Solving Schubert Problems with Littlewood-Richardson Homotopies
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1 Solving Schubert Problems with Littlewood-Richardson Homotopies Jan Verschelde joint work with Frank Sottile and Ravi Vakil University of Illinois at Chicago Department of Mathematics, Statistics, and Computer Science jan ISSAC th International Symposium on Symbolic and Algebraic Computation. T.U. München, July Jan Verschelde (UIC) Littlewood-Richardson Homotopies ISSAC 2010, July 1 / 37
2 Outline 1 Numerical Schubert Calculus Schubert varieties and Schubert problems homotopies for enumerative geometry 2 The Moving Flag in Littlewood-Richardson Homotopies specialization and generalization encoding the moves by black checkers 3 2 Lines meeting 4 given general Lines in 3-space checker games encode moving flag and coordinates for solutions coordinate transformations and moving coordinates in homotopy 4 Localization Patterns and Checker Movements Jan Verschelde (UIC) Littlewood-Richardson Homotopies ISSAC 2010, July 2 / 37
3 Schubert Varieties A Schubert variety is defined by an n-dimensional flag F: F =[f 1 f 2 f n ] C n n f 1 f 1, f 2 f 1, f 2,...,f n and a k-dimensional bracket ω N k,1 ω 1 <ω 2 < <ω k n: { } Ω ω (F) = X C n k dim(x f 1,...,f ωi )=i, i = 1, 2,...,k. For example: for F C 6 6, Ω [246] (F) contains X = 0 x 21 x 31 x 32 1 x 41 x 42 x 43 0 x 52 x x 63 dim(x f 1, f 2 )=1 dim(x f 1, f 2, f 3, f 4 )=2 dim(x f 1, f 2, f 3, f 4, f 5, f 6 )=3 expressed via conditions on minors system of 13 polynomials in 9 variables Jan Verschelde (UIC) Littlewood-Richardson Homotopies ISSAC 2010, July 3 / 37
4 Schubert Problems A triple intersection [2 46] 3 =[246][2 46][2 46] means Ω [246] (I) Ω [246] (M) Ω [246] (F) where I : the identity matrix represents the standard flag, M : a matrix represents the moving flag, F : another matrix represents the fixed flag. The Littlewood-Richardson rule computes the number of solutions: [2 46] 3 = ([246][2 46])[2 46] = ([234]+2[1 35]+[126])[2 46] = [234][2 46]+2[1 35][2 46]+[126][2 46] = 0 + 2[1 23]+0 there are 2 isolated 3-planes in Ω [246] (I) Ω [246] (M) Ω [246] (F). Jan Verschelde (UIC) Littlewood-Richardson Homotopies ISSAC 2010, July 4 / 37
5 poset to resolve general problems Resolving [2 46][2 56] 3 : [246][256] 3 = (1[235]+1[145]+1[136])[256] 2 = (2[134]+2[125])[256] = 2[123]. Using a poset: specialize 2[246] 0[235] 1[136] 1[145] 0[134] 1[125] 1[123] generalize Jan Verschelde (UIC) Littlewood-Richardson Homotopies ISSAC 2010, July 5 / 37
6 a Geometric Littlewood-Richardson Rule William Fulton: Young Tableau. With Applications to Representation Theory and Geometry. Cambridge University Press, The first geometric proof and interpretation was given by Ravi Vakil: a geometric Littlewood-Richardson rule. Ann of Math, A combinatorial checker game for the Littlewood-Richardson coefficients implies that we can count (enumerate) the solutions to Schubert problems, compute these solutions via explicit deformations. Littlewood-Richardson homotopies Motivation: experimental study of reality conjectures secant Christopher Hillar, Luis Garcia-Puente, Abraham Martin del Campo, James Ruffo, Zach Teitler, Stephen L. Johnson, Frank Sottile: Experimentation at the Frontiers of Reality in Schubert Calculus. Contemporary Math. AMS Jan Verschelde (UIC) Littlewood-Richardson Homotopies ISSAC 2010, July 6 / 37
