Littlewood–Richardson Homotopy

• Littlewood–Richardson Homotopy is a numerical continuation algorithm that leverages Vakil’s geometric LR rule to efficiently solve Schubert problems using minimal combinatorial paths.
• It computes zero-dimensional intersections in Grassmannians by tracking exactly the number of solution paths equal to the classical intersection number, ensuring optimal performance even for large-scale instances.
• The method employs local Stiefel coordinates, checkerboard configurations, and predictor–corrector techniques to robustly track solution paths in computational algebraic geometry.

The Littlewood–Richardson homotopy is an optimal numerical homotopy continuation algorithm designed for solving all solutions to Schubert problems in Grassmannians. Developed by leveraging Vakil’s geometric proof of the Littlewood–Richardson (LR) rule, it enables the efficient computation of zero-dimensional intersections of Schubert varieties, a fundamental task in numerical Schubert calculus. Unlike generic path-tracking approaches, the Littlewood–Richardson homotopy is combinatorially minimal—tracking exactly the number of solution paths equals the classical intersection number—with versatility to solve problem instances involving tens of thousands of solutions (Sottile et al., 2010, Leykin et al., 2018).

1. Geometric and Algebraic Foundations

The Grassmannian $\operatorname{Gr}(k,n)$ parametrizes $k$-dimensional complex subspaces $X \subset \mathbb{C}^n$. A point in $\operatorname{Gr}(k,n)$ can be represented as a full-rank $n \times k$ matrix, up to the right action of $\operatorname{GL}(k)$. A complete flag $F_\bullet$ in $\mathbb{C}^n$ is a nested sequence $$F_1 \subset F_2 \subset \cdots \subset F_n = \mathbb{C}^n$$ with $\dim F_i = i$.

A bracket (Schubert symbol) $\omega = [\omega_1 < \cdots < \omega_k]$ encodes a Schubert condition: $$\dim(X \cap F_{\omega_i}) \ge i, \quad i = 1,\ldots,k.$$ The corresponding Schubert variety $\Omega_\omega(F_\bullet)$ is defined as the subset of $\operatorname{Gr}(k,n)$ satisfying these rank conditions, with codimension $|\omega| = \sum_{i=1}^k (n-k+i-\omega_i)$. A Schubert problem consists of a list of brackets $\omega^1,\ldots,\omega^s$ such that

$|\omega^1| + \cdots + |\omega^s| = k(n-k),$

so that the intersection $\bigcap_{i=1}^s \Omega_{\omega^i}(F^i)$ is finite and transverse for general flags $F^1,\ldots,F^s$. The cardinality of this intersection is a classical cohomological invariant of $\operatorname{Gr}(k,n)$ (Sottile et al., 2010, Leykin et al., 2018).

2. Vakil’s Geometric Littlewood–Richardson Rule

Vakil’s geometric LR rule provides the backbone for the homotopy’s combinatorial structure. The intersection of two Schubert varieties $\Omega_\lambda(F_\bullet)$, $\Omega_\mu(M_\bullet)$—with $F_\bullet$, $M_\bullet$ general—is degenerated via a 1-parameter family as $M_\bullet$ is moved into a special position relative to $F_\bullet$.

This process is encoded by placing $n$ black checkers (for $\operatorname{Gr}(k,n)$) on an $n \times n$ grid (permutation matrix interpretation), with the movement realized as a bubble-sort that brings the configuration from the anti-diagonal to diagonal in $n(n-1)/2$ elementary swaps. K red checkers capture the geometric data of $\operatorname{Gr}(k,n)$ at each degeneration stage (Sottile et al., 2010, Leykin et al., 2018).

Each swap of black checkers affects the associated families of $k$-planes, resulting in local phenomena: in most cases, the solution variety deforms birationally, but in specific configurations (corresponding to a Littlewood–Richardson coefficient $>1$), the intersection splits into several components. Tracking the movement and branching of red checkers underpins the local deformation in the homotopy, with the multiplicity and branching governed by combinatorial LR coefficients.

