64 research outputs found
A new characterization for the m-quasiinvariants of S_n and explicit basis for two row hook shapes
In 2002, Feigin and Veselov defined the space of m-quasiinvariants for any
Coxeter group, building on earlier work of Chalykh and Veselov. While many
properties of those spaces were proven from this definition, an explicit
computation of a basis was only done in certain cases. In particular, Feigin
and Veselov computed bases for the m-quasiinvariants of dihedral groups,
including S_3, and Felder and Veselov computed the non-symmetric
m-quasiinvariants of lowest degree for general S_n. In this paper, we provide a
new characterization of the m-quasiinvariants of S_n, and use this to provide a
basis for the isotypic component indexed by the partition [n-1,1]. This builds
on a previous paper in which we computed a basis for S_3 via combinatorial
methods.Comment: 26 pages, uses youngtab.st
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
Higher Cluster Categories and QFT Dualities
We present a unified mathematical framework that elegantly describes
minimally SUSY gauge theories in even dimension, ranging from to , and
their dualities. This approach combines recent developments on graded quiver
with potentials, higher Ginzburg algebras and higher cluster categories (also
known as -cluster categories). Quiver mutations studied in the context of
mathematics precisely correspond to the order dualities of the gauge
theories. Our work suggests that these equivalences of quiver gauge theories
sit inside an infinite family of such generalized dualities, whose physical
interpretation is yet to be understood.Comment: 61 pages, 30 figure
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