This seems worth passing along to the Math-Fun crowd. Read the bottom half first. My apologies to Propp & Speyer. -- Rich ----- Forwarded message from speyer@UMICH.EDU ----- Date: Mon, 2 Oct 2017 12:34:10 -0400 From: David Speyer <speyer@UMICH.EDU> Reply-To: The Robbins Forum <ROBBINS@LISTSERV.UML.EDU> Subject: Re: [ROBBINS] A 7-cycle To: ROBBINS@LISTSERV.UML.EDU This provides no insight but, yes, the composition has order 7. Define the seven cones 1: 0 < x,y 2: 0 < -x < y 3: 0 < y < -x 4: x<y<0 5: y<x<0 6: 0 < x < -y 7: 0 < y < -x Then one can check that h and v are linear on each cone, with image another cone. Specifically, h(k) = 4-k mod 7 and v(k) = -k mod 7. So hv permutes the cones by k+4 mod 7 and (hv)^7 is the identity on the set of cones; it is then easy to check that it is the identity on each cone. I'm curious about the tropicalization question -- seven isn't a number which tends to come out of cluster algebras. On Mon, Oct 2, 2017 at 12:20 PM, James Propp <jamespropp@gmail.com> wrote:
Apologies for the double posting to the DAC listserv and Robbins listserv; both seem equally appropriate.
If we define piecewise-linear continuous maps h and v from R^2 to itself with h((x,y)) = (-x-max(y,0),y) and v((x,y)) = (x,-y-max(x,0)), then the composition of h and v appears to be of order 7.
Is this true?
Usually the way I prove a PL identity like this is by finding a birational identity that tropicalizes to it (and having Mathematica do the heavy lifting), but in this case that strategy failed; I tried the obvious de-tropicalization H((x,y)) = (1/(x (1 + y)), y) and V((x,y)) = (x, 1/(y (1 + x))) but the composition of these maps is not of order 7.
Is the composition of h and v the tropicalization of some birational map of order 7?
(If you're wondering where h and v come from: they're the horizontal and vertical fliber-flipping maps for the polygon bounded by the lines x=-1, x=1, y=-1, y=1, and x+y=1; the composition of h and v leaves the origin fixed and cyclically permutes the 7 lattice points on the boundary. If the term "fiber-flipping" is unfamiliar, see the bottom of page 3 and the top of page 4 of https://arxiv.org/pdf/1404.3455.pdf, where the fiber-flipping maps are called piecewise-linear toggles.)
Jim Propp
----- End forwarded message ----- This provides no insight but, yes, the composition has order 7. Define the seven cones 1: 0 < x,y 2: 0 < -x < y 3: 0 < y < -x 4: x<y<0 5: y<x<0 6: 0 < x < -y 7: 0 < y < -x Then one can check that h and v are linear on each cone, with image another cone. Specifically, h(k) = 4-k mod 7 and v(k) = -k mod 7. So hv permutes the cones by k+4 mod 7 and (hv)^7 is the identity on the set of cones; it is then easy to check that it is the identity on each cone. I'm curious about the tropicalization question -- seven isn't a number which tends to come out of cluster algebras. On Mon, Oct 2, 2017 at 12:20 PM, James Propp <jamespropp@gmail.com> wrote:
Apologies for the double posting to the DAC listserv and Robbins listserv; both seem equally appropriate.
If we define piecewise-linear continuous maps h and v from R^2 to itself with h((x,y)) = (-x-max(y,0),y) and v((x,y)) = (x,-y-max(x,0)), then the composition of h and v appears to be of order 7.
Is this true?
Usually the way I prove a PL identity like this is by finding a birational identity that tropicalizes to it (and having Mathematica do the heavy lifting), but in this case that strategy failed; I tried the obvious de-tropicalization H((x,y)) = (1/(x (1 + y)), y) and V((x,y)) = (x, 1/(y (1 + x))) but the composition of these maps is not of order 7.
Is the composition of h and v the tropicalization of some birational map of order 7?
(If you're wondering where h and v come from: they're the horizontal and vertical fliber-flipping maps for the polygon bounded by the lines x=-1, x=1, y=-1, y=1, and x+y=1; the composition of h and v leaves the origin fixed and cyclically permutes the 7 lattice points on the boundary. If the term "fiber-flipping" is unfamiliar, see the bottom of page 3 and the top of page 4 of https://arxiv.org/pdf/1404.3455.pdf, where the fiber-flipping maps are called piecewise-linear toggles.)
Jim Propp