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Leclerc Bernard | Leclerc M. | Leclerc Arnaud | Leclerc N. | Leclerc B. | Leclerc Sarah | Leclerc Guillaume | Leclerc Anthony P. | Leclerc Hugo | LeClerc J.

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Noncommutative symmetric functions

Gelfand Israel, Krob D., Lascoux Alain, Leclerc B., Retakh V. S., Thibon J. -Y.
20 Jul 1994 hep-th math.QA arxiv.org/abs/hep-th/9407124

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

Minor Identities for Quasideterminants and Quantum Determinants

Krob D., Leclerc B.
26 Nov 1994 hep-th math.QA q-alg arxiv.org/abs/hep-th/9411194

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

Crystal Graphs and $q$-Analogues of Weight Multiplicities for the Root System $A_n$

Lascoux A., Leclerc B., Thibon J. -Y.
04 Mar 1995 q-alg math.QA arxiv.org/abs/q-alg/9503001

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.

The Robinson-Schensted correspondence as the quantum straightening at $q=0$

Leclerc B., Thibon J. -Y.
07 Apr 1995 q-alg math.QA arxiv.org/abs/q-alg/9504004

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.

Spontaneous breaking of CP symmetry by orbifold moduli

Acharya B., D , Bailin , Love A., Sabra W. A., Thomas S.
21 Jun 1995 hep-th arxiv.org/abs/hep-th/9506143

CP-violating phases which contribute to the electric dipole moment(EDM) of the neutron are considered in the context of orbifold compactificationof the heterotic string. In particular, we study the situation where CP is spontaneously broken by moduli fields acquiring, in general, complex expectation values at the minimum of duality invariant low energy effective potentials. We show, by explicit minimization of such a potential in the case of the ${\bf Z}{6}-{\rm IIb}$ orbifold, that it is the presence of so called Green-Schwarz anomaly coefficients $\delta{\rm GS}^{i} $, that leads to significant CP violating expectation values of the moduli. By evaluating the soft supersymmetry breaking moduli dependent $A$ and $B$ terms in this model, we find that the experimental bounds $\Phi (A) $, $ \Phi (B) $ $\leq 5 \times 10^{-3} $ are exceeded for a particular range of values of the auxiliary field of the $S$ modulus.

Ribbon Tableaux, Hall-Littlewood Functions, Quantum Affine Algebras and Unipotent Varieties

Lascoux Alain, Leclerc Bernard, Thibon Jean-Yves
27 Dec 1995 q-alg math.QA arxiv.org/abs/q-alg/9512031

We introduce a new family of symmetric functions, which are $q$-analogues of products of Schur functions defined in terms of ribbon tableaux. These functions can be interpreted in terms of the Fock space representation of the quantum affine algebra of type $A_{n-1}^{(1)}$ and are related to Hall-Littlewood functions via the geometry of flag varieties. We present a series of conjectures, and prove them in special cases. The essential step in proving that these functions are actually symmetric consists in the calculation of a basis of highest weight vectors of the $q$-Fock space using ribbon tableaux.

Canonical bases of q-deformed Fock spaces

Leclerc Bernard, Thibon Jean-Yves
16 Feb 1996 q-alg math.QA arxiv.org/abs/q-alg/9602025

We define a canonical basis of the $q$-deformed Fock space representation of the affine Lie algebra $\glchap_n$. We conjecture that the entries of the transition matrix between this basis and the natural basis of the Fock space are $q$-analogues of decomposition numbers of the $v$-Schur algebras for $v$ specialized to a $n$th root of unity.

Flag varieties and the Yang-Baxter equation

Lascoux Alain, Leclerc Bernard, Thibon Jean-Yves
14 Jul 1996 q-alg math.QA arxiv.org/abs/q-alg/9607015

We investigate certain bases of Hecke algebras defined by means of the Yang-Baxter equation, which we call Yang-Baxter bases. These bases are essentially self-adjoint with respect to a canonical bilinear form. In the case of the degenerate Hecke algebra, we identify the coefficients in the expansion of the Yang-Baxter basis on the usual basis of the algebra with specializations of double Schubert polynomials. We also describe the expansions associated to other specializations of the generic Hecke algebra.

RSOS models and Jantzen-Seitz representations of Hecke algebras at roots of unity

Foda Omar, Leclerc Bernard, Okado Masato, Thibon Jean-Yves, Welsh Trevor A.
18 Jan 1997 q-alg hep-th math.QA arxiv.org/abs/q-alg/9701020

A special family of partitions occurs in two apparently unrelated contexts: the evaluation of 1-dimensional configuration sums of certain RSOS models, and the modular representation theory of symmetric groups or their Hecke algebras $H_m$. We provide an explanation of this coincidence by showing how the irreducible $H_m$-modules which remain irreducible under restriction to $H_{m-1}$ (Jantzen-Seitz modules) can be determined from the decomposition of a tensor product of representations of affine $\sl_n$.

Combinatorics of solvable lattice models, and modular representations of Hecke algebras

Foda Omar, Leclerc Bernard, Okado Masato, Thibon Jean-Yves, Welsh Trevor A.
18 Jan 1997 q-alg hep-th math.QA arxiv.org/abs/q-alg/9701021

We review and motivate recently-observed relationships between exactly solvable lattice models and modular representations of Hecke algebras. Firstly, we describe how the set of $n$-regular partitions label both of the following classes of objects: 1. The spectrum of unrestricted solid-on-solid lattice models based on level-1 representations of the affine algebras $\sl_n$, 2. The irreducible representations of type-A Hecke algebras at roots of unity: $H_m(\sqrt[n]{1})$. Secondly, we show that a certain subset of the $n$-regular partitions label both of the following classes of objects: 1. The spectrum of restricted solid-on-solid lattice models based on cosets of affine algebras $(sl(n)^1 \times sl(n)^_1)/ sl(n)^_2$. 2. Jantzen-Seitz (JS) representations of $H_m(\sqrt[n]{1})$: irreducible representations that remain irreducible under restriction to $H{m-1}(\sqrt[n]{1})$. Using the above relationships, we characterise the JS representations of $H_m(\sqrt[n]{1})$ and show that the generating series that count them are branching functions of affine $\sl_n$.