Séminaire Lotharingien de Combinatoire, 93B.130 (2025), 12 pp.

Yifeng Zhang

Molecules of Affine Fixed-Point-Free$W-Graphs

Abstract. RSK correspondence gives a bijection from a permutation to a pair of semistandard Young tableaux with same shape. For the symmetric groups, Kazhdan and Lusztig use the RSK correspondence to classify the molecules and cells of the S_n-graphs. To generalize this to the affine case, Chmutov, Pylyavskyy and Yudovina set up an affine RSK correspondence, giving a bijection from an affine permutation to a triple (P,Q,ρ) where P and Q are tabloids of same shape and ρ is a dominant weight. Moreover, Chmutov, Lewis and Pylyavskyy use it to classify the affine molecules. Marberg generalizes Kazhdan and Lusztig's W-graphs, using affine fixed-point-free involutions as their indices. In this paper, we use affine RSK correspondence to classify the molecules of Marberg's S^~_n-graphs.

Received: November 15, 2024. Accepted: February 15, 2025. Final version: April 1, 2025.

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