schur-weyl

Tools from Schur–Weyl duality, built from scratch on top of numpy and the Python standard library — no representation-theory or combinatorics dependencies.

Schur–Weyl duality decomposes $(\mathbb{C}^d)^{\otimes n}$ under the commuting actions of $S_n$ (permuting tensor factors) and $GL_d$ (acting diagonally) into isotypic blocks indexed by partitions $\lambda \vdash n$ with $\ell(\lambda) \le d$:

$$(\mathbb{C}^d)^{\otimes n} \cong \bigoplus_{\lambda} S^\lambda \otimes V_\lambda^d$$

This package implements the combinatorics and linear algebra needed to work with that decomposition directly: partitions and Young tableaux, dimension formulas, symmetric-group characters via Murnaghan–Nakayama, Schur polynomials, Kronecker coefficients, isotypic projectors on tensor space, the Schur–Weyl measure, and Young's orthogonal form for the irreducible representations of $S_n$.

Installation

Requires Python ≥ 3.9.

pip install schur-weyl

The package itself only depends on numpy. For local development (cloning the repo, running the test suite), install it editable with the dev extra instead, which pulls in pytest and ruff:

pip install -e ".[dev]"

Modules

Usage

import numpy as np

from schur_weyl import (
    partitions, dim_specht, dim_weyl, kostka,
    schur_polynomial, kronecker_coefficient,
    isotypic_proj, apply_isotypic_proj,
)
from schur_weyl.character import character, character_table
from schur_weyl.sw_measure import schur_weyl_measure

# dimension formulas
dim_specht((2, 1))          # -> 2, the standard rep of S_3
dim_weyl((2, 1), d=3)       # -> 8

# characters
character((2, 1), (3,))     # -> -1

# the Schur-Weyl measure for n i.i.d. copies of a state with given spectrum
schur_weyl_measure(spectrum=[0.7, 0.3], k=4)
# -> {(4,): 0.4141, (3, 1): 0.4977, (2, 2): 0.0882, (2, 1, 1): 0.0, (1, 1, 1, 1): 0.0}

# Kronecker coefficients: the multiplicity of S^lam inside a tensor product
# of Specht modules, g(lam, mu, nu, ...), for any number of partitions of n
kronecker_coefficient((3,), (2, 1), (2, 1))       # -> 1
kronecker_coefficient((3,), (2, 1), (1, 1, 1))    # -> 0

# the isotypic projector on (C^d)^{\otimes n}
P = isotypic_proj((2, 1), d=2)   # np.ndarray, P @ P == P

# ... or apply it along chosen axes of a tensor, without ever building
# the d^k x d^k matrix. Axes outside `axes` are carried along untouched.
T = np.random.default_rng(0).normal(size=(4, 2, 2, 2, 5))
apply_isotypic_proj(T, (2, 1), axes=(1, 2, 3))    # -> same shape as T

Testing

pytest

Tests live in tests/ and mirror the module layout; most assert the closed-form identities from the roadmap (e.g. $\sum_\lambda (f^\lambda)^2 = n!$, character-table orthogonality, $P_\lambda^2 = P_\lambda$, $tr\rho^\lambda(X_k)$ diagonal with content eigenvalues) rather than fixed expected outputs.

License

MIT — see LICENSE.