74 research outputs found
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
Combinatorial, piecewise-linear, and birational homomesy for products of two chains
This article illustrates the dynamical concept of in three kinds
of dynamical systems -- combinatorial, piecewise-linear, and birational -- and
shows the relationship between these three settings. In particular, we show how
the rowmotion and promotion operations of Striker and Williams can be lifted to
(continuous) piecewise-linear operations on the order polytope of Stanley, and
then lifted to birational operations on the positive orthant in
and indeed to a dense subset of . When the
poset is a product of a chain of length and a chain of length ,
these lifted operations have order , and exhibit the homomesy phenomenon:
the time-averages of various quantities are the same in all orbits. One
important tool is a concrete realization of the conjugacy between rowmotion and
promotion found by Striker and Williams; this allows us
to use homomesy for promotion to deduce homomesy for rowmotion.
NOTE: An earlier draft showed that Stanley's transfer map between the order
polytope and the chain polytope arises as the tropicalization of an analogous
map in the bilinear realm; in 2020 we removed this material for the sake of
brevity, especially after Joseph and Roby generalized our proof to the
noncommutative realm (see arXiv:1909.09658v3). Readers who nonetheless wish to
see our proof can find the September 2018 draft of this preprint through the
arXiv
Promotion and Rowmotion
We present an equivariant bijection between two actions--promotion and
rowmotion--on order ideals in certain posets. This bijection simultaneously
generalizes a result of R. Stanley concerning promotion on the linear
extensions of two disjoint chains and recent work of D. Armstrong, C. Stump,
and H. Thomas on root posets and noncrossing partitions. We apply this
bijection to several classes of posets, obtaining equivariant bijections to
various known objects under rotation. We extend the same idea to give an
equivariant bijection between alternating sign matrices under rowmotion and
under B. Wieland's gyration. Finally, we define two actions with related orders
on alternating sign matrices and totally symmetric self-complementary plane
partitions.Comment: 25 pages, 22 figures; final versio
Orbits of Plane Partitions of Exceptional Lie Type
For each minuscule flag variety , there is a corresponding minuscule
poset, describing its Schubert decomposition. We study an action on plane
partitions over such posets, introduced by P. Cameron and D. Fon-der-Flaass
(1995). For plane partitions of height at most , D. Rush and X. Shi (2013)
proved an instance of the cyclic sieving phenomenon, completely describing the
orbit structure of this action. They noted their result does not extend to
greater heights in general; however, when is one of the two minuscule flag
varieties of exceptional Lie type , they conjectured explicit instances of
cyclic sieving for all heights.
We prove their conjecture in the case that is the Cayley-Moufang plane of
type . For the other exceptional minuscule flag variety, the Freudenthal
variety of type , we establish their conjecture for heights at most ,
but show that it fails generally. We further give a new proof of an unpublished
cyclic sieving of D. Rush and X. Shi (2011) for plane partitions of any height
in the case is an even-dimensional quadric hypersurface. Our argument uses
ideas of K. Dilks, O. Pechenik, and J. Striker (2017) to relate the action on
plane partitions to combinatorics derived from -theoretic Schubert calculus.Comment: 25 pages, 7 figures, 3 tables. Section 5 rewritten and simplifie
Paths to Understanding Birational Rowmotion on Products of Two Chains
Birational rowmotion is an action on the space of assignments of rational
functions to the elements of a finite partially-ordered set (poset). It is
lifted from the well-studied rowmotion map on order ideals (equivariantly on
antichains) of a poset , which when iterated on special posets, has
unexpectedly nice properties in terms of periodicity, cyclic sieving, and
homomesy (statistics whose averages over each orbit are constant) [AST11, BW74,
CF95, Pan09, PR13, RuSh12,RuWa15+,SW12, ThWi17, Yil17. In this context,
rowmotion appears to be related to Auslander-Reiten translation on certain
quivers, and birational rowmotion to -systems of type
described in Zamolodchikov periodicity.
We give a formula in terms of families of non-intersecting lattice paths for
iterated actions of the birational rowmotion map on a product of two chains.
This allows us to give a much simpler direct proof of the key fact that the
period of this map on a product of chains of lengths and is
(first proved by D.~Grinberg and the second author), as well as the first proof
of the birational analogue of homomesy along files for such posets.Comment: 31 pages, to appear in Algebraic Combinatoric
- …