42 research outputs found
Tsirelson's problem and an embedding theorem for groups arising from non-local games
Tsirelson's problem asks whether the commuting operator model for two-party
quantum correlations is equivalent to the tensor-product model. We give a
negative answer to this question by showing that there are non-local games
which have perfect commuting-operator strategies, but do not have perfect
tensor-product strategies. The weak Tsirelson problem, which is known to be
equivalent to Connes embedding problem, remains open.
The examples we construct are instances of (binary) linear system games. For
such games, previous results state that the existence of perfect strategies is
controlled by the solution group of the linear system. Our main result is that
every finitely-presented group embeds in some solution group. As an additional
consequence, we show that the problem of determining whether a linear system
game has a perfect commuting-operator strategy is undecidable.Comment: Update to match published version, including a new background section
on quantum correlation sets, fixes to the definition of pictures in Section 6
(the previous definition using smooth curves allowed some pathological
behaviour), and other minor changes throughout. 66 pages, 36 figure
A pattern avoidance criterion for free inversion arrangements
We show that the hyperplane arrangement of a coconvex set in a finite root
system is free if and only if it is free in corank 4. As a consequence, we show
that the inversion arrangement of a Weyl group element w is free if and only if
w avoids a finite list of root system patterns. As a key part of the proof, we
use a recent theorem of Abe and Yoshinaga to show that if the root system does
not contain any factors of type C or F, then Peterson translation of coconvex
sets preserves freeness. This also allows us to give a
Kostant-Shapiro-Steinberg rule for the coexponents of a free inversion
arrangement in any type.Comment: 20 pages. Corrects some errors from a preliminary version that was
privately circulate
A Brylinski filtration for affine Kac-Moody algebras
Braverman and Finkelberg have recently proposed a conjectural analogue of the
geometric Satake isomorphism for untwisted affine Kac-Moody groups. As part of
their model, they conjecture that (at dominant weights) Lusztig's q-analog of
weight multiplicity is equal to the Poincare series of the principal nilpotent
filtration of the weight space, as occurs in the finite-dimensional case. We
show that the conjectured equality holds for all affine Kac-Moody algebras if
the principal nilpotent filtration is replaced by the principal Heisenberg
filtration. The main body of the proof is a Lie algebra cohomology vanishing
result. We also give an example to show that the Poincare series of the
principal nilpotent filtration is not always equal to the q-analog of weight
multiplicity. Finally, we give some partial results for indefinite Kac-Moody
algebras.Comment: Typos and reference correcte
The set of quantum correlations is not closed
We construct a linear system non-local game which can be played perfectly
using a limit of finite-dimensional quantum strategies, but which cannot be
played perfectly on any finite-dimensional Hilbert space, or even with any
tensor-product strategy. In particular, this shows that the set of
(tensor-product) quantum correlations is not closed. The constructed non-local
game provides another counterexample to the "middle" Tsirelson problem, with a
shorter proof than our previous paper (though at the loss of the universal
embedding theorem). We also show that it is undecidable to determine if a
linear system game can be played perfectly with a finite-dimensional strategy,
or a limit of finite-dimensional quantum strategies.Comment: 31 pages; v2 adds result on undecidability for finite-dimensional
strategie
Rationally smooth Schubert varieties and inversion hyperplane arrangements
We show that an element of a finite Weyl group is rationally smooth
if and only if the hyperplane arrangement associated to the inversion set
of is inductively free, and the product of the
coexponents is equal to the size of the Bruhat interval
, where is the identity in . As part of the proof, we describe
exactly when a rationally smooth element in a finite Weyl group has a chain
Billey-Postnikov decomposition. For finite Coxeter groups, we show that chain
Billey-Postnikov decompositions are connected with certain modular coatoms of
.Comment: 26 pages. Revised for publication, examples adde
Staircase diagrams and enumeration of smooth Schubert varieties
We enumerate smooth and rationally smooth Schubert varieties in the classical
finite types A, B, C, and D, extending Haiman's enumeration for type A. To do
this enumeration, we introduce a notion of staircase diagrams on a graph. These
combinatorial structures are collections of steps of irregular size, forming
interconnected staircases over the given graph. Over a Dynkin-Coxeter graph,
the set of "nearly-maximally labelled" staircase diagrams is in bijection with
the set of Schubert varieties with a complete Billey-Postnikov (BP)
decomposition. We can then use an earlier result of the authors showing that
all finite-type rationally smooth Schubert varieties have a complete BP
decomposition to finish the enumeration.Comment: 42 pages, 3 table
Entanglement in non-local games and the hyperlinear profile of groups
We relate the amount of entanglement required to play linear-system non-local
games near-optimally to the hyperlinear profile of finitely-presented groups.
By calculating the hyperlinear profile of a certain group, we give an example
of a finite non-local game for which the amount of entanglement required to
play -optimally is at least , for some
. Since this function approaches infinity as approaches
zero, this provides a quantitative version of a theorem of the first author.Comment: 27 pages. v2: improved results based on a suggestion by N. Ozaw
Annular embeddings of permutations for arbitrary genus
In the symmetric group on a set of size 2n, let P_{2n} denote the conjugacy
class of involutions with no fixed points (equivalently, we refer to these as
``pairings'', since each disjoint cycle has length 2). Harer and Zagier
explicitly determined the distribution of the number of disjoint cycles in the
product of a fixed cycle of length 2n and the elements of P_{2n}. Their famous
result has been reproved many times, primarily because it can be interpreted as
the genus distribution for 2-cell embeddings in an orientable surface,of a
graph with a single vertex attached to n loops. In this paper we give a new
formula for the cycle distribution when a fixed permutation with two cycles
(say the lengths are p,q, where p+q=2n) is multiplied by the elements of
P_{2n}. It can be interpreted as the genus distribution for 2-cell embeddings
in an orientable surface, of a graph with two vertices, of degrees p and q. In
terms of these graphs, the formula involves a parameter that allows us to
specify, separately, the number of edges between the two vertices and the
number of loops at each of the vertices. The proof is combinatorial, and uses a
new algorithm that we introduce to create all rooted forests containing a given
rooted forest.Comment: 17 pages, 9 figure
The Nash blow-up of a cominuscule Schubert variety
We compute the Nash blow-up of a cominuscule Schubert variety. In particular,
we show that the Nash blow-up is algebraically isomorphic to another Schubert
variety of the same Lie type. As a consequence, we give a new characterization
of the smooth locus and, for Grassmannian Schubert varieties, determine when
the Nash blow-up is a resolution of singularities. We also study the induced
torus action on the Nash blow-up and give a bijection between its torus fixed
points and Peterson translates on the Schubert variety.Comment: 16 page
Perfect Commuting-Operator Strategies for Linear System Games
Linear system games are a generalization of Mermin's magic square game
introduced by Cleve and Mittal. They show that perfect strategies for linear
system games in the tensor-product model of entanglement correspond to
finite-dimensional operator solutions of a certain set of non-commutative
equations. We investigate linear system games in the commuting-operator model
of entanglement, where Alice and Bob's measurement operators act on a joint
Hilbert space, and Alice's operators must commute with Bob's operators. We show
that perfect strategies in this model correspond to
possibly-infinite-dimensional operator solutions of the non-commutative
equations. The proof is based around a finitely-presented group associated to
the linear system which arises from the non-commutative equations