13,456 research outputs found
Operads with compatible CL-shellable partition posets admit a Poincar\'e-Birkhoff-Witt basis
In 2007, Vallette built a bridge across posets and operads by proving that an
operad is Koszul if and only if the associated partition posets are
Cohen-Macaulay. Both notions of being Koszul and being Cohen-Macaulay admit
different refinements: our goal here is to link two of these refinements. We
more precisely prove that any (basic-set) operad whose associated posets admit
isomorphism-compatible CL-shellings admits a Poincar\'e-Birkhoff-Witt basis.
Furthermore, we give counter-examples to the converse
Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables
We introduce a natural Hopf algebra structure on the space of noncommutative
symmetric functions which was recently studied as a vector space by Rosas and
Sagan. The bases for this algebra are indexed by set partitions. We show that
there exist a natural inclusion of the Hopf algebra of noncommutative symmetric
functions indexed by compositions in this larger space. We also consider this
algebra as a subspace of noncommutative polynomials and use it to understand
the structure of the spaces of harmonics and coinvariants with respect to this
collection of noncommutative polynomials.Comment: 30 page
Mixed Bruhat operators and Yang-Baxter equations for Weyl groups
We introduce and study a family of operators which act in the span of a Weyl
group and provide a multi-parameter solution to the quantum Yang-Baxter
equations of the corresponding type. Our operators generalize the "quantum
Bruhat operators" that appear in the explicit description of the multiplicative
structure of the (small) quantum cohomology ring of .
The main combinatorial applications concern the "tilted Bruhat order," a
graded poset whose unique minimal element is an arbitrarily chosen element
. (The ordinary Bruhat order corresponds to the case .) Using the
mixed Bruhat operators, we prove that these posets are lexicographically
shellable, and every interval in a tilted Bruhat order is Eulerian. This
generalizes well known results of Verma, Bjorner, Wachs, and Dyer.Comment: 19 page
Symmetry-protected self-correcting quantum memories
A self-correcting quantum memory can store and protect quantum information
for a time that increases without bound with the system size and without the
need for active error correction. We demonstrate that symmetry can lead to
self-correction in 3D spin-lattice models. In particular, we investigate codes
given by 2D symmetry-enriched topological (SET) phases that appear naturally on
the boundary of 3D symmetry-protected topological (SPT) phases. We find that
while conventional on-site symmetries are not sufficient to allow for
self-correction in commuting Hamiltonian models of this form, a generalized
type of symmetry known as a 1-form symmetry is enough to guarantee
self-correction. We illustrate this fact with the 3D "cluster-state" model from
the theory of quantum computing. This model is a self-correcting memory, where
information is encoded in a 2D SET-ordered phase on the boundary that is
protected by the thermally stable SPT ordering of the bulk. We also investigate
the gauge color code in this context. Finally, noting that a 1-form symmetry is
a very strong constraint, we argue that topologically ordered systems can
possess emergent 1-form symmetries, i.e., models where the symmetry appears
naturally, without needing to be enforced externally.Comment: 39 pages, 16 figures, comments welcome; v2 includes much more
explicit detail on the main example model, including boundary conditions and
implementations of logical operators through local moves; v3 published
versio
On the representation theory of finite J-trivial monoids
In 1979, Norton showed that the representation theory of the 0-Hecke algebra
admits a rich combinatorial description. Her constructions rely heavily on some
triangularity property of the product, but do not use explicitly that the
0-Hecke algebra is a monoid algebra.
The thesis of this paper is that considering the general setting of monoids
admitting such a triangularity, namely J-trivial monoids, sheds further light
on the topic. This is a step to use representation theory to automatically
extract combinatorial structures from (monoid) algebras, often in the form of
posets and lattices, both from a theoretical and computational point of view,
and with an implementation in Sage.
Motivated by ongoing work on related monoids associated to Coxeter systems,
and building on well-known results in the semi-group community (such as the
description of the simple modules or the radical), we describe how most of the
data associated to the representation theory (Cartan matrix, quiver) of the
algebra of any J-trivial monoid M can be expressed combinatorially by counting
appropriate elements in M itself. As a consequence, this data does not depend
on the ground field and can be calculated in O(n^2), if not O(nm), where n=|M|
and m is the number of generators. Along the way, we construct a triangular
decomposition of the identity into orthogonal idempotents, using the usual
M\"obius inversion formula in the semi-simple quotient (a lattice), followed by
an algorithmic lifting step.
Applying our results to the 0-Hecke algebra (in all finite types), we recover
previously known results and additionally provide an explicit labeling of the
edges of the quiver. We further explore special classes of J-trivial monoids,
and in particular monoids of order preserving regressive functions on a poset,
generalizing known results on the monoids of nondecreasing parking functions.Comment: 41 pages; 4 figures; added Section 3.7.4 in version 2; incorporated
comments by referee in version
T-structures on some local Calabi-Yau varieties
Let be a Fano variety satisfying the condition that the rank of the
Grothendieck group of is one more than the dimension of . Let
denote the total space of the canonical line bundle of , considered as a
non-compact Calabi-Yau variety. We use the theory of exceptional collections to
describe t-structures on the derived category of coherent sheaves on
. The combinatorics of these t-structures is determined by a natural
action of an affine braid group, closely related to the well-known action of
the Artin braid group on the set of exceptional collections on .Comment: 30 page
Fixed points of involutive automorphisms of the Bruhat order
Applying a classical theorem of Smith, we show that the poset property of
being Gorenstein over is inherited by the subposet of fixed
points under an involutive poset automorphism. As an application, we prove that
every interval in the Bruhat order on (twisted) involutions in an arbitrary
Coxeter group has this property, and we find the rank function. This implies
results conjectured by F. Incitti. We also show that the Bruhat order on the
fixed points of an involutive automorphism induced by a Coxeter graph
automorphism is isomorphic to the Bruhat order on the fixed subgroup viewed as
a Coxeter group in its own right.Comment: 16 pages. Appendix added, minor revisions; to appear in Adv. Mat
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