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Ribbon tableaux, Hall-Littlewood functions, quantum affine algebras, and unipotent varieties. (English) Zbl 0869.05068
Summary: 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 \({\mathcal F}_q\) of \(U_q(\widehat{\mathfrak{sl}}_n)\), 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 \({\mathcal F}_q\) using ribbon tableaux.
See also the authors’ paper [Ribbon tableaux, Hall-Littlewood functions and unipotent varieties, Sémin. Lothar. Comb. 34, 23 p. (1995; Zbl 0855.05099)].

MSC:
05E05 Symmetric functions and generalizations
05E10 Combinatorial aspects of representation theory
33D80 Connections of basic hypergeometric functions with quantum groups, Chevalley groups, \(p\)-adic groups, Hecke algebras, and related topics
17B37 Quantum groups (quantized enveloping algebras) and related deformations
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