schubmult

Fast Schubert calculus in Python: Littlewood-Richardson coefficients, Schubert/Grothendieck rings, and the combinatorics behind them.

schubmult multiplies single, double, quantum, and quantum double Schubert polynomials (with parabolic variants), expands products in the Schubert basis, and gives you the combinatorial objects that index them: permutations, RC graphs (pipe dreams), bumpless pipe dreams, tableaux, and their crystal structures. It is built on SymEngine for speed and SymPy for display, so results drop straight into SymPy or Sage.

Documentation: https://matthematics.github.io/schubmult/

Installation

pip install schubmult

Development version:

pip install git+https://github.com/matthematics/schubmult.git

Requires Python 3.10+. To build the documentation locally, install the docs extra: pip install -e ".[docs]".

Quick start

Python

from schubmult import Sx, DSx, QSx, QDSx, Gx, DGx, Permutation
from schubmult.abc import x, y, z

# Ordinary Schubert polynomials: S_312 * S_132 = S_321 + S_4123
Sx([3, 1, 2]) * Sx([1, 3, 2])
# 𝔖_(3, 2, 1)(x) + 𝔖_(4, 1, 2, 3)(x)

# Expand to a polynomial, or go the other way
Sx([3, 1, 2]).expand()                       # x_1**2
Sx.from_expr(x[1]**2 + x[1] * x[2])          # 𝔖_(2, 3, 1)(x) + 𝔖_(3, 1, 2)(x)

# Double Schubert polynomials S_w(x; y); coefficients are polynomials in y
DSx([1, 3, 2]) * DSx([1, 3, 2])
# (-y_2 + y_3)*𝔖_(1, 3, 2)(x; y) + 𝔖_(1, 4, 2, 3)(x; y) + 𝔖_(2, 3, 1)(x; y)

# Mixed variables: S_w(x; y) * S_v(x; z)
DSx([1, 3, 2]) * DSx([2, 1, 3], z)
# (y_1 - z_1)*𝔖_(1, 3, 2)(x; y) + 𝔖_(2, 3, 1)(x; y) + 𝔖_(3, 1, 2)(x; y)

# Quantum Schubert polynomials
QSx([2, 1, 3]) * QSx([2, 1, 3])              # q_1 + 𝕼𝔖_312(x)

# Quantum double Schubert polynomials
QDSx([2, 1, 3]) * QDSx([2, 1, 3])
# q_1 + (-y_1 + y_2)*𝕼𝔖_21(x; y) + 𝕼𝔖_312(x; y)

# Grothendieck polynomials (K-theory)
Gx([2, 1, 3]) * Gx([2, 1, 3])                # 𝔊_(3, 1, 2)(x)

# Double Grothendieck polynomials; .simplify() puts the rational coefficients in y, β in normal form
(DGx([1, 3, 2]) * DGx([2, 1, 3])).simplify()
# β*𝔊_(3, 2, 1)(x; y) + 𝔊_(2, 3, 1)(x; y) + 𝔊_(3, 1, 2)(x; y)
(DGx([2, 1, 3]) * DGx([2, 1, 3])).simplify()
# (y_1 - y_2)*𝔊_(2, 1)(x; y)   (y_1*β + 1)*𝔊_(3, 1, 2)(x; y)
# -------------------------- + -----------------------------
#         y_2*β + 1                      y_2*β + 1

# Permutations use one-line notation; Lehmer codes are available too
w = Permutation([3, 1, 4, 2])
w.code, w.inv, w.descents()                  # ([2, 0, 1], 3, {0, 2})

Ring objects: Sx (single), DSx (double), QSx/QDSx (quantum, quantum double), QPSx/QPDSx (parabolic quantum), Gx/DGx (single/double Grothendieck). Elements are dictionaries {Permutation: coefficient} with *, +, .expand(), .coproduct(), and change of basis between them. The underlying multiplication kernels live in schubmult.mult (single, double, quantum, quantum_double, groth, groth_double).

Command line

Each ring has a CLI. Permutations are space-separated, factors are separated by -:

schubmult_py 3 1 2 - 2 1 3                         # 1  (4, 1, 2, 3)
schubmult_py --code 2 0 - 1 0                      # same product via Lehmer codes
schubmult_double 1 3 2 - 1 3 2 --display-positive  # double Schubert, coefficients displayed positively
schubmult_q 2 1 3 - 2 1 3                          # quantum
schubmult_q_double 2 1 3 - 2 1 3 --parabolic 1     # parabolic quantum double
grothmult_py 2 1 3 - 2 1 3                         # Grothendieck
grothmult_double 2 1 3 - 2 1 3                     # double Grothendieck (coefficients in y and β)
grothmult_q 2 1 - 2 1                              # quantum Grothendieck (conjectural quantum K-Pieri rule; coefficients in q and β)
grothmult_q_double 2 1 - 2 1 --mixed-var           # quantum double Grothendieck (coefficients in y, z, q and β)

--display-positive writes double and quantum double coefficients as manifestly positive expressions in the differences y_i - z_j (Graham positivity), using integer programming to find a positive representative. Run any script with --help for the full option list.

