On certain formulas of Karlin and Szegö.
A Littlewood-Richardson rule for evaluation representations of .
Domino tableaux, functions and plethysms. (Tableaux de dominos, fonctions et plethysmes.)
The Robinson-Schensted correspondence, crystal bases, and the quantum straightening at .
Ribbon tableaux, Hall-Littlewood functions and unipotent varieties.
A cluster algebra approach to -characters of Kirillov–Reshetikhin modules
We describe a cluster algebra algorithm for calculating -characters of Kirillov–Reshetikhin modules for any untwisted quantum affine algebra . This yields a geometric -character formula for tensor products of Kirillov–Reshetikhin modules. When is of type , this formula extends Nakajima’s formula for -characters of standard modules in terms of homology of graded quiver varieties.
Semicanonical bases and preprojective algebras
Partial flag varieties and preprojective algebras
Let be a preprojective algebra of type , and let be the corresponding semisimple simply connected complex algebraic group. We study rigid modules in subcategories for an injective -module, and we introduce a mutation operation between complete rigid modules in . This yields cluster algebra structures on the coordinate rings of the partial flag varieties attached to .
Twisted action of the symmetric group on the cohomology of a flag manifold
Classes dual to Schubert cycles constitute a basis on the cohomology ring of the flag manifold F, self-adjoint up to indexation with respect to the intersection form. Here, we study the bilinear form (X,Y) :=〈X·Y, c(F)〉 where X,Y are cocycles, c(F) is the total Chern class of F and〈,〉 is the intersection form. This form is related to a twisted action of the symmetric group of the cohomology ring, and to the degenerate affine Hecke algebra. We give a distinguished basis for this form, which is a...
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