# Ribbon tableaux, Hall–Littlewood functions, quantum affine algebras, and unipotent varieties

@article{Lascoux1995RibbonTH, title={Ribbon tableaux, Hall–Littlewood functions, quantum affine algebras, and unipotent varieties}, author={Alain Lascoux and Bernard Leclerc and Jean-Yves Thibon}, journal={Journal of Mathematical Physics}, year={1995}, volume={38}, pages={1041-1068} }

We introduce a new family of symmetric functions, which are q analogs of products of Schur functions, defined in terms of ribbon tableaux. These functions can be interpreted in terms of the Fock space representation Fq of Uq(sln), 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…

## 235 Citations

### A unipotent realization of the chromatic quasisymmetric function

- Mathematics
- 2022

This paper realizes of two families of combinatorial symmetric functions via the complex character theory of the finite general linear group GLnpFqq: the chromatic quasisymmetric functions and the…

### Combinatorial Hopf algebras, noncommutative Hall–Littlewood functions, and permutation tableaux

- Mathematics
- 2008

### Demazure crystals and the Schur positivity of Catalan functions

- Mathematics
- 2020

Catalan functions, the graded Euler characteristics of certain vector bundles on the flag variety, are a rich class of symmetric functions which include $k$-Schur functions and parabolic…

### SYMMETRIC POLYNOMIALS AND Uq ( ̂ sl2)

- Mathematics
- 1999

We study the explicit formula of Lusztig’s integral forms of the level one quantum affine algebra Uq(ŝl2) in the endomorphism ring of symmetric functions in infinitely many variables tensored with…

### LLT polynomials in the Schiffmann algebra

- Mathematics
- 2021

We identify certain combinatorially defined rational functions which, under the shuffle to Schiffmann algebra isomorphism, map to LLT polynomials in any of the distinguished copies Λ(X) ⊂ E of the…

### Standard Young tableaux for finite root systems

- Mathematics
- 2004

The study of representations of affine Hecke algebras has led to a new notion of shapes and standard Young tableaux which works for the root system of any finite Coxeter group. This paper is…

### Ribbon tableaux and q-analogues of fusion rulesin WZW conformal

- Mathematics

Starting from known q-analogues of ordinary SU(n) tensor products multiplicities, we introduce q-analogues of the fusion coeecients of the WZW conformal eld theories associated with SU(n). We…

### Ribbon tableaux and q-analogues of fusion rules in WZW conformal field theories

- Mathematics
- 1998

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…

### Arithmetic geometry of character varieties with regular monodromy

- Mathematics
- 2022

. We study character varieties arising as moduli of representations of an orientable surface group into a reductive group G . We ﬁrst show that if G/Z acts freely on the representation variety, then…

## References

SHOWING 1-10 OF 62 REFERENCES

### Euler-Poincare Characteristic and Polynomial Representations of Iwahori-Hecke Algebras

- Mathematics
- 1995

The Hecke algebras of type A „ admit faithful representations by symmetrization operators acting on polynomial rings. These operators are related to the geometry of flag manifolds and in particular…

### Green Polynomials and Hall-Littlewood Functions at Roots of Unity

- MathematicsEur. J. Comb.
- 1994

The two conjectures of N. Sultana 17 on specializations of Green polynomials are proved, and classical results concerning characters of the symmetric group induced by maximal cyclic subgroups are generalized.

### Hecke algebras at roots of unity and crystal bases of quantum affine algebras

- Mathematics
- 1996

AbstractWe present a fast algorithm for computing the global crystal basis of the basic
$$U_q (\widehat{\mathfrak{s}\mathfrak{l}}_n )$$
-module. This algorithm is based on combinatorial techniques…

### Symmetric polynomials and divided differences in formulas of intersection theory

- Mathematics
- 1996

The goal of the paper is two-fold. At first, we attempt to give a survey of some recent applications of symmetric polynomials and divided differences to intersection theory. We discuss: polynomials…

### Splitting the Square of a Schur Function into its Symmetric and Antisymmetric Parts

- Mathematics
- 1995

We propose a new combinatorial description of the product of two Schur functions. In the particular case of the square of a Schur function SI, it allows to discriminate in a very natural way between…

### Formulas for Lagrangian and orthogonal degeneracy loci; the Q-polynomials approach

- Mathematics
- 1996

Let V be a vector bundle on a scheme X endowed with a nondegenerate symplectic or orthogonal form. Let G be a Grassmannian bundle parametrizing maximal isotropic subbundles of V. The main goal of the…

### Hall–Littlewood polynomials at roots of 1 and modular representations of the symmetric group

- MathematicsMathematical Proceedings of the Cambridge Philosophical Society
- 1991

We first give a brief introduction to Hall–Littlewood functions; we follow closely the notation used in Macdonald [3]. Let λ = (λ1,…,λm) be a be a partition of n; that is λ1 + … + λm = n, λ1 ≥ λ2 ≥ ……

### Hall-Littlewood functions and Kostka-Foulkes polynomials in representation theory.

- Mathematics
- 1994

This paper presents a survey of recent applications of Hall-Littlewood functions and Kostka-Foulkes polynomials to the representation theory of the general linear group GL(n,C) and of the symmetric…

### Decomposition ofq-deformed Fock spaces

- Mathematics
- 1995

A decomposition of the level-oneq-deformed Fock space representations ofUq(sln) is given. It is found that the action ofU′q(sln) on these Fock spaces is centralized by a Heisenberg algebra, which…