# Noncommutative Symmetric Functions Iv: Quantum Linear Groups and Hecke Algebras at q = 0

@article{Krob1997NoncommutativeSF, title={Noncommutative Symmetric Functions Iv: Quantum Linear Groups and Hecke Algebras at q = 0}, author={Daniel Krob and Jean-Yves Thibon}, journal={Journal of Algebraic Combinatorics}, year={1997}, volume={6}, pages={339-376} }

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.

## 221 Citations

### Noncommutative Symmetric Functions V: a degenerate Version of UQ(Gln)

- MathematicsInt. J. Algebra Comput.
- 1999

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).

### Symmetric Functions, Noncommutative Symmetric Functions and Quasisymmetric Functions II

- Mathematics
- 2002

Abstract
Like its precursor this paper is concerned with the Hopf algebra of noncommutative symmetric functions and its graded dual, the Hopf algebra of quasisymmetric functions. It complements and…

### The peak algebra and the Hecke – Clifford algebras at q 1⁄4 0

- Mathematics
- 2004

Using the formalism of noncommutative symmetric functions, we derive the basic theory of the peak algebra of symmetric groups and of its graded Hopf dual. Our main result is to provide a…

### Noncommutative Symmetric Functions Vi: Free Quasi-Symmetric Functions and Related Algebras

- MathematicsInt. J. Algebra Comput.
- 2002

This article is devoted to the study of several algebras related to symmetric functions, which admit linear bases labelled by various combinatorial objects: permutations (free quasi-symmetric…

### Some Generalizations of Quasi-symmetric Functions and Noncommutative Symmetric Functions

- Mathematics
- 2000

In this paper, we investigate various kinds of generalisations of symmetric functions. The classical algebra Sym of symmetric functions is embedded in QSym, the algebra of quasi-symmetric functions,…

### Noncommutative Symmetric Bessel Functions

- MathematicsCanadian Mathematical Bulletin
- 2008

Abstract The consideration of tensor products of 0-Hecke algebramodules leads to natural analogs of the Bessel $J$ -functions in the algebra of noncommutative symmetric functions. This provides a…

### Noncommutative Bessel symmetric functions

- Mathematics
- 2006

The consideration of tensor products of 0-Hecke algebra modules leads to natural analogs of the Bessel J-functions in the algebra of noncommutative symmetric functions. This provides a simple…

### Crystal graphs for general linear Lie superalgebras and quasi-symmetric functions

- MathematicsJ. Comb. Theory, Ser. A
- 2009

### Grothendieck Bialgebras, Partition Lattices, and Symmetric Functions in Noncommutative Variables

- MathematicsElectron. J. Comb.
- 2006

We show that the Grothendieck bialgebra of the semi-tower of partition lattice algebras is isomorphic to the graded dual of the bialgebra of symmetric functions in noncommutative variables. In…

## References

SHOWING 1-10 OF 79 REFERENCES

### Noncommutative symmetric functions III: Deformations of Cauchy and convolution algebras

- MathematicsDiscret. Math. Theor. Comput. Sci.
- 1997

The main examples are deformations of noncommutative symmetric functions related to families of idempotents in descent algebras, and a simple q -analogue of the shuffle product, which has unexpected connections with quantum groups, hyperplane arrangements, and certain questions in mathematical physics.

### 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…

### CHARACTER TABLE OF HECKE ALGEBRA OF TYPE AN-1 AND REPRESENTATIONS OF THE QUANTUM GROUP Uq(gln+1)

- Mathematics
- 1992

A q-analogue of the Frobenius formula is proved by means of the quantum groups Uq(gln+1), Aq(GLn+1) and Iwahori's Hecke algebra of type AN-1, and then, the character table of this Hecke algebra is…

### 0-Hecke algebras

- MathematicsJournal of the Australian Mathematical Society
- 1979

Abstract The structure of a 0-Hecke algebra H of type (W, R) over a field is examined. H has 2n distinct irreducible representations, where n = ∣R∣, all of which are one-dimensional, and correspond…

### Noncommutative Symmetric Functions II: Transformations of Alphabets

- MathematicsInt. J. Algebra Comput.
- 1997

Noncommutative analogues of classical operations on symmetric functions are investigated, and several q-analogues of the Eulerian idempotents and of the Garsia-Reutenauer idempotsents are obtained.

### Noncommutative symmetric functions

- Mathematics
- 1994

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…

### Representations and traces of the Hecke algebras Hn(q) of type An−1

- Mathematics
- 1992

The notion of the connectivity class of minimal words in the algebra Hn(q) is introduced and a method of explicitly constructing irreducible representation matrices is described and implemented.…

### Symmetric functions and Hall polynomials

- Mathematics
- 1979

I. Symmetric functions II. Hall polynomials III. HallLittlewood symmetric functions IV. The characters of GLn over a finite field V. The Hecke ring of GLn over a finite field VI. Symmetric functions…