Cell Classification of Gelfand $S_n$-Graphs

Kazhdan and Lusztig introduced the $W$-graphs, which represent the multiplication action of the standard basis on the canonical bais in the Iwahori-Hecke algebra. In the Hecke algebra module, Marberg defined two generalied $W$-graphs, called the Gelfand $W$-graphs. The classification of the molecules of the type $A$ Gelfand $S_n$-graphs are determined by two RSK-like insertion algorithms. We finish the classification of cells by proving that every molecule in the $S_n$-graphs is indeed a cell.

💡 Research Summary

The paper addresses the cell structure of two Gelfand W‑graphs associated with the symmetric group Sₙ: the “row” graph Γ_row and the “column” graph Γ_col. These graphs arise from Marberg’s generalized Gelfand W‑graphs, which encode the action of the Iwahori–Hecke algebra on canonical bases. The authors first recall the necessary background on quasiparabolic W‑sets, Hecke‑module constructions, and the definition of cells as strongly connected components of the directed graphs. They then introduce two insertion procedures—row Beissinger insertion and column Beissinger insertion—both of which are analogues of the Robinson–Schensted–Knuth (RSK) insertion algorithm.

The main result, Theorem 1.1, states that any two vertices belonging to the same cell of either Γ_row or Γ_col must also belong to the same molecule (the connected component of the underlying undirected graph). In other words, molecules and cells coincide for both graphs. The proof proceeds in two parallel parts. For the row graph, the authors show that the row insertion algorithm defines a dominance order on vertices that is respected by the one‑direction edges of the graph. By analyzing the Hecke‑module multiplication formulas, they construct explicit directed paths linking any two vertices within the same molecule, thereby establishing that the molecule is contained in a single cell. A symmetric argument using column Beissinger insertion handles the column graph.

Key technical tools include the Bruhat order on quasiparabolic sets, the characterization of minimal and maximal elements in each W‑orbit, and detailed recurrence relations for the Kazhdan–Lusztig polynomials governing the Hecke action. The paper also leverages the perfect model theory for finite Coxeter groups, showing that the unique perfect model for Sₙ produces the distinguished pair of Gelfand graphs under consideration.

By proving that every molecule is a cell, the authors complete the classification of cells in Gelfand Sₙ‑graphs. This result aligns the combinatorial structure of these graphs with the classical Kazhdan–Lusztig cell theory and provides a concrete, insertion‑based description of the cell decomposition. Consequently, the work offers a robust framework for further investigations into Hecke algebra representations, combinatorial models of symmetric groups, and the interplay between canonical bases and insertion algorithms.