Shifted insertion algorithms for primed words

Abstract

This article studies some new insertion algorithms that associate pairs of shifted tableaux to finite integer sequences in which certain terms may be primed. When primes are ignored in the input word these algorithms reduce to known correspondences, namely, a shifted form of Edelman-Greene insertion, Sagan-Worley insertion, and Haiman's shifted mixed insertion. These maps have the property that when the input word varies such that one output tableau is fixed, the other output tableau ranges over all (semi)standard tableaux of a given shape with no primed diagonal entries. Our algorithms have the same feature, but now with primes allowed on the main diagonal. One application of this is to give another Littlewood-Richardson rule for products of Schur $Q$-functions. It is hoped that there will exist set-valued generalizations of our bijections that can be used to understand products of $K$-theoretic Schur $Q$-functions.

arXiv comment72 pages, 3 figures; v2: some additional results in section 3, section 5 migrated to arXiv:2108.12020, minor corrections; v3: major revision, extensive reorganization, many corrections, added exposition; v4: updated references, final version

References in corpus (13)

• K-theoretic analogues of factorial Schur P- and Q-functions
• Involution words: counting problems and connections to Schubert calculus for symmetric orbit closures
• Schur P-positivity and involution Stanley symmetric functions
• Genomic Tableaux
• On some properties of symplectic Grothendieck polynomials
• K-theory formulas for orthogonal and symplectic orbit closures
• A symplectic refinement of shifted Hecke insertion
• Symmetric subgroup orbit closures on flag varieties: Their equivariant geometry, combinatorics, and connections with degeneracy loci
• Bumping operators and insertion algorithms for queer supercrystals
• Enriched set-valued P-partitions and shifted stable Grothendieck polynomials
• Atoms for signed permutations
• Extending a word property for twisted Coxeter systems
• Shifted dual equivalence and Schur P-positivity

Cited by in corpus (4)

• Highest weight crystals for Schur Q-functions
• Combinatorial formulas for shifted dual stable Grothendieck polynomials
• Shifted combinatorial Hopf algebras from $K$-theory
• Primed decomposition tableaux and extended queer crystals