Journal of Algebraic Combinatorics 

About: Journal of Algebraic Combinatorics is an academic journal published by Springer Science+Business Media. The journal publishes majorly in the area(s): Mathematics & Combinatorics. It has an ISSN identifier of 0925-9899. Over the lifetime, 1584 publications have been published receiving 29519 citations.

Some Combinatorial Properties of Schubert Polynomials

Abstract: Schubert polynomials were introduced by Bernstein et al. and Demazure, and were extensively developed by Lascoux, Schutzenberger, Macdonald, and others. We give an explicit combinatorial interpretation of the Schubert polynomial {\mathfrak S}_w in terms of the reduced decompositions of the permutation w. Using this result, a variation of Schensted's correspondence due to Edelman and Greene allows one to associate in a natural way a certain set {\cal M}_w of tableaux with w, each tableau contributing a single term to {\mathfrak S}_w. This correspondence leads to many problems and conjectures, whose interrelation is investigated. In Section 2 we consider permutations with no decreasing subsequence of length three (or 321-avoiding permutations). We show for such permutations that {\mathfrak S}_w is a flag skew Schur function. In Section 3 we use this result to obtain some interesting properties of the rational function s_{\lambda/\mu}(1,q,q^2,\cdots), where s_{\lambda/\mu} denotes a skew Schur function.

The Subconstituent Algebra of an Association Scheme, (Part I)

Abstract: We introduce a method for studying commutative association schemes with “many” vanishing intersection numbers and/or Krein parameters, and apply the method to the P- and Q-polynomial schemes. Let Y denote any commutative association scheme, and fix any vertex x of Y. We introduce a non-commutative, associative, semi-simple \Bbb {C}-algebra T e T(x) whose structure reflects the combinatorial structure of Y. We call T the subconstituent algebra of Y with respect to x. Roughly speaking, T is a combinatorial analog of the centralizer algebra of the stabilizer of x in the automorphism group of Y. In general, the structure of T is not determined by the intersection numbers of Y, but these parameters do give some information. Indeed, we find a relation among the generators of T for each vanishing intersection number or Krein parameter. We identify a class of irreducible T-moduIes whose structure is especially simple, and say the members of this class are thin. Expanding on this, we say Y is thin if every irreducible T(y)-module is thin for every vertex y of Y. We compute the possible thin, irreducible T-modules when Y is P- and Q-polynomial. The ones with sufficiently large dimension are indexed by four bounded integer parameters. If Y is assumed to be thin, then “sufficiently large dimension” means “dimension at least four”. We give a combinatorial characterization of the thin P- and Q-polynomial schemes, and supply a number of examples of these objects. For each example, we show which irreducible T-modules actually occur. We close with some conjectures and open problems.

Conjectures on the Quotient Ring by Diagonal Invariants

Abstract: We formulate a series of conjectures (and a few theorems) on the quotient of the polynomial ring {\Bbb Q}[x_1,\ldots , x_n, y_1, \ldots , y_n] in two sets of variables by the ideal generated by all Sn invariant polynomials without constant term The theory of the corresponding ring in a single set of variables X e lx1, …, xnr is classical Introducing the second set of variables leads to a ring about which little is yet understood, but for which there is strong evidence of deep connections with many fundamental results of enumerative combinatorics, as well as with algebraic geometry and Lie theory

On the Fully Commutative Elements of Coxeter Groups

Abstract: Let W be a Coxeter group. We define an element w ∈ W to be fully commutative if any reduced expression for w can be obtained from any other by means of braid relations that only involve commuting generators. We give several combinatorial characterizations of this property, classify the Coxeter groups with finitely many fully commutative elements, and classify the parabolic quotients whose members are all fully commutative. As applications of the latter, we classify all parabolic quotients with the property that (1) the Bruhat ordering is a lattice, (2) the Bruhat ordering is a distributive lattice, (3) the weak ordering is a distributive lattice, and (4) the weak ordering and Bruhat ordering coincide.

Quasi-Shuffle Products

Abstract: Given a locally finite graded set A and a commutative, associative operation on A that adds degrees, we construct a commutative multiplication a on the set of noncommutative polynomials in A which we call a quasi-shuffle products it can be viewed as a generalization of the shuffle product III. We extend this commutative algebra structure to a Hopf algebra (U, a, Δ)s in the case where A is the set of positive integers and the operation on A is addition, this gives the Hopf algebra of quasi-symmetric functions. If rational coefficients are allowed, the quasi-shuffle product is in fact no more general than the shuffle products we give an isomorphism exp of the shuffle Hopf algebra (U, III, Δ) onto (U, a, Δ) the set L of Lyndon words on A and their images l exp(w) m w e Lr freely generate the algebra (U, a). We also consider the graded dual of (U, a, Δ). We define a deformation a_q of a that coincides with a when q e 1 and is isomorphic to the concatenation product when q is not a root of unity. Finally, we discuss various examples, particularly the algebra of quasi-symmetric functions (dual to the noncommutative symmetric functions) and the algebra of Euler sums.