Séminaire Lotharingien de Combinatoire, 93B.89 (2025), 12 pp.
Joshua Arroyo, Zachary Hamaker, Graham Hawkes and Jianping Pan
Type C K-Stanley Symmetric Functions and Kraśkiewicz-Hecke Insertion
Abstract.
We study Type C K-Stanley symmetric functions, which are K-theoretic extensions of the Type C Stanley symmetric functions.
Our main contribution is Kraśkiewicz-Hecke insertion (KH), a K-theoretic analogue of Kraśkiewicz insertion.
Much like Kraśkiewicz insertion enumerates reduced words for signed permutations, KH enumerates their 0-Hecke expressions.
The former enumeration witnesses the Type C Stanley expansion into Schur-Q functions.
We conjecture that KH extends this to give the Type C K-Stanley expansion into GQ functions, which are K-theory representatives for the Lagrangian Grassmannian introduced by Ikeda and Naruse.
We also show Type C K-Stanleys of top fully commutative signed permutations are skew GQ's.
This allows us to prove a conjecture of Lewis and Marberg and to give the first conjectural formulas for the expansion of a skew GQ into GQ's.
The latter specializes to a rule for multiplying two GQ functions where one has trapezoid shape.
This would extend Buch and Ravikumar's Pieri rule, the only known product rule for GQ's.
Received: November 15, 2024.
Accepted: February 15, 2025.
Final version: April 1, 2025.
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