoidal categories and show that they are in one-to-one correspondence with quantum categories with a separable Frobenius object-of-objects. Weak Hopf monoids are shown to be quantum groupoids. Each separable Frobenius monoid R leads to a weak Hopf monoid R⊗R.
TL;DR: In this paper, a relationship between continued fractions and Weyl groupoids of Cartan schemes of rank two was found, which allows to decide easily if a given Cartan scheme of rank 2 admits a finite root system.
Abstract: A relationship between continued fractions and Weyl groupoids of Cartan schemes of rank two is found. This allows to decide easily if a given Cartan scheme of rank two admits a finite root system. We obtain obstructions and sharp bounds for the entries of the Cartan matrices.
Key words: Cartan matrix, continued fraction, Nichols algebra, Weyl groupoid
TL;DR: In this paper, it was shown that a Lie algebra of type E8 over a field of characteristic 5 does not have a trace symmetric bilinear form, answering a question posed in the 1960s.
Abstract: A finite-dimensional representation of an algebraic group G gives a trace symmetric bilinear form on the Lie algebra of G. We give a criterion in terms of the root system data for this form to vanish. As a corollary, we show that a Lie algebra of type E8 over a field of characteristic 5 does not have a so-called “quotient trace form”, answering a question posed in the 1960s. Let G be an algebraic group over a field F , acting on a finite-dimensional vector space V via a homomorphism ρ : G → GL(V ). The differential dρ of ρ maps the Lie algebra Lie(G) of G into gl(V ), and we put Trρ for the symmetric bilinear form Trρ(x, y) := trace(dρ(x) dρ(y)) for x, y ∈ Lie(G). We call Trρ a trace form of G. Such forms appear, for example, in the hypotheses for the Jacobson-Morozov Theorem [Ca, 5.3.1]. We prove: Theorem A. Assume G is simply connected, split, and almost simple. Then the following are equivalent: (a) The characteristic of F is a torsion prime for G. (b) Every trace form of G is zero. The set of torsion primes for G is given by the following table, cf. e.g. [St 75, 1.13]: type of G torsion primes An, Cn none Bn (n ≥ 3), Dn (n ≥ 4), G2 2 F4, E6, E7 2, 3 E8 2, 3, 5 A prime p is called a torsion prime for G if the corresponding group G(C) over C (or, equivalently, its compact form) is such that one of its homology groups, with coefficients in Z, contains an element of order p. We also prove a generalization of Theorem A that removes the hypotheses “simply connected” and “split”; it is somewhat more complicated, so we leave the statement until Th. D (and Remark 4.6). Replacing the simply connected group G with a nontrivial quotient G changes the situation in two ways: the group G has “fewer” representations and the Lie algebras of G and G may be different. These two changes are reflected in the integers N(G) and E(G) defined below. As a particular example of Th. A, for G of type E8 over a field of characteristic 2, 3, or 5, Trρ is zero for every representation ρ of G. One may ask whether the same is true for the representations of the Lie algebra Lie(G). That is, for a representation ψ of Lie(G), we write Trψ for the bilinear form (x, y) 7→ trace(ψ(x)ψ(y)), and ask 2000 Mathematics Subject Classification. 20G05 (17B50, 17B25).
TL;DR: In this article, the abstract group G(k[t]) for any semisimple (connected) simply connected k-group G is presented, and the Whitehead group and the naive group of connected components of G coincide.
Abstract: Following Soule’s ideas [14] we give a presentation of the abstract group G(k[t]) for any semisimple (connected) simply connected absolutely almost simple k–group G. As an application, we give a description of G(k[t]) in terms of direct limits, and show that the Whitehead group and the naive group of connected components of G coincide.
TL;DR: In this paper, the authors compute dimensions and characters of the components of the operad of two compatible associative products and give an explicit combinatorial construction of the corresponding free algebras in terms of planar rooted trees.
Abstract: We compute dimensions and characters of the components of the operad of two compatible associative products and give an explicit combinatorial construction of the corresponding free algebras in terms of planar rooted trees.
TL;DR: In this paper, a characteristic free version of Frobenius splittings for toric varieties is discussed and a polyhedral criterion for a toric variety to be diagonally split is given.
Abstract: We discuss a characteristic free version of Frobenius splittings for toric varieties and give a polyhedral criterion for a toric variety to be diagonally split. We apply this criterion to show that section rings of nef line bundles on diagonally split toric varieties are normally presented and Koszul, and that Schubert varieties are not diagonally split in general.
TL;DR: In this paper, it was shown that if all the higher Tor sheaves of F against G-orbits vanish, then for generic g in G, the sheaf Tor^X_j(gF, E) vanishes for all j > 0.
