Computational group theory and representation theory in Clojure
Harmonica is a library for working with finite groups, their representations, and Fourier analysis. It covers cyclic, dihedral, symmetric, and product groups, with applications to combinatorics, signal processing, and card shuffling.
| Website | https://scicloj.github.io/harmonica/ |
| Source |  |
| Deps |  |
| License | MIT |
| Status | 🛠alpha🛠 |
- Cyclic groups Z/nZ, dihedral groups D_n, symmetric groups S_n, product groups G₁ × G₂
- Uniform protocol —
op,inv,id,elements,order,conjugacy-classes
- Cycle decomposition, cycle type, sign, composition, inverse
- Partition enumeration, conjugate partitions
- Standard Young tableaux, hook-length formula
- Cyclic (DFT matrix), dihedral, symmetric (Murnaghan-Nakayama rule), product (Kronecker product)
- Character inner product, orthogonality relations
- Forward and inverse Fourier transform on any finite abelian group
- Convolution via pointwise multiplication in the Fourier domain
- Total variation distance between distributions
- Irreducible representations of S_n via Young's orthogonal form
- Matrix Fourier transform (matrix-valued coefficients)
- Tensor product, direct sum, restriction, induction, branching rule
- Orbits, stabilizers, fixed points
- Burnside's lemma, cycle index, Pólya enumeration
- Subset actions, coloring actions
- SVG: Young diagrams, hook diagrams, standard Young tableaux, cycle diagrams
- SVG: Cayley tables, Cayley graphs
Add to your deps.edn:
{:deps {org.scicloj/harmonica {:mvn/version "0.1.0"}}}The Harmonica book is organized for incremental learning, interleaving theory with applications — from the DFT and rosette patterns through necklace counting, music theory, and card shuffling.
See the API reference for the full list of functions with examples.
Most users need only two namespaces:
(require '[scicloj.harmonica :as hm])
(require '[scicloj.harmonica.linalg.complex :as cx])scicloj.harmonica is the public API — groups, character tables, Fourier
transforms, representations, group actions, and visualization. cx provides
ComplexTensor operations for working with complex-valued results.
Internal namespaces (analysis.characters, analysis.fourier,
analysis.representations, linalg.ejml, protocols, action, etc.) are
available but rarely needed directly.
- dtype-next — array/tensor numerics
- fastmath — matrix operations
The book notebooks also use
tablecloth,
tableplot, and
kindly (kindly is also a top-level dependency; tablecloth and tableplot are in the :dev and :test aliases).
clojure -M:dev -m nrepl.cmdline # start REPL
./run_tests.sh # run tests
clojure -T:build ci # test + build JAR- Diaconis, P. (1988). Group Representations in Probability and Statistics. IMS Lecture Notes.
- Diaconis, P. & Shahshahani, M. (1981). Generating a random permutation with random transpositions. Z. Wahrscheinlichkeitstheorie, 57, 159–179.
- Bayer, D. & Diaconis, P. (1992). Trailing the dovetail shuffle to its lair. Annals of Applied Probability, 2(2), 294–313.
- Sagan, B. (2001). The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions. Springer.
- James, G. & Kerber, A. (1981). The Representation Theory of the Symmetric Group. Addison-Wesley.
MIT License — see LICENSE file.
Part of the scicloj ecosystem for scientific computing in Clojure.