Structure of the Malvenuto-Reutenauer Hopf algebra of permutations (Extended Abstract)

We analyze the structure of the Malvenuto-Reutenauer Hopf algebra of permutations in detail. We give explicit formulas for its antipode, prove that it is a cofree coalgebra, determine its primitive elements and its coradical filtration and show that it decomposes as a crossed product over the Hopf algebra of quasi-symmetric functions. We also describe the structure constants of the multiplication as a certain number of facets of the permutahedron. Our results reveal a close relationship between the structure of this Hopf algebra and the weak order on the symmetric groups.Comment: 12 pages, 2 .eps figures. (minor revisions) Extended abstract for Formal Power Series and Algebraic Combinatorics, Melbourne, July 200

The algebra of spatio-temporal intervals

[[abstract]]The relations among temporal intervals can be used to model all time dependent objects. We propose a fast mechanism for temporal relation compositions. A temporal transitive closure table is derived, and an interval-based temporal relation algebraic system is constructed. Thus, we propagate the time constraints of arbitrary two objects across long distances n by linear time. We also give a complete discussion of different possible domains of interval relations. A set of algorithms is proposed to detect time conflicts and to derive reasonable interval relations. The algorithms are extended for time-based media in an arbitrary n-dimensional space[[notice]]補正完畢[[conferencetype]]國際[[conferencedate]]19980121~19980121[[conferencelocation]]Tokyo, Japa

A uniform model for Kirillov-Reshetikhin crystals. Extended abstract

We present a uniform construction of tensor products of one-column Kirillov-Reshetikhin (KR) crystals in all untwisted affine types, which uses a generalization of the Lakshmibai-Seshadri paths (in the theory of the Littelmann path model). This generalization is based on the graph on parabolic cosets of a Weyl group known as the parabolic quantum Bruhat graph. A related model is the so-called quantum alcove model. The proof is based on two lifts of the parabolic quantum Bruhat graph: to the Bruhat order on the affine Weyl group and to Littelmann's poset on level-zero weights. Our construction leads to a simple calculation of the energy function. It also implies the equality between a Macdonald polynomial specialized at t=0 and the graded character of a tensor product of KR modules.Comment: 10 pages, 1 figur

Moduli space actions on the Hochschild Co-Chains of a Frobenius algebra I: Cell Operads

This is the first of two papers in which we prove that a cell model of the moduli space of curves with marked points and tangent vectors at the marked points acts on the Hochschild co--chains of a Frobenius algebra. We also prove that a there is dg--PROP action of a version of Sullivan Chord diagrams which acts on the normalized Hochschild co-chains of a Frobenius algebra. These actions lift to operadic correlation functions on the co--cycles. In particular, the PROP action gives an action on the homology of a loop space of a compact simply--connected manifold. In this first part, we set up the topological operads/PROPs and their cell models. The main theorems of this part are that there is a cell model operad for the moduli space of genus $g$ curves with $n$ punctures and a tangent vector at each of these punctures and that there exists a CW complex whose chains are isomorphic to a certain type of Sullivan Chord diagrams and that they form a PROP. Furthermore there exist weak versions of these structures on the topological level which all lie inside an all encompassing cyclic (rational) operad.Comment: 50 pages, 7 figures. Newer version has minor changes. Some material shifted. Typos and small things correcte