cahlen/kronecker-coefficients

Kronecker Coefficients — S_5 through S_40

Complete Kronecker coefficient tables and character tables for symmetric groups, computed via GPU-accelerated Murnaghan-Nakayama rule + slab-parallel triple-sum.

Part of the bigcompute.science project.

Datasets

Group	Partitions	Nonzero Kronecker	Max Coefficient	Char Table	GPU Time
S_5	7	206	6	392 B	<1s
S_20	627	32.7M	6.1B	3.0 MB	3.7s
S_30	5,604	26.4B	24.2T	240 MB	7.4 min
S_40	37,338	TBD	TBD	4.6 GB	Char table done, GPU triple-sum pending

S_40 — Largest Character Table Computed (2026-04-01)

The S_40 character table contains 37,338 x 37,338 = 1.394 billion entries, computed via the Murnaghan-Nakayama rim-path method using Python arbitrary-precision integers (int64 overflows at n >= 35).

• Validation: Row orthogonality OK, column orthogonality OK, sum of dim² = 40! (exact match)
• Size: 4.6 GB (text format — values exceed int64, stored as decimal strings)
• Computation: ~6.5 hours on CPU (Intel Xeon Platinum 8570)
• Note: GPU Kronecker triple-sum for S_40 requires int128 arithmetic. Standard int64 kernel will overflow. Kernel upgrade in progress.

Files

s40/
  char_table_n40.txt      # 4.6 GB — character table (37,338 x 37,338)
  z_inv_n40.bin           # 292 KB — centralizer inverse table
  partitions_n40.txt      # partition list (37,338 partitions of 40)

S_30 — Largest Kronecker Computation Published

26.4 billion nonzero Kronecker coefficients for S_30, computed in 7.4 minutes on a single NVIDIA B200. To our knowledge, the largest Kronecker coefficient computation ever published.

Files

s30/
  char_table_n30.bin      # 240 MB — character table (5,604 x 5,604)
  nonzero/part_*.bin      # 370 GB — nonzero triples in binary chunks
  z_inv_n30.bin           # centralizer inverse table
  partitions_n30.txt      # 5,604 partitions of 30

S_20 — Complete Tensor

32.7 million nonzero Kronecker coefficients, full tensor available as NPZ.

Files

s20/
  char_table_n20.bin      # 3.0 MB
  kronecker_n20_full_tensor.npz
  kronecker_n20_nonzero.csv
  z_inv_n20.bin
  partitions_n20.txt

Method

1. Character table: Murnaghan-Nakayama rule via rim-path border strip enumeration. Python arbitrary-precision ints for n >= 35 (int64 overflows).
2. Kronecker triple-sum: g(λ,μ,ν) = (1/n!) Σ_ρ χ^λ(ρ) χ^μ(ρ) χ^ν(ρ) / z_ρ. GPU-parallel: one thread per (i,k) pair for fixed j, atomic reduction.
3. Validation: Row/column orthogonality, Σ dim² = n!, spot checks against known values.

Hardware

• Character tables: CPU (Intel Xeon Platinum 8570, 112 cores)
• Kronecker triple-sum: NVIDIA B200 (183 GB VRAM)
• Part of 8×B200 DGX cluster

Applications

Kronecker coefficients are central to:

• Geometric Complexity Theory (GCT) — Mulmuley-Sohoni approach to P vs NP
• Quantum information — entanglement and quantum marginal problem
• Algebraic combinatorics — plethysm, Schur functions, symmetric function theory

Source

• Code: char_table.py, kronecker_gpu.cu
• MCP Server: mcp.bigcompute.science (22 tools, no auth)
• AGENTS.md: Contribution guide

Citation

@misc{humphreys2026kronecker,
  author = {Humphreys, Cahlen and Claude (Anthropic)},
  title = {Kronecker Coefficients: Complete Tables for S_20, S_30, and S_40},
  year = {2026},
  publisher = {Hugging Face},
  url = {https://huggingface.co/datasets/cahlen/kronecker-coefficients}
}

Human-AI collaborative work. Not independently peer-reviewed. All code and data open for verification. CC BY 4.0.