7 Homotopies for Enumerative Geometry B. Huber, F. Sottile, and B. Sturmfels: Numerical Schubert calculus. J. of Symbolic Computation, 26(6): , J. Verschelde: Numerical evidence for a conjecture in real algebraic geometry. Experimental Mathematics 9(2): , B. Huber and J. Verschelde: Pieri homotopies for problems in enumerative geometry applied to pole placement in linear systems control. SIAM J. Control Optim. 38(4): , F. Sottile and B. Sturmfels: A sagbi basis for the quantum Grassmannian. J. Pure and Appl. Algebra 158(2-3): , T.Y. Li, X. Wang, and M. Wu: Numerical Schubert calculus by the Pieri homotopy algorithm. SIAM J. Numer. Anal. 20(2): , J. Verschelde and Y. Wang: Computing dynamic output feedback laws. IEEE Trans. Automat. Control. 49(8): , A. Leykin and F. Sottile: Galois group of Schubert problems via homotopy continuation. Math. Comp. 78(267): , Jan Verschelde (UIC) Littlewood-Richardson Homotopies ISSAC 2010, July 7 / 37
8 Degenerating the moving Flag Given I : the identity matrix represents the standard flag, M : a matrix represents the moving flag, F : another matrix represents the fixed flag, we consider a triple intersection for some bracket ω: general problem: Ω ω (I) Ω ω (M) Ω ω (F) degeneration generalization degenerate problem: Ω ω (I) Ω ω (I) Ω ω (F) The degeneration M I allows to satisfy the intersection condition by solving some linear systems. Littlewood-Richardson homotopies generalize I to M via invertible transformations involving a parameter t. Jan Verschelde (UIC) Littlewood-Richardson Homotopies ISSAC 2010, July 8 / 37
9 Generalizing the moving Flag first three moves for n = 4, random γ ij C γ γ 21 0 γ γ 31 t γ 21 t γ 11 t = = = γ 31 t γ 21 t 0 γ γ 11 t 0 γ 21 0 γ Jan Verschelde (UIC) Littlewood-Richardson Homotopies ISSAC 2010, July 9 / 37
10 Generalizing the moving Flag last three moves for n = 4, random γ ij C γ 11 0 γ 21 0 γ γ 11 0 γ 2 γ 22 1 γ γ 11 γ 21 γ 21 γ 22 γ γ 22 t γ 21 t γ 13 t = = = γ 11 0 γ 2 γ 22 t 1 γ γ 11 γ 21 t γ 21 γ 22 γ γ 11 γ 21 γ 13 t 1 γ 21 γ 22 γ Jan Verschelde (UIC) Littlewood-Richardson Homotopies ISSAC 2010, July 10 / 37
11 Encoding the Moves bubble sort on nn 1 21 I γ γ 11 0 γ 2 γ 22 1 γ γ 21 0 γ γ 11 γ 21 γ 21 γ 22 γ γ 11 0 γ 21 0 γ γ 11 γ 21 γ 13 1 γ 21 γ 22 γ Jan Verschelde (UIC) Littlewood-Richardson Homotopies ISSAC 2010, July 11 / 37
12 Specialization in P Jan Verschelde (UIC) Littlewood-Richardson Homotopies ISSAC 2010, July 12 / 37
13 Littlewood-Richardson Homotopies ( ) n Degeneration of general flag from M to I in moves. 2 Three flag intersection condition Ω ω (I) Ω ω (M) Ω ω (F) is at the special position for M = I reduced to the equations imposed on X Ω ω (F) : P(X) =0. Generalizing the moving flag M leads to homotopies of the form P(M(t)X) =0, t [0, 1]. The solution k-plane X is represented in this moving basis M(t) in suitable local coordinates, via a localization pattern. Jan Verschelde (UIC) Littlewood-Richardson Homotopies ISSAC 2010, July 13 / 37
14 The Problem of Four General Lines Classical problem in projective 3-space P 3 : Given four general lines in P 3 (no triplet is coplanar), find all lines that meet those four given lines nontrivially. Identify P 3 with C 4 with natural basis (e 1, e 2, e 3, e 4 ). Let L 1 = span(e 1, e 2 ) and L 2 = span(e 3, e 4 ), consider X = x 2, 0 x 4,2 P(X) = where X meets L 1 and L 2 by its pattern. { det([x L3 ]) = 0 det([x L 4 ]) = 0 Specializing L 2 to coincide with L 1 makes X change pattern so P(X) =0 becomes a linear system. Jan Verschelde (UIC) Littlewood-Richardson Homotopies ISSAC 2010, July 14 / 37
15 Specializing Moving Flag to Fixed Flag stage Jan Verschelde (UIC) Littlewood-Richardson Homotopies ISSAC 2010, July 15 / 37
16 Specializing Moving Flag to Fixed Flag stage Jan Verschelde (UIC) Littlewood-Richardson Homotopies ISSAC 2010, July 16 / 37
17 Specializing Moving Flag to Fixed Flag stage Jan Verschelde (UIC) Littlewood-Richardson Homotopies ISSAC 2010, July 17 / 37
18 Specializing Moving Flag to Fixed Flag stage Jan Verschelde (UIC) Littlewood-Richardson Homotopies ISSAC 2010, July 18 / 37
