A ``flip-and-reversal" involution arising in the study of quasisymmetric Schur functions provides a passage between what we term ``Young" and ``reverse" variants of bases of polynomials or quasisymmetric functions. Building on this perspective, which has found recent application in the study of q-analogues of combinatorial Hopf algebras and generalizations of dual immaculate functions, we develop and explore Young analogues of well-known bases for polynomials. We prove several combinatorial formulas for the Young analogue of the key polynomials, show that they form the generating functions for left keys, and provide a representation-theoretic interpretation of Young key polynomials as traces on certain modules. We also give combinatorial formulas for the Young analogues of Schubert polynomials, including their crystal graph structure. We moreover determine the intersections of (reverse) bases and their Young counterparts, further clarifying their relationships to one another.
@article{10_37236_10579,
author = {Sarah Mason and Dominic Searles},
title = {The {``Young''} and ``reverse'' dichotomy of polynomials},
journal = {The electronic journal of combinatorics},
year = {2022},
volume = {29},
number = {3},
doi = {10.37236/10579},
zbl = {1506.05212},
url = {http://geodesic.mathdoc.fr/articles/10.37236/10579/}
}
TY - JOUR
AU - Sarah Mason
AU - Dominic Searles
TI - The ``Young'' and ``reverse'' dichotomy of polynomials
JO - The electronic journal of combinatorics
PY - 2022
VL - 29
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.37236/10579/
DO - 10.37236/10579
ID - 10_37236_10579
ER -
%0 Journal Article
%A Sarah Mason
%A Dominic Searles
%T The ``Young'' and ``reverse'' dichotomy of polynomials
%J The electronic journal of combinatorics
%D 2022
%V 29
%N 3
%U http://geodesic.mathdoc.fr/articles/10.37236/10579/
%R 10.37236/10579
%F 10_37236_10579
Sarah Mason; Dominic Searles. The ``Young'' and ``reverse'' dichotomy of polynomials. The electronic journal of combinatorics, Tome 29 (2022) no. 3. doi: 10.37236/10579