3. Algorithmic Construction of the Homotopy

To solve a generic Schubert problem $S = \bigcap_{i=1}^s \Omega_{\omega^i}(F^i)$, the algorithm proceeds inductively by iteratively applying the geometric LR rule: $$\Omega_{\omega^1}(F^1) \cap \Omega_{\omega^2}(F^2) \sim \sum_{\sigma} c^\sigma_{\omega^1, \omega^2} \Omega_\sigma(F^1),$$ where $c^\sigma_{\omega^1, \omega^2}$ are the LR coefficients, and continuing recursively on smaller Schubert problems for each component. At the recursive leaf, intersections of complementary brackets lead to elementary problems solvable by linear algebra (the “start solutions”). The algorithm then reverses the sequence of geometric degenerations via a series of homotopy continuations parameterized by $t\in[0,1]$ to transfer solutions up the combinatorial hierarchy (the “checkerboard poset”) to recover all solutions to the original problem (Sottile et al., 2010, Leykin et al., 2018).

4. Local and Global Coordinates: Stiefel Forms and Checkerboards

The computational implementation employs local Stiefel coordinates: the Stiefel manifold $$\operatorname{St}_{k,n} \subset \mathbb{C}^{n\times k}$$ of full-rank $n \times k$ matrices, mapping to the Grassmannian by column-span. Echelon forms are selected for each Schubert cell, with the local equations for Schubert varieties formulated as rank conditions: $\rank\,([H | F_{\alpha_i}]) \le k + \alpha_i - i.$ For optimality, the intersection conditions are reformulated using the Plücker map and Cauchy–Binet to obtain a set of bilinear complete-intersection equations in minors, significantly reducing computational complexity (Leykin et al., 2018).

Checkerboard configurations encode both the combinatorial structure (black checkers: permutation; red checkers: particular subspace) and the parametric system of equations at each homotopy stage. Transitions correspond to coordinate changes and, in splitting cases, to true path-tracking in a small, canonical set of local variables (Sottile et al., 2010, Leykin et al., 2018).

5. Numerical Homotopy, Path-Tracking, and Optimality

Each checker move (simple transposition) induces a one-parameter family of flags, realized as $M(t)$ for $t\in[0,1]$, altering the system’s defining polynomials. The process is guided by three geometric cases, depending on the positions of red checkers (no move, affine shift, or two-path split). The solution paths are tracked numerically using predictor–corrector algorithms (e.g., Runge–Kutta–Fehlberg predictors with Newton correctors), with “start systems” always arising from elementary, linearly solvable subproblems.

For generic instances, all solution paths are regular: no path coalesces, bifurcates, or misses a solution unless prescribed by the LR coefficients. The total number of paths exactly equals the degree of the Schubert problem, ensuring no redundant tracking occurs (Sottile et al., 2010, Leykin et al., 2018).

6. Implementation and Performance

The Littlewood–Richardson homotopy is implemented in both PHCpack (compiled C++ via the phcpy interface) and in Macaulay2’s NumericalSchubertCalculus (interpreted; using NAG4M2). PHCpack employs sparse matrices for localization patterns, random complex numerical “γ-trick” entries for flag generality, and adaptive step-size control. In the overdetermined setting, random linear combinations are used to reduce systems to square form at each Newton step, a standard stabilization technique (Sottile et al., 2010, Leykin et al., 2018).

Reported timings demonstrate high efficiency for moderate-scale problems:

• On $\operatorname{Gr}(2,7)$, a degree $42$ problem is solved in $1.4\,\mathrm{s}$ (PHCpack).
• For large-scale problems ($d=30,459,\, 24,024,\, 8,860$), PHCpack executes in hours to days, but remains tractable, a first for such combinatorially optimal methods (Leykin et al., 2018).
• Classical problems such as four lines in $\mathbb{P}^3$ (on $\operatorname{Gr}(2,4)$) are solved in milliseconds, with detailed numerical outputs for each solution provided (Sottile et al., 2010).

7. Significance and Outlook

The Littlewood–Richardson homotopy constitutes the first general-purpose, combinatorially optimal homotopy algorithm for Schubert calculus on Grassmannians. Its solution path count is minimal and precisely matches the classical degree, with robust handling of large and complicated instances. The method’s reliance on deep geometric and combinatorial insights—via Vakil’s rule, efficient local coordinates, and checkerboard encoding—has shaped both the theory and the practice of numerical intersection theory. Independent software implementations in PHCpack and Macaulay2 are publicly available and have demonstrated the approach’s scaling properties and correctness for tens of thousands of solutions (Sottile et al., 2010, Leykin et al., 2018).