19 Specializing Moving Flag to Fixed Flag stage Jan Verschelde (UIC) Littlewood-Richardson Homotopies ISSAC 2010, July 19 / 37
20 Specializing Moving Flag to Fixed Flag stage Jan Verschelde (UIC) Littlewood-Richardson Homotopies ISSAC 2010, July 20 / 37
21 Specializing Moving Flag to Fixed Flag stage Jan Verschelde (UIC) Littlewood-Richardson Homotopies ISSAC 2010, July 21 / 37
22 Resolving the Problem of Four Lines stage Jan Verschelde (UIC) Littlewood-Richardson Homotopies ISSAC 2010, July 22 / 37
23 Resolving the Problem of Four Lines stage Jan Verschelde (UIC) Littlewood-Richardson Homotopies ISSAC 2010, July 23 / 37
24 Resolving the Problem of Four Lines stage Jan Verschelde (UIC) Littlewood-Richardson Homotopies ISSAC 2010, July 24 / 37
25 Resolving the Problem of Four Lines stage Jan Verschelde (UIC) Littlewood-Richardson Homotopies ISSAC 2010, July 25 / 37
26 Resolving the Problem of Four Lines stage Jan Verschelde (UIC) Littlewood-Richardson Homotopies ISSAC 2010, July 26 / 37
27 Resolving the Problem of Four Lines stage Jan Verschelde (UIC) Littlewood-Richardson Homotopies ISSAC 2010, July 27 / 37
28 Resolving the Problem of Four Lines stage Jan Verschelde (UIC) Littlewood-Richardson Homotopies ISSAC 2010, July 28 / 37
29 Checker Games resolving [2 4][2 4][2 4][2 4] Jan Verschelde (UIC) Littlewood-Richardson Homotopies ISSAC 2010, July 29 / 37
30 no homotopy, only change of coordinates 0 0 Jan Verschelde (UIC) Littlewood-Richardson Homotopies ISSAC 2010, July 30 / 37
31 corresponding coordinate transformation x 1 0 x 32 = x 1 0 x 32 1 x 1 /(x 32 1) Jan Verschelde (UIC) Littlewood-Richardson Homotopies ISSAC 2010, July 31 / 37
32 homotopy, as red checkers swap rows 0 Jan Verschelde (UIC) Littlewood-Richardson Homotopies ISSAC 2010, July 32 / 37
33 homotopy as red checkers swap Similar to the case of a line meeting two lines and a fixed point, we use a homotopy: X(t) = At t = 0, X(0) fits the pattern. x 12 t x 12 x 32 0 x 32 t x 32. At t = 1, a coordinate change makes X(1) to fit the pattern. Jan Verschelde (UIC) Littlewood-Richardson Homotopies ISSAC 2010, July 33 / 37
34 Localization Patterns B E A C F D x 1,1 x 2,1 x 2,3 1 x 3,3 x 3,4 x 4,3 x 4,4 x 5,3 x 5,4 x 5,6 x 6,2 x 6,3 x 6,4 x 6,5 x 6,6 x 6,7 1 1 x 8,4 x 8,5 x 8,6 x 8,7 x 9,4 x 9,5 x 9,6 x 9,7 1 x 10,6 x 10,7 x 11,5 x 11,6 x 11,7 1 1 x 13,7 1 Jan Verschelde (UIC) Littlewood-Richardson Homotopies ISSAC 2010, July 34 / 37
35 Rising and Falling Checkers critical diagonal critical row To resolve Ω ω (F) Ω τ (M), 9 cases to consider: 1 Where is the top red checker in the critical diagonal? (a) In the rising checker s square. (b) Elsewhere in the critical diagonal. (c) There is no red checker in the critical diagonal. 2 Where is the red checker in the critical row? (α) In the descending checker s square. (β) Elsewhere in the critical row. (γ) There is no red checker in the critical row. Jan Verschelde (UIC) Littlewood-Richardson Homotopies ISSAC 2010, July 35 / 37
36 Movement of Red Checkers a α β γ b or c Jan Verschelde (UIC) Littlewood-Richardson Homotopies ISSAC 2010, July 36 / 37
37 An Implementation in PHCpack Since v Littlewood-Richardson homotopies are in PHCpack, at jan/download.html phc -e option #4 allows to resolve intersection conditions, e.g.: in C 10 :[6810] 7 = 720[1 23], in C 11 :[7911] 8 = 3598[1 23], in C 12 :[91112][8 1112] 13 = [123], etc... Solving small Schubert problems on a Mac OS X 2.2 Ghz: [2 4] 4 = 2 takes 5 milliseconds, [2 46] 3 = 2 takes 169 milliseconds, [2 58] 2 [468]=2 takes seconds, [2 468] 2 [2578]=3 takes seconds. Jan Verschelde (UIC) Littlewood-Richardson Homotopies ISSAC 2010, July 37 / 37
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