arXiv subjects
Publications and source records attributed to J. -Y. Thibon.
We define two-parameter families of noncommutative symmetric functions and quasi-symmetric functions, which appear to be the proper analogues of the Macdonald symmetric functions in these settings.
We exhibit a vertex operator which implements multiplication by power-sums of Jucys-Murphy elements in the centers of the group algebras of all symmetric groups simultaneously. The coefficients of this operator generate a representation of ${\cal W}_{1+\infty}$, to which operators multiplying by normalized conjugacy classes are also shown to belong. A new derivation of such operators based on matrix integrals is proposed, and our vertex operator is used to give an alternative approach to the polynomial functions on Young diagrams introduced by Kerov and Olshanski.
This article is devoted to the study of several algebras which are related to symmetric functions, and which admit linear bases labelled by various combinatorial objects: permutations (free quasi-symmetric functions), standard Young tableaux (free symmetric functions) and packed integer matrices (matrix quasi-symmetric functions). Free quasi-symmetric functions provide a kind of noncommutative Frobenius characteristic for a certain category of modules over the 0-Hecke algebras. New examples of indecomposable $H_n(0)$-modules are discussed, and the homological properties of $H_n(0)$ are computed for small $n$. Finally, the algebra of matrix quasi-symmetric functions is interpreted as a convolution algebra.
Starting from known $q$-analogues of ordinary SU(n) tensor products multiplicities, we introduce $q$-analogues of the fusion coefficients of the WZW conformal field theories associated with SU(n). We conjecture combinatorial interpretations of these polynomials, which can be proved in special cases. This allows us to derive in a simple way various kinds of branching functions, the simplest ones being the characters of the minimal unitary series of the Virasoro algebra. We also obtain $q$-analogues of the dimensions of spaces of nonabelian theta functions.
We give a combinatorial algorithm for computing Zelevinsky's involution of the set of isomorphism classes of irreducible representations of the affine Hecke algebra $\H_m(t)$ when $t$ is a primitive $n$th root of 1. We show that the same map can also be interpreted in terms of aperiodic nilpotent orbits of $\Zb/n\Zb$-graded vector spaces.
We study the restrictions of simple modules of Ariki-Koike algebras $\H_m(\v)$ with set of parameters $\v= (ζ;ζ^{v_0},... ,ζ^{v_{l-1}})$, where $ζ$ is an $n$th root of unity, to their subalgebras $\H_{m-j}(\v)$. Using a theorem of Ariki and the crystal basis theory of Kashiwara, we relate this problem to the calculation of tensor product multiplicities of highest weight irreducible representations of the affine Lie algebra $A_{n-1}^{(1)}$. These multiplicities have a combinatorial description in terms of higher level paths or highest-lift multipartitions. This enables us to solve the Jantzen-Seitz problem for Ariki-Koike algebras, that is, to determine which irreducible representations of $\H_m(\v)$ restrict to irreducible representations of $\H_{m-1}(\v)$. From a combinatorial point of view, this problem is identical to that of computing the tensor product of an $A_{n-1}^{(1)}$-module of level $l$ and one of level 1. We also consider natural generalisations of the Jantzen-Seitz problem corresponding to the product of a level $l$ module by a level $l'>1$ module, and from the commutativity of tensor products, we deduce a remarkable symmetry between the generalised Jantzen-Seitz conditions and the sets of parameters of the Ariki-Koike algebras.
We use the Fock space representation of the quantum affine algebra of type $A^{(2)}_{2n}$ to obtain a description of the global crystal basis of its basic level 1 module. We formulate a conjecture relating this basis to decomposition matrices of spin symmetric groups in characteristic $2n+1$.
We show that the quantum straightening algorithm for Young tableaux and Young bitableaux reduces in the crystal limit $q \mapsto 0$ to the Robinson-Schensted algorithm.
We give an expression of the $q$-analogues of the multiplicities of weights in irreducible $\sl_{n+1}$-modules in terms of the geometry of the crystal graph attached to the corresponding $U_q(\sl_{n+1})$-modules. As an application, we describe multivariate polynomial analogues of the multiplicities of the zero weight, refining Kostant's generalized exponents.
This paper presents a noncommutative theory of symmetric functions, based on the notion of quasi-determinant. We begin with a formal theory, corresponding to the case of symmetric functions in an infinite number of independent variables. This allows us to endow the resulting algebra with a Hopf structure, which leads to a new method for computing in descent algebras. It also gives unified reinterpretation of a number of classical constructions. Next, we study the noncommutative analogs of symmetric polynomials. One arrives at different constructions, according to the particular kind of application under consideration. For example, when a polynomial with noncommutative coefficients in one central variable is decomposed as a product of linear factors, the roots of these factors differ from those of the expanded polynomial. Thus, according to whether one is interested in the construction of a polynomial with given roots or in the expansion of a product of linear factors, one has to consider two distinct specializations of the formal symmetric functions. A third type appears when one looks for a noncommutative generalization of applications related to the notion of characteristic polynomial of a matrix. This construction can be applied, for instance, to the noncommutative matrices formed by the generators of the universal enveloping algebra $U(gl_n)$ or of