CAG@LOCS#7
Sometimes at Bollo's, and occasionally elsewhere, a sensation of nearly complete tranquility will permeate me. The fogged-up front window closing the place off from the presumed snow outside, evoking microcosmic short stories; the lukewarm trickle of mocha; the playlist falling from the Meters into a sparkling puddle of Sigur Rós . . . and a mathematical epiphany drawing my eyes, for once, away from the screen and onto the paper where they belong. I feel like i've found something i lost that never existed in the first place.
Today's epiphany should've come to me a year ago. What i've been calling the "shape" of a Grassmannian affine permutation w (and cleverly denoting |w|) is not the Grassmannian (finite) permutation i've been assuming, but actually in some sense the complement of it; i should more appropriately label it |w|-perp (can't do a "perp" superscript in html), for as a (k,n)-Grassmannian permutation it is obtained by swapping the two increasing strings of |w|, effectively changing the length from l(|w|) to k(n - k) - l(|w), but also producing the correct subset of generating equations for the matrix affine Schubert variety. (For anyone up to speed on Plücker coordinates, as an example it sends p1, p23, p245, and p246 (with n = 7) to p12, p134, and p234.)
So i'm content today, despite stressing like Hannelore over my hopeful advancement to candidacy Tuesday.
Today's epiphany should've come to me a year ago. What i've been calling the "shape" of a Grassmannian affine permutation w (and cleverly denoting |w|) is not the Grassmannian (finite) permutation i've been assuming, but actually in some sense the complement of it; i should more appropriately label it |w|-perp (can't do a "perp" superscript in html), for as a (k,n)-Grassmannian permutation it is obtained by swapping the two increasing strings of |w|, effectively changing the length from l(|w|) to k(n - k) - l(|w), but also producing the correct subset of generating equations for the matrix affine Schubert variety. (For anyone up to speed on Plücker coordinates, as an example it sends p1, p23, p245, and p246 (with n = 7) to p12, p134, and p234.)
So i'm content today, despite stressing like Hannelore over my hopeful advancement to candidacy Tuesday.
