298 research outputs found
Combinatorial representation theory of Lie algebras. Richard Stanley's work and the way it was continued
Richard Stanley played a crucial role, through his work and his students, in
the development of the relatively new area known as combinatorial
representation theory. In the early stages, he has the merit to have pointed
out to combinatorialists the potential that representation theory has for
applications of combinatorial methods. Throughout his distinguished career, he
wrote significant articles which touch upon various combinatorial aspects
related to representation theory (of Lie algebras, the symmetric group, etc.).
I describe some of Richard's contributions involving Lie algebras, as well as
recent developments inspired by them (including some open problems), which
attest the lasting impact of his work.Comment: 11 page
Analogs of Schur functions for rank two Weyl groups obtained from grid-like posets
In prior work, the authors, along with M. McClard, R. A. Proctor, and N. J.
Wildberger, studied certain distributive lattice models for the "Weyl
bialternants" (aka "Weyl characters") associated with the rank two root
systems/Weyl groups. These distributive lattices were uniformly described as
lattices of order ideals taken from certain grid-like posets, although the
arguments connecting the lattices to Weyl bialternants were case-by-case
depending on the type of the rank two root system. Using this connection with
Weyl bialternants, these lattices were shown to be rank symmetric and rank
unimodal, and their rank generating functions were shown to have beautiful
quotient-of-products expressions. Here, these results are re-derived from
scratch using completely uniform and elementary combinatorial reasoning in
conjunction with some new combinatorial methodology developed elsewhere by the
second listed author.Comment: 15 page
The toggle group, homomesy, and the Razumov-Stroganov correspondence
The Razumov-Stroganov correspondence, an important link between statistical
physics and combinatorics proved in 2011 by L. Cantini and A. Sportiello,
relates the ground state eigenvector of the O(1) dense loop model on a
semi-infinite cylinder to a refined enumeration of fully-packed loops, which
are in bijection with alternating sign matrices. This paper reformulates a key
component of this proof in terms of posets, the toggle group, and homomesy, and
proves two new homomesy results on general posets which we hope will have
broader implications.Comment: 14 pages, 10 figures, final versio
A Generalized Macaulay Theorem and Generalized Face Rings
We prove that the -vector of members in a certain class of meet
semi-lattices satisfies Macaulay inequalities. We construct a large family of
meet semi-lattices belonging to this class, which includes all posets of
multicomplexes, as well as meet semi-lattices with the "diamond property",
discussed by Wegner, as spacial cases. Specializing the proof to that later
family, one obtains the Kruskal-Katona inequalities and their proof as in
Wegner's.
For geometric meet semi lattices we construct an analogue of the exterior
face ring, generalizing the classic construction for simplicial complexes. For
a more general class, which include also multicomplexes, we construct an
analogue of the Stanley-Reisner ring. These two constructions provide algebraic
counterparts (and thus also algebraic proofs) of Kruskal-Katona's and
Macaulay's inequalities for these classes, respectively.Comment: Final version: 13 pages, 2 figures. Improved presentation, more
detailed proofs, same results. To appear in JCT
- …