Shuffle Algebras, Homology and Consecutive Pattern Avoidance
This paper has been started as a particular application of the method of resolutions via Grobner bases we suggested here. We introduce a notion of a shuffle algebra. A shuffle algebra is a Z+-graded vector space V=∪∞i=1 such that for any pair (i,j) there exists a collection of operations ∗σ:Vi⊗Vj→Vi+j numbered by (i,j)-shuffle permutations σ∈Si+j (i.e. σ preserves the order of the first i elements and the order of the last j elements) yielding the natural associativity conditions. Enumerative problems for monomial shuffle algebras are in one-to-one correspondence with the pattern avoidance problems for permutations. We present two homological results on shuffle algebras with monomial relations, and use them to prove exact and asymptotic results on consecutive pattern avoidance in permutations. Both results generalizes the classical results for associative algebras. The first homological result is a generalization of the Golod-Shafarevich theorem and the second one generalizes the theory of Anick chains. It seems that most of particular applications we discuss are known to specialists but the general method was definitely not known. We hope that it will simplify a lot of work in this area. It is not hard to see that shuffle algebras form an interesting class of binary shuffle operads and illustrates quite well the importance of the latter notion.
This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.
The main goal of this paper is to present a way to compute Quillen homology of a shuffle operad with a known Grobner basis. Similar to the strategy taken in a celebrated paper of David Anick, our approach goes in several steps. We define a combinatorial resolution for the ``monomial replacement'' of a shuffle operad, explain how to ``deform'' the differential to handle the general case, and find explicit representatives of Quillen homology for a large class of operads with monomial relations. We present various applications, including a new proof of Hoffbeck's PBW criterion, a proof of Koszulness for a class of operads coming from commutative algebras, and a homology computation for the operads of Batalin--Vilkovisky algebras and of Rota--Baxter algebras. The method of writing a resolution presented in this paper is very general. Namely, whenever you have a category which admits a theory of monomials (including Grobner bases and Buchberger algorithm) you can do the same procedure: First, take an object with a chosen Grobner basis. Second, define a resolution for an object with monomial relations (the monomial replacement of the starting object). Third, lower terms of relations will affect additional summands in the description of the differential in the resolution.
This survey paper presents general approach to the wellknown F5 algorithm for calculating Gröbner bases, which was created by Faugère in 2002.
Sequential and parallel implementations of the F4 algorithm for computing Gr¨obner bases of polynomial ideals are discussed.
We prove a simple homological expression for the homology of a connected spectrum represented by an infinite loop space via the Segal machine. The expression is essentially due to Pirashvili but not stated explicitly in his work; we give an independent proof.
A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.
Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.
Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.