13 research outputs found
Minuscule posets from neighbourly graph sequences
AbstractWe construct minuscule posets, an interesting family of posets arising in Lie theory, algebraic geometry and combinatorics, from sequences of vertices of a graph with particular neighbourly properties
Minuscule reverse plane partitions via quiver representations
A nilpotent endomorphism of a quiver representation induces a linear
transformation on the vector space at each vertex. Generically among all
nilpotent endomorphisms, there is a well-defined Jordan form for these linear
transformations, which is an interesting new invariant of a quiver
representation. If is a Dynkin quiver and is a minuscule vertex, we
show that representations consisting of direct sums of indecomposable
representations all including in their support, the category of which we
denote by , are determined up to isomorphism by this
invariant. We use this invariant to define a bijection from isomorphism classes
of representations in to reverse plane partitions whose
shape is the minuscule poset corresponding to and . By relating the
piecewise-linear promotion action on reverse plane partitions to
Auslander-Reiten translation in the derived category, we give a uniform proof
that the order of promotion equals the Coxeter number. In type , we show
that special cases of our bijection include the Robinson-Schensted-Knuth and
Hillman-Grassl correspondences.Comment: Comments welcom
Eriksson's numbers game and finite Coxeter groups
The numbers game is a one-player game played on a finite simple graph with
certain ``amplitudes'' assigned to its edges and with an initial assignment of
real numbers to its nodes. The moves of the game successively transform the
numbers at the nodes using the amplitudes in a certain way. This game and its
interactions with Coxeter/Weyl group theory and Lie theory have been studied by
many authors. In particular, Eriksson connects certain geometric
representations of Coxeter groups with games on graphs with certain real number
amplitudes. Games played on such graphs are ``E-games.'' Here we investigate
various finiteness aspects of E-game play: We extend Eriksson's work relating
moves of the game to reduced decompositions of elements of a Coxeter group
naturally associated to the game graph. We use Stembridge's theory of fully
commutative Coxeter group elements to classify what we call here the
``adjacency-free'' initial positions for finite E-games. We characterize when
the positive roots for certain geometric representations of finite Coxeter
groups can be obtained from E-game play. Finally, we provide a new Dynkin
diagram classification result of E-game graphs meeting a certain finiteness
requirement.Comment: 18 page