Abstract
In his study of Kazhdan-Lusztig cells in affine type A, Shi has introduced an affine analog of Robinson- Schensted correspondence. We generalize the Matrix-Ball Construction of Viennot and Fulton to give a more combi- natorial realization of Shi's algorithm. As a biproduct, we also give a way to realize the affine correspondence via the usual Robinson-Schensted bumping algorithm. Next, inspired by Honeywill, we extend the algorithm to a bijection between extended affine symmetric group and triples (P, Q, ρ) where P and Q are tabloids and ρ is a dominant weight. The weights ρ get a natural interpretation in terms of the Affine Matrix-Ball Construction. Finally, we prove that fibers of the inverse map possess a Weyl group symmetry, explaining the dominance condition on weights.
Highlights
This is an extended abstract; the full version of the paper is [CPY15].1.1 Cells in Kazhdan-Lusztig theoryIn their groundbreaking paper [KL89] Kazhdan and Lusztig gave an approach to the representation theory of Hecke algebras
In type A, i.e. when W is a symmetric group, the Kazhdan-Lusztig cell structure corresponds to the Robinson-Schensted correspondence, a bijective correspondence between elements of the symmetric group and pairs of standard Young tableaux of the same shape
We introduce an algorithm called Affine Matrix-Ball Construction (AMBC) which from an affine permutation w produces two tabloids (P (w), Q(w)) of common shape λ, and ρ ∈ Z (λ)
Summary
In their groundbreaking paper [KL89] Kazhdan and Lusztig gave an approach to the representation theory of Hecke algebras. Right) cells are equivalence classes corresponding to a certain pre-order ≤L Two-sided cells are larger classes with both left and right equivalences allowed. In type A, i.e. when W is a symmetric group, the Kazhdan-Lusztig cell structure corresponds to the Robinson-Schensted correspondence, a bijective correspondence between elements of the symmetric group and pairs of standard Young tableaux of the same shape. It is well known ([BV82], [KL89], [GM88], [Ari99]) that two permutations lie in the same left It is well known ([BV82], [KL89], [GM88], [Ari99]) that two permutations lie in the same left (resp. right) cell whenever they have the same recording tableau Q (resp. insertion tableau P ), and in the same two-sided cell whenever the four tableaux have the same shape
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