For si-stable and almost-si-stable condition sets, the dot-action module H_C decomposes as a direct sum of a fixed subspace and a multiply-shifted copy, realizing the modular relation at the representation level.
Ribbon Tableaux, Hall-Littlewood Functions, Quantum Affine Algebras and Unipotent Varieties
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abstract
We introduce a new family of symmetric functions, which are $q$-analogues of products of Schur functions defined in terms of ribbon tableaux. These functions can be interpreted in terms of the Fock space representation of the quantum affine algebra of type $A_{n-1}^{(1)}$ 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 the $q$-Fock space using ribbon tableaux.
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Divided difference operators for Hessenberg representations
For si-stable and almost-si-stable condition sets, the dot-action module H_C decomposes as a direct sum of a fixed subspace and a multiply-shifted copy, realizing the modular relation at the representation level.