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Noncommutative symmetric functions. IV: Quantum linear groups and Hecke algebras at $$q=0$$. (English) Zbl 0881.05120
This paper is devoted to the representation theoretical interpretation of noncommutative symmetric functions and quasi-symmetric functions. These objects, which are two different generalizations of ordinary symmetric functions, built up two Hopf algebras dual to each other, and have been shown to provide a Frobenius type theory for Hecke algebras of type $$A$$ at $$q=0$$, playing the same rôle as the classical correspondence between symmetric functions and characters of symmetric groups (which extends to the case of the generic Hecke algebra).
The paper is structured as follows. We first recall the basic definitions concerning noncommutative symmetric functions and quasi-symmetric functions (Section 2) and review the Frobenius correspondence for the generic Hecke algebras (Section 3). Next, we introduce the Dipper-Donkin version of the quantized function algebra of the space of $$n\times n$$ matrices (Section 4). We describe some interesting subspaces (Sections 4.5 and 4.6), and prove that the $$q=0$$ specialization of the diagonal subalgebra is a quotient of the plactic algebra, which we call the hypoplactic algebra (Section 4.7). Next we review the representation theory of the 0-Hecke algebra and its interpretation in terms of quasi-symmetric functions and noncommutative symmetric functions, providing the details which were omitted in G. Duchamp, D. Krob, B. Leclerc and J.-Y. Thibon [C. R. Acad. Sci., Paris, Sér. I 322, No. 2, 107-112 (1996; Zbl 0839.20017)]. In Section 6, we introduce a notion of noncommutative character for $$A_q(n)$$-comodules, and prove that these characters live in the diagonal subalgebra. For generic $$q$$, the characters of irreducible comodules are quantum analogues of Schur functions. For $$q=0$$, we show that hypoplactic analogues of the fundamental quasi-symmetric functions $$F_I$$ (quasi-ribbons) can be obtained as the characters of irreducible $$A_0(n)$$ comodules, and give a similar construction for the ribbon Schur functions. These constructions lead to degenerate versions of the Robinson-Schensted correspondence, which are discussed in Section 7.

##### MSC:
 05E05 Symmetric functions and generalizations 20C30 Representations of finite symmetric groups
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