Abstract: Let G be a smooth algebraic group acting on a variety X. Let F and E be coherent sheaves on X. We show that if all the higher Tor sheaves of F against G-orbits vanish, then for generic g in G, the sheaf Tor^X_j(gF, E) vanishes for all j >0. This generalizes a result of Miller and Speyer for transitive group actions and a result of Speiser, itself generalizing the classical Kleiman-Bertini theorem, on generic transversality, under a general group action, of smooth subvarieties over an algebraically closed field of characteristic 0.
TL;DR: In this paper, a general formalism for equivariant Schubert calculus consisting of a basis basis is developed, which can be adapted to give a formalism that is applicable to equivariance.
Abstract: In previous work we have developed a general formalism for Schubert calculus. Here we show how this theory can be adapted to give a formalism for equivariant Schubert calculus consisting of a basis ...
TL;DR: In this article, it was shown that the secant variety of a linearly normal smooth curve of degree at least 2g+3 is arithmetically Cohen-Macaulay.
Abstract: We show the secant variety of a linearly normal smooth curve of degree at least 2g+3 is arithmetically Cohen–Macaulay, and we use this information to study the graded Betti numbers of the secant variety.
TL;DR: In this paper, the effectiveness of the log Iitaka fibration in Kodaira codimension two for varieties of dimension ≤ 4 was proved, and it was shown that if the fiber is of dimension two, the denominator of the moduli part is bounded.
Abstract: We prove the effectiveness of the log Iitaka fibration in Kodaira codimension two for varieties of dimension ≤4. In particular, we finish the proof of effective log Iitaka fibration in dimension two. Also, we show that for the log Iitaka fibration, if the fiber is of dimension two, the denominator of the moduli part is bounded.
TL;DR: In this article, a pencil of Enriques surfaces with no section and index 1 was shown to be a pencil-of-Enriques, and the same technique was used to prove the existence of witness curves for dimension d.
Abstract: Monodromy arguments and deformation-and-specialization are used to prove existence of a pencil of Enriques surfaces with no section and index 1. The same technique completes the strategy from [GHMS05, §7.3] proving the family of witness curves for dimension d depends on the integer d.
TL;DR: In this paper, it was shown that termination of terminal log flips in dimension d implies that any klt pair of dimension d has a log minimal model or a Mori fibre space.
Abstract: Following Shokurov’s ideas, we give a short proof of the following klt version of his result: termination of terminal log flips in dimension d implies that any klt pair of dimension d has a log minimal model or a Mori fibre space Thus, in particular, any klt pair of dimension 4 has a log minimal model or a Mori fibre space
TL;DR: In this paper, the Burnside quotient Green ring of a Mackey functor is defined and studied, together with applications to computation results for K-theory and Ltheory of finite and infinite groups.
Abstract: We define and study the Burnside quotient Green ring of a Mackey functor, introduced in our MSRI preprint (18). Some refinements of Dress induction theory are presented, together with applications to computation results for K-theory and L-theory of finite and infinite groups.
TL;DR: In this article, an additive dilogarithm Li2,n : B2(k[e]n) → k⊕(n−1), where B2 is the Bloch group which is crucial in studying weight two motivic cohomology.
Abstract: Let k be a field of characteristic zero, and k[e]n := k[e]/(e ). We construct an additive dilogarithm Li2,n : B2(k[e]n) → k⊕(n−1), where B2 is the Bloch group which is crucial in studying weight two motivic cohomology. We use this construction to show that the Bloch complex of k[e]n has cohomology groups expressed in terms of the K-groups K·(k[e]n) as expected. Finally we compare this construction to the construction of the additive dilogarithm by Bloch and Esnault [5] defined on the complex TnQ(2)(k).
TL;DR: The recent proof of the Boij-Soderberg conjectures reveals new structure about Betti diagrams of modules, giving a complete description of the cone of Betti diagram.
Abstract: The recent proof of the Boij-Soderberg conjectures reveals new structure about Betti diagrams of modules, giving a complete description of the cone of Betti diagrams. We begin to expand on this new structure by investigating the semigroup of Betti diagrams. We prove that this semigroup is finitely gener- ated, and we answer several other fundamental questions about this semigroup.
TL;DR: In this paper, the Artin braid groups and Bergman's non-locally-indicable right orderable groups are considered, and necessary and sufficient conditions for right-orderable groups to be discretely orderable.
Abstract: We consider group orders and right-orders which are discrete, meaning there is a least element which is greater than the identity. We note that free groups cannot be given discrete orders, although they do have right-orders which are discrete. More generally, we give necessary and sufficient conditions that a given orderable group can be endowed with a discrete order. In particular, every orderable group G embeds in a discretely orderable group. We also consider conditions on right-orderable groups to be discretely right-orderable. Finally, we discuss a number of illustrative examples involving discrete orderability, including the Artin braid groups and Bergman's non-locally-indicable right orderable groups.