abakus

abakus

Sparse, vectorized algebra over countably infinite bases.

Many algebras in mathematics have a countable basis: polynomials in infinitely many variables, symmetric functions, Clifford and exterior algebras, Weyl algebras, class algebras of symmetric groups. abakus computes in such algebras directly in a chosen basis, with numpy arrays and Numba-compiled kernels.

Ideas

Vectors and bases. An element of an algebra is a Vector: a finite linear combination of basis elements, stored as an integer index array and a coefficient array. A basis is a class that knows what its indices mean. It defines multiplication, comultiplication and changes of basis on whole arrays of indices at once. Tensor products, smash products and linear maps are built uniformly on top of this.

Countably infinite bases. Most bases are infinite: all partitions, all monomials in infinitely many variables. Each vector touches only finitely many basis elements, and indices grow as needed. Beyond the int64 range, indices turn into exact Python integers on their own.

Tensor products are built uniformly: index rows are placed side by side, a @ b is a⊗b, and Koszul signs appear when two factors are superalgebras. Smash products such as Weyl algebras are tensor products with a twisted multiplication.

Partitions as integers. An integer partition is encoded by the boundary of its Ferrers diagram: a step up is a 1 bit, a step right is a 0 bit. The partition (8, 5, 2, 2) becomes 0b10001000110 == 1094. Partitions are then plain integers that can live in numpy arrays. Containment of diagrams implies x <= y. Horizontal and vertical strips, rim hooks, and transposition become bit operations. So do the Pieri, Littlewood–Richardson and Murnaghan–Nakayama rules (see docs/BinaryPartitions.md).

Example

>>> from abakus.symm import s, h, p
>>> s(2, 1) * s(1)              # Littlewood–Richardson rule
s{2,2} + s{3,1} + s{2,1,1}
>>> h(s(2, 1))                  # Jacobi–Trudi
-h{3} + h{2,1}
>>> s(p(3))                     # Murnaghan–Nakayama
s{3} - s{2,1} + s{1,1,1}
>>> s(100) * s(100) == s(h(100, 100))
True

>>> from abakus import bpart
>>> x = bpart.from_lambda([8, 5, 2, 2])
>>> x
1094
>>> bpart.to_lambda(bpart.transpose(x))
[4, 4, 2, 2, 2, 1, 1, 1]

>>> from abakus.clifford import CliffordAlgebraBasis, ExteriorAlgebraBasis as E
>>> C = CliffordAlgebraBasis[3, 0]
>>> C.u * C.v, C.v * C.u, (C.u + C.v) ** 2
(uv, -uv, 2)

>>> from abakus.polynomial import PolynomialBasis
>>> from abakus.weyl import polynomial_weyl_basis
>>> W = polynomial_weyl_basis(PolynomialBasis["x"])
>>> W.dx * W.x
1 + x∂

>>> s(2) @ E.u
s{2}⊗u

More in the tutorial, in docs/SymmetricFunctions.md and in docs/TensorProducts.md.

Modules

Installation

pip install abakus

Requires Python 3.14 or later, numpy and Numba.

For development:

pip install -e ".[dev]"
pytest

License

MIT, see LICENSE.