The up-operators $u_i$ and down-operators $d_i$ (introduced as Schur operators by Fomin) act on partitions by adding/removing a box to/from the $i$th column if possible. It is well known that the $u_i$ alone satisfy the relations of the (local) plactic monoid, and the present authors recently showed that relations of degree at most 4 suffice to describe all relations between the up-operators. Here we characterize the algebra generated by the up- and down-operators together, showing that it can be presented using only quadratic relations.
@article{10_37236_10099,
author = {Ricky Liu and Christian Smith},
title = {Up- and down-operators on {Young's} lattice},
journal = {The electronic journal of combinatorics},
year = {2021},
volume = {28},
number = {3},
doi = {10.37236/10099},
zbl = {1470.05171},
url = {http://geodesic.mathdoc.fr/articles/10.37236/10099/}
}
TY - JOUR
AU - Ricky Liu
AU - Christian Smith
TI - Up- and down-operators on Young's lattice
JO - The electronic journal of combinatorics
PY - 2021
VL - 28
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.37236/10099/
DO - 10.37236/10099
ID - 10_37236_10099
ER -
%0 Journal Article
%A Ricky Liu
%A Christian Smith
%T Up- and down-operators on Young's lattice
%J The electronic journal of combinatorics
%D 2021
%V 28
%N 3
%U http://geodesic.mathdoc.fr/articles/10.37236/10099/
%R 10.37236/10099
%F 10_37236_10099
Ricky Liu; Christian Smith. Up- and down-operators on Young's lattice. The electronic journal of combinatorics, Tome 28 (2021) no. 3. doi: 10.37236/10099
