Shifted insertion algorithms for primed words

Eric Marberg

Published 2021-04-23Version 1

This article studies a few 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 mixed shifted insertion. The latter maps have the property that when the input word varies such that one of the output tableaux is fixed, the other output tableau ranges over all (semi)standard tableau of a given shape with no primed entries on the diagonal. Our algorithms have the same feature, but now with primes allowed on the 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 similarly to understand products of $K$-theoretic Schur $Q$-functions. We also discuss how Hansson and Hultman's analogue of Matsumoto's theorem for involution words extends to the primed setting.