50 research outputs found
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
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