The totally nonnegative part of G/P is a ball

Galashin, Pavel; Karp, Steven N.; Lam, Thomas

doi:10.1016/j.aim.2019.05.009

Cited by 22 publications

(25 citation statements)

References 27 publications

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“…In , Galashin, Karp, and Lam regard

σ \in {prefixfrakturgl}_{n} false(double-struckC false)

as a vector field on

{prefixGr}_{k, n} false(double-struckC false)

(viewed as a quotient of

{prefixGL}_{n} false(double-struckC false)

), whose integral curves are

t \mapsto exp (t σ) (V)

for

t \in R

and

V \in {prefixGr}_{k, n} false(double-struckC false)

. They show that if

V \in {prefixGr}_{k, n}^{⩾ 0}

, then

exp (t σ) (V)

lies in the interior of

{Gr}_{k, n}^{⩾ 0}

for

t > 0

, and converges to

V_{0}

t \to \infty

.…”

Section: Related Workmentioning

confidence: 99%

“…In [14], Galashin, Karp, and Lam regard σ ∈ gl n (C) as a vector field on Gr k,n (C) (viewed as a quotient of GL n (C)), whose integral curves are t → exp(tσ)(V ) for t ∈ R and V ∈ Gr k,n (C). They show that if V ∈ Gr 0 k,n , then exp(tσ)(V ) lies in the interior of Gr 0 k,n for t > 0, and converges to V 0 as t → ∞.…”

Section: Control Theory and The Topology Of Gr 0 Knmentioning

confidence: 99%

See 1 more Smart Citation

Moment curves and cyclic symmetry for positive Grassmannians

Karp

2019

Bull. London Math. Soc.

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We show that for each k and n, the cyclic shift map on the Grassmannian prefixGrk,nfalse(double-struckCfalse) has exactly ()nk fixed points. There is a unique totally nonnegative fixed point, given by taking n equally spaced points on the trigonometric moment curve (if k is odd) or the symmetric moment curve (if k is even). We introduce a parameter q∈double-struckC×, and show that the fixed points of a q‐deformation of the cyclic shift map are precisely the critical points of the mirror‐symmetric superpotential Fq on prefixGrk,nfalse(double-struckCfalse). This follows from results of Rietsch about the quantum cohomology ring of prefixGrk,nfalse(double-struckCfalse). We survey many other diverse contexts which feature moment curves and the cyclic shift map.

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“…In , Galashin, Karp, and Lam regard

σ \in {prefixfrakturgl}_{n} false(double-struckC false)

as a vector field on

{prefixGr}_{k, n} false(double-struckC false)

(viewed as a quotient of

{prefixGL}_{n} false(double-struckC false)

), whose integral curves are

t \mapsto exp (t σ) (V)

for

t \in R

and

V \in {prefixGr}_{k, n} false(double-struckC false)

. They show that if

V \in {prefixGr}_{k, n}^{⩾ 0}

, then

exp (t σ) (V)

lies in the interior of

{Gr}_{k, n}^{⩾ 0}

for

t > 0

, and converges to

V_{0}

t \to \infty

.…”

Section: Related Workmentioning

confidence: 99%

Section: Control Theory and The Topology Of Gr 0 Knmentioning

confidence: 99%

Moment curves and cyclic symmetry for positive Grassmannians

Karp

2019

Bull. London Math. Soc.

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“…. , b n ) ∈ R n be a generator of the kernel of Z 0 (it follows from the cyclic symmetry of Z 0 that b i = (−1) i−1 for 1 ≤ i ≤ n, see [7]). Choose them in such a way that a 1 and b 1 have the same sign.…”

Section: Proof Of Theorem 18mentioning

confidence: 99%

Some more amplituhedra are contractible

Blagojević

Galashin

Palić

et al. 2019

Sel. Math. New Ser.

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The amplituhedra arise as images of the totally nonnegative Grassmannians by projections that are induced by linear maps. They were introduced in Physics by Arkani-Hamed & Trnka (Journal of High Energy Physics, 2014) as model spaces that should provide a better understanding of the scattering amplitudes of quantum field theories. The topology of the amplituhedra has been known only in a few special cases, where they turned out to be homeomorphic to balls. The amplituhedra are special cases of Grassmann polytopes introduced by Lam (Current Developments in Mathematics 2014, Int. Press). In this paper we show that that some further amplituhedra are homeomorphic to balls, and that some more Grassmann polytopes and amplituhedra are contractible.

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“…A new property related to the discrete integrability called the Devron property is proposed in [8], whose definition is related to the anti-confined singularities [21]. The notion of strong τ -sequence is based on the irreducibility and the coprimeness of Laurent systems [6]. An observation on the integrability using the factorization of each iterate is given in [24].…”

Section: Introductionmentioning

confidence: 99%

On the Coprimeness Property of Discrete Systems without the Irreducibility Condition

Kanki¹,

Mase²,

Tokihiro³

2018

SIGMA

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In this article we investigate the coprimeness properties of one and twodimensional discrete equations, in a situation where the equations are decomposable into several factors of polynomials. After experimenting on a simple equation, we shall focus on some higher power extensions of the Somos-4 equation and the (1-dimensional) discrete Toda equation. Our previous results are that all of the equations satisfy the irreducibility and the coprimeness properties if the r.h.s. is not factorizable. In this paper we shall prove that the coprimeness property still holds for all of these equations even if the r.h.s. is factorizable, although the irreducibility property is no longer satisfied.

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The totally nonnegative part of G/P is a ball

Abstract: We show that the totally nonnegative part of a partial flag variety (in the sense of Lusztig) is homeomorphic to a closed ball.

Cited by 22 publications

References 27 publications

Moment curves and cyclic symmetry for positive Grassmannians

Moment curves and cyclic symmetry for positive Grassmannians

Some more amplituhedra are contractible

On the Coprimeness Property of Discrete Systems without the Irreducibility Condition

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