this paper, we consider the problem of enumerating the fully commutative elements of these groups. The main result (Theorem 2.6) is that for six of the seven infinite families (we omit the trivial dihedral family I 2 (m)), the generating function for the number of fully commutative elements can be expressed in terms of three simpler generating functions for certain formal languages over an alphabet with at most four letters. The languages in question vary from family to family, but have a uniform description. The resulting generating function one obtains for each family is algebraic, although in some cases quite complicated. (See (3.7) and (3.11).) In a general Coxeter group, the fully commutative elements index a basis for a natural quotient of the corresponding Iwahori-Hecke algebra [G]. (See also [F1] for the simplylaced case.) For An , this quotient is the Temperley-Lieb algebra. Recently, Fan [F2] has shown that for types A, B, D, E and (in a sketched proof) F , this quotient is generically semisimple, and gives recurrences for the dimensions of the irreducible representations. (For type H, the question of semisimplicity remains open.) This provides another possible approach to computing the number of fully commutative elements in these cases; namely, as the sum of the squares of the dimensions of these representations. Interestingly, Fan also shows that the sum of these dimensions is the number of fully commutative involutions

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The main results of the present paper are the equivalence of definability by monadic second-order logic and recognizability for real trace languages, and that first-order definable, star-free, and aperiodic real trace languages form the same class of languages. This generalizes results on infinite words and on finite traces to infinite traces. It closes an important gap in the different characterizations of recognizable languages of infinite traces. 1 Introduction In the late 70's, A. Mazurkiewicz introduced the notion of trace as a suitable mathematical model for concurrent systems [16] (for surveys on this topic see also [1, 6, 10, 17]). In this framework, a concurrent system is seen as a set \Sigma of atomic actions together with a fixed irreflexive and symmetric independence relation I ` \Sigma \Theta \Sigma. The relation I specifies pairs of actions which can be carried out in parallel. It generates an equivalence relation on the set of sequential observations of the system. As ...

Introduction This work was started as an attempt to apply theory from noncommutative graded algebra to questions about the holonomy algebra of a hyperplane arrangement. We soon realized these algebras and their deformations form a class of quadratic graded algebras that have not been studied much and are interesting to algebra, arrangement theory and combinatorics. Let X be a topological space having homotopy type of a finite cell complex and H (X) its homology coalgebra with coefficients in a field and comultiplication dual to the cup product. Then the holonomy Lie algebra GX of X is the quotient of the free Lie algebra on H 1 (X) over the ideal generated by the image of the comultiplication H 2 (X) ! 2 (H 1 (X)). The universal

The most fundamental precept of the mathematical faith is thou shalt prove everything rigorously. While the practitioners of mathematics differ in their view of what constitutes a rigorous proof, and there are fundamentalists who insist on even a more rigorous rigor than the one practiced by the mainstream, the belief in this principle could be taken as the defining property of mathematician. The Day After Tomorrow There are writings on the wall that, now that the silicon savior has arrived, a new testament is going to be written. Although there will always be a small group of “rigorous ” old-style mathematicians(e.g. [JQ]) who will insist that the true religion is theirs, and that the computer is a false Messiah, they may be viewed by future mainstream mathematicians as a fringe sect of harmless eccentrics, like mathematical physicists are viewed by regular physicists today. The computer has already started doing to mathematics what the telescope and microscope did to astronomy and biology. In the future, not all mathematicians will care about absolute certainty, since there will be so many exciting new facts to discover: mathematical pulsars and quasars that will make the Mandelbrot set seem like a mere Jovian moon. We will have (both human and machine 2) professional theoretical mathematicians, who will develop conceptual paradigms to make

Summary. Markov chains on finite sets are used in a great variety of situations to approximate, understand and sample from their limit distribution. A familiar example is provided by card shuffling methods. From this viewpoint, one is interested in the “mixing time ” of the chain, that is, the time at which the chain gives a good approximation of the limit distribution. A remarkable phenomenon known as the cut-off phenomenon asserts that this often happens abruptly so that it really makes sense to talk about “the mixing time”. Random walks on finite groups generalize card shuffling models by replacing the symmetric group by other finite groups. One then would like to understand how the structure of a particular class of groups relates to the mixing time of natural random walks on those groups. It turns out that this is an extremely rich problem which is very far to be understood. Techniques from a great

Given a finite simplicial graph % the graph group G< § is the group with generators in one-to-one correspondence with the vertices of ^ and with relations stating that two generators commute if their associated vertices are adjacent in % The Bieri-Neumann-Strebel invariant can be explicitly described in terms of the original graph ^ and hence there is an explicit description of the distribution of finitely generated normal subgroups of G^S with abelian quotient. We construct Eilenberg-MacLane spaces for graph groups and find partial extensions of this work to the higher-dimensional invariants.

This paper is dedicated to Gian-Carlo Rota, on his millionth2’s birthday. Abstract. We derive combinatorial proofs of the main two evaluations of the Ihara-Selberg zeta function associated with a graph. We give three proofs of the first evaluation all based on the algebra of Lyndon words. In the third proof it is shown that the first evaluation is an immediate consequence of Amitsur’s identity on the characteristic polynomial of a sum of matrices. The second evaluation of the Ihara-Selberg zeta function is first derived by means of a sign-changing involution technique. Our second approach makes use of a short matrix-algebra argument. 1.

We investigate the Koszul property for quotients of affine semigroup rings by semigroup ideals. Using a combinatorial and topological interpretation for the Koszul property in this context, we recover known results asserting that certain of these rings are Koszul. In the process, we prove a stronger fact, suggesting a more general definition of Koszul rings. This more general definition of Koszulness turns out to be satisfied by all Cohen-Macaulay rings of minimal multiplicity.

This paper shows the equivalence between the family of recognizable languages over infinite traces and the family of languages which are recognized by deterministic asynchronous cellular Muller automata. We thus give a proper generalization of McNaughton's Theorem from infinite words to infinite traces. Thereby we solve one of the main open problems in this field. As a special case we obtain that every closed (w.r.t. the independence relation) word language is accepted by some I-diamond deterministic Muller automaton. 1 Introduction A. Mazurkiewicz introduced the concept of traces as a suitable semantics for concurrent systems [Maz77]. A concurrent system is given by a set of atomic actions \Sigma = fa; b; c; : : :g together with an independence relation I ` \Sigma \Theta \Sigma, which specifies pairs of actions which can be performed concurrently. This leads to an equivalence relation on \Sigma generated by the independence relation I. More precisely, if a and b denote independent...