6 research outputs found
Sortable Elements for Quivers with Cycles
Each Coxeter element c of a Coxeter group W defines a subset of W called the
c-sortable elements. The choice of a Coxeter element of W is equivalent to the
choice of an acyclic orientation of the Coxeter diagram of W. In this paper, we
define a more general notion of Omega-sortable elements, where Omega is an
arbitrary orientation of the diagram, and show that the key properties of
c-sortable elements carry over to the Omega-sortable elements. The proofs of
these properties rely on reduction to the acyclic case, but the reductions are
nontrivial; in particular, the proofs rely on a subtle combinatorial property
of the weak order, as it relates to orientations of the Coxeter diagram. The
c-sortable elements are closely tied to the combinatorics of cluster algebras
with an acyclic seed; the ultimate motivation behind this paper is to extend
this connection beyond the acyclic case.Comment: Final version as published. An error corrected in the previous
counterexample, other minor improvement
Universal geometric cluster algebras
We consider, for each exchange matrix B, a category of geometric cluster
algebras over B and coefficient specializations between the cluster algebras.
The category also depends on an underlying ring R, usually the integers,
rationals, or reals. We broaden the definition of geometric cluster algebras
slightly over the usual definition and adjust the definition of coefficient
specializations accordingly. If the broader category admits a universal object,
the universal object is called the cluster algebra over B with universal
geometric coefficients, or the universal geometric cluster algebra over B.
Constructing universal coefficients is equivalent to finding an R-basis for B
(a "mutation-linear" analog of the usual linear-algebraic notion of a basis).
Polyhedral geometry plays a key role, through the mutation fan F_B, which we
suspect to be an important object beyond its role in constructing universal
geometric coefficients. We make the connection between F_B and g-vectors. We
construct universal geometric coefficients in rank 2 and in finite type and
discuss the construction in affine type.Comment: Final version to appear in Math. Z. 49 pages, 5 figure
An affine almost positive roots model
We generalize the almost positive roots model for cluster algebras from
finite type to a uniform finite/affine type model. We define the almost
positive Schur roots and a compatibility degree, given by a formula
that is new even in finite type. The clusters define a complete fan
. Equivalently, every vector has a unique cluster
expansion. We give a piecewise linear isomorphism from the subfan of
induced by real roots to the -vector
fan of the associated cluster algebra. We show that is the set of
denominator vectors of the associated acyclic cluster algebra and conjecture
that the compatibility degree also describes denominator vectors for
non-acyclic initial seeds. We extend results on exchangeability of roots to the
affine case.Comment: 45 pages. *Version 4 addresses concerns from a referee * Version 3
corrects typesetting errors caused by the order of packages in the preamble *
Version 2 is a major revision and contains only the results concerning the
affine almost positive roots model; the discussion on orbits of coxeter
elements is now arXiv:1808.0509
Stack-Sorting for Coxeter Groups
Given an essential semilattice congruence on the left weak order of
a Coxeter group , we define the Coxeter stack-sorting operator by , where
is the unique minimal element of the congruence
class of containing . When is the sylvester congruence on
the symmetric group , the operator is West's
stack-sorting map. When is the descent congruence on , the
operator is the pop-stack-sorting map. We establish several
general results about Coxeter stack-sorting operators, especially those acting
on symmetric groups. For example, we prove that if is an essential
lattice congruence on , then every permutation in the image of has at most right
descents; we also show that this bound is tight.
We then introduce analogues of permutree congruences in types and
and use them to isolate Coxeter stack-sorting operators
and that serve as
canonical type- and type- counterparts of West's stack-sorting
map. We prove analogues of many known results about West's stack-sorting map
for the new operators and
. For example, in type , we
obtain an analogue of Zeilberger's classical formula for the number of
-stack-sortable permutations in .Comment: 39 pages, 11 figure