Combinatorics

The schubmult.combinatorics package provides the objects that index Schubert calculus, with conversions between them:

• Permutation -- one-line notation, Lehmer codes (uncode), reduced words, Bruhat order, descents, dominant/Grassmannian tests.
• RCGraph -- reduced compatible sequences / pipe dreams, with enumeration (RCGraph.all_rc_graphs(w, n)), Kashiwara crystal operators (raising_operator(i), lowering_operator(i)), Edelman-Greene insertion, and the crystal products used in the transition formulas.
• BPD -- bumpless pipe dreams with droop moves, the Gao-Huang bijection to RC graphs (BPD.from_rc_graph, .to_rc_graph()), and marked/unreduced variants for Grothendieck polynomials.
• WCGraph, PipeDream, HPD -- K-theoretic and hybrid pipe dream models.
• Tableaux -- Plactic (semistandard, jeu de taquin), NilPlactic (Edelman-Greene), RootTableau, IncreasingTableau, SetValuedTableau, HeckePlactic.
• Forests -- indexed forests and the Thompson monoid factorization behind the forest and grove bases.

from schubmult import RCGraph, BPD, Permutation

w = Permutation([1, 4, 2, 3])
rcs = RCGraph.all_rc_graphs(w, 3)        # all RC graphs of w in 3 rows

rc = next(rc for rc in rcs if rc.lowering_operator(1) is not None)
rc.lowering_operator(1)                  # crystal operator f_1 (None when undefined)

bpd = BPD.from_rc_graph(rc)              # Gao-Huang bijection
assert bpd.to_rc_graph() == rc

Rings and algebras

Beyond the Schubert rings, schubmult.rings includes:

• PolynomialAlgebra with interchangeable bases: monomials, Schubert, key (Demazure), Lascoux, fundamental/monomial slide, glide, forest, grove, Grothendieck, and elementary symmetric bases.
• FreeAlgebra -- the graded dual of the polynomial algebra (a word (a_1, ..., a_n) is dual to x_1^a_1 ... x_n^a_n), with a dual basis for each of the above; ASx is the dual Schubert basis.
• Combinatorial rings -- RCGraphRing, WCGraphRing, CrystalGraphRing, and related algebras whose basis elements are the combinatorial objects themselves.
• Tensor and direct products, NSym, QSym, and the nil-Hecke algebra.

from schubmult import ASx, FA

ASx([2, 1, 3]) * ASx([1, 3, 2])   # dual Schubert basis of the free algebra
FA(1, 0) * FA(2)                  # word basis: concatenation, [102]

Symbolic layer

schubmult.symbolic wraps SymEngine and SymPy behind one interface (sympify, expand, Add, Mul, S), provides indexed variable families (GeneratingSet("x") gives x_1, x_2, ...), and unevaluated elementary/complete symmetric polynomial atoms (E, e, H, h) so Schubert polynomials can be manipulated in the SEM basis. schubmult.abc exposes ready-made x, y, z, q, beta in the spirit of sympy.abc.

Using with SageMath

Sage is not a dependency: everything above works in plain Python. If you do have Sage, install schubmult into that environment (sage -pip install schubmult, or pip install schubmult inside a conda sage env) and schubmult.sage provides Sage parents whose arithmetic is delegated to the kernels, alongside Sage's own SchubertPolynomialRing:

sage: from schubmult.sage import DoubleSchubertPolynomialRing, QuantumSchubertPolynomialRing
sage: X = DoubleSchubertPolynomialRing(QQ)
sage: X([3, 1, 2]) * X([2, 1])
(y_2-y_0)*X_y[3, 1, 2] + X_y[4, 1, 2, 3]
sage: X([3, 1, 2]).expand()
x0^2 - x0*y0 - x0*y1 + y0*y1
sage: G = QuantumSchubertPolynomialRing(QQ, parabolic=(2, 3))   # QH^*(Gr(2, 5))
sage: G([3, 5, 1, 2, 4]) * G([1, 3, 2])
q_0*Xq[1, 3, 2] + Xq[3, 6, 1, 2, 4, 5] + Xq[4, 5, 1, 2, 3]
sage: from schubmult.sage import DoubleGrothendieckPolynomialRing
sage: GD = DoubleGrothendieckPolynomialRing(QQ)                  # K_T of the flag variety
sage: GD([2, 1]) * GD([2, 1])
-((y_1-y_0)/(beta*y_1+1))*G_y[2, 1] + ((beta*y_0+1)/(beta*y_1+1))*G_y[3, 1, 2]
sage: from schubmult.sage import PolynomialAlgebra
sage: A = PolynomialAlgebra(QQ); k = A.key(); F = A.fundamental_slide()   # bases of QQ[x0, x1, ...]
sage: F(k[1, 0, 2])
F[1, 0, 2] + F[2, 0, 1]
sage: k(A.schubert()[3, 1, 2])
k[2]

DoubleSchubertPolynomialRing, QuantumSchubertPolynomialRing, QuantumDoubleSchubertPolynomialRing (with parabolic= block sizes), GrothendieckPolynomialRing, and DoubleGrothendieckPolynomialRing accept permutations, Sage polynomials, and elements of SchubertPolynomialRing and KeyPolynomials; a ring in another alphabet (DoubleSchubertPolynomialRing(QQ, 'z')) coerces in for mixed products, and from_symmetric_function(f, n) / to_symmetric_function() go back and forth with SymmetricFunctions. PolynomialAlgebra(R) (or PolynomialAlgebra(R, n) in n variables) is the polynomial ring with its combinatorial bases as mutually coercing realizations: Schubert and Grothendieck indexed by permutations, monomial/key/slide/forest/glide/Lascoux/grove by weak compositions, and in n variables the elementary symmetric basis; its K-theoretic bases are at beta = -1, whereas the Grothendieck rings keep beta as a variable. Variables are 0-indexed as in Sage (x0, y0, q0).

Documentation

Full API documentation, generated from the source docstrings, is at https://matthematics.github.io/schubmult/. To build it locally:

pip install -e ".[docs]"
python docs/generate_docs.py
mkdocs serve

Development

git clone https://github.com/matthematics/schubmult.git
cd schubmult
pip install -e .
pytest

Tests live in tests/; the script tests compare CLI output against stored JSON cases in tests/scripts/data. ruff check src/schubmult should pass.

License

GPL-3.0. See LICENSE.