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Hecke algebras at roots of unity and crystal bases of quantum affine algebras. (English) Zbl 0874.17009
The authors present a fast algorithm for computing the global crystal basis of the basic \(U_q (\widehat {sl}_n)\)-module. This algorithm is based on combinatorial techniques which have been developed for dealing with modular representations of symmetric groups, and more generally with representations of Hecke algebras of type \(A\) at roots of unity. The main conjecture, suggested in this paper, is that upon specialization \(q\to 1\), the algorithm computes the decomposition matrices of all Hecke algebras at an \(n\)th root of 1.
The article is structured as follows. Sections 2 and 3 recall the necessary background on modular representations of symmetric groups and Hecke algebras at roots of unity. Section 4 describes the Fock representation of \(U_q (\widehat {sl}_n)\) with conventions differing slightly from the usual ones in order to be compatible with those of the modular representation theory. Section 5 describes the crystal graph of the Fock representation and gives some applications to the combinatorics of \(n\)-cores and \(n\)-quotients.
Section 6 is devoted to the global lower crystal basis and presents a fast algorithm to compute it. Some properties of the transition matrices are established and the main conjecture is stated. Other properties of these matrices, also supporting the conjecture, are given in Section 7 where the Mullineux-Kleshchev involution is studied from the point of view of crystal bases. Section 8 is devoted to a reformulation of the main conjecture in terms of the upper crystal basis. Section 9 discusses, following an idea of Rouquier, a possible interpretation of these \(q\)-decomposition numbers in terms of the Jantzen filtration of a Specht module. Finally, this paper is concluded by an appendix of tables giving the global crystal basis, the Gram matrices of the Shapovalov form and some examples of crystal graphs.
The main results of this paper have been announced in [C. R. Acad. Sci., Paris, Sér. I 321, 511-516 (1995; Zbl 0840.05102)].
Reviewer: Li Fang (Nanjing)

17B37 Quantum groups (quantized enveloping algebras) and related deformations
17-08 Computational methods for problems pertaining to nonassociative rings and algebras
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
05E15 Combinatorial aspects of groups and algebras (MSC2010)
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