Daniel Krob

THE EQUALITY PROBLEM FOR RATIONAL SERIES WITH MULTIPLICITIES IN THE TROPICAL SEMIRING IS UNDECIDABLE

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Noncommutative Symmetric Functions Iv: Quantum Linear Groups andHecke Algebras at {\bi q}\,{\bf =}\,{\bf 0}

We present representation theoretical interpretations ofquasi-symmetric functions and noncommutative symmetric functions in terms ofquantum linear groups and Hecke algebras at q = 0. We obtain inthis way a noncommutative realization of quasi-symmetric functions analogousto the plactic symmetric functions of Lascoux and Schützenberger. Thegeneric case leads to a notion of quantum Schur function.

Noncommutative symmetric functions II : Transformations of alphabets

Noncommutative analogues of classical operations on symmetric functions are investigated, and applied to the description of idempotents and nilpotents in descent algebras. It is shown that any sequence of Lie idempotents (one in each descent algebra) gives rise to a complete set of indecomposable orthogonal idempotents of each descent algebra, and various deformations of the classical sequences of Lie idempotents are obtained. In particular, we obtain several q-analogues of the Eulerian idempotents and of the Garsia-Reutenauer idempotents.

Minor identities for quasi-determinants and quantum determinants

We present several identities involving quasi-minors of noncommutative generic matrices. These identities are specialized to quantum matrices, yieldingq-analogues of various classical determinantal formulas.

The lower central series of the free partially commutative group

This paper is devoted to the study of the lower central series of the free partially commutative groupF(A, ϑ) in connection with the associated free partially commutative Lie algebra. Using a convenient Magnus transformation, we show that the quotients of the lower central series ofF(A, ϑ) are free abelian groups and thatF(A, ϑ) can be fully ordered.

NONCOMMUTATIVE SYMMETRIC FUNCTIONS V: A DEGENERATE VERSION OF Uq(glN)

We interpret quasi-symmetric functions and noncommutative symmetric functions as characters of a degenerate quantum group obtained by putting q=0 in a variant of Uq(glN).

A complete system of identities for one-letter rational expressions with multiplicities in the tropical semiring

Abstract We construct a complete system of rational identities for the equational theory of the algebra of one-letter rational expressions with multiplicities in the tropical semiring.

The free partially commutative Lie algebra: Bases and ranks

Abstract In this paper, we study the free partially commutative Lie K -algebra L ( A , θ ) defined by a commutation relation θ on an alphabet A . Its behavior is very similar to that of the free Lie algebra. Indeed, we obtain in particular a partially commutative version of Lazards elimination process which allows us to prove that the K -module L ( A , θ ) is free and to construct explicitly K -bases for it. We show also how the classical Witts calculus can be extended to L ( A , θ ).

PARTIALLY COMMUTATIVE MAGNUS TRANSFORMATIONS

Grâce a l’introduction de la notion de forme exponentielle reduite, nous montrons l’injectivite de la transformation de Magnus associee au groupe partiellement commutatif libre F(A, ϑ). Nous en deduisons la separation de la topologie p-adique de F(A, ϑ), l’existence de plus petites racines dans F(A, ϑ) et enfin la structure des centralisateurs dans F(A, ϑ). We introduce the notion of reduced exponential form in the free partially commutative group F(A, ϑ). This allows us to prove that the Magnus transformation associated with F(A, ϑ) is injective. We deduce from this result that the p-adic topology on F(A, ϑ) is Hausdorff, that there exist least roots in F(A, ϑ) and finally the exact structure of centralizers in F(A, ϑ).

Some consequences of a Fatou property of the tropical semiring

Abstract We show that the equatorial semiring is a Fatou extension of the tropical semiring . This property allows us to give partial decidability results for the equality problem for rational series with multiplicities in the tropical semiring. We also deduce from it the decidability of the limitedness problem for the equatorial semiring, solving therefore a question of I. Simon.