README.md

---
license: cc-by-4.0
tags:
  - mathematics
  - representation-theory
  - symmetric-groups
  - kronecker-coefficients
  - gpu-computation
  - geometric-complexity-theory
pretty_name: "Kronecker Coefficients — S_5 through S_40"
---

# 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](https://bigcompute.science) project.

## Datasets

| Group | Partitions | Nonzero Kronecker | Max Coefficient | Char Table | Time |
|-------|-----------|-------------------|-----------------|------------|------|
| S_5 | 7 | 206 | 6 | 392 B | <1s |
| S_20 | 627 | 32.7M (79.5%) | 6,408,361 | 3.0 MB | 3.7s |
| S_30 | 5,604 | 26.4B (89.9%) | 24.2T | 240 MB | 7.4 min |
| **S_40** | **37,338** | **~94.9% (sampled)** | **≥ 1.3 × 10¹⁸ (sampled)** | **4.6 GB** | **9.5 hr (char table)** |

## S_40 — Complete Character Table + Targeted Kronecker Analysis (2026-04-03)

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

- **Max |χ|**: 5.9 × 10²² (exceeds int64 max of 9.2 × 10¹⁸)
- **Char table nonzero**: 64.0% (891.7M / 1.394B entries)
- **Validation**: Row orthogonality OK, column orthogonality OK, Σ dim² = 40! (exact match)
- **Size**: 4.6 GB (text format — values exceed int64, stored as decimal strings)
- **Computation**: 9.5 hours on CPU (Intel Xeon Platinum 8570)

### Targeted Kronecker Coefficients (Exact Arithmetic)

Since the full triple-sum (8.68 trillion triples) requires int128 GPU arithmetic, we computed targeted Kronecker coefficients using Python exact `Fraction` arithmetic:

- **Hook × hook × hook** (11,480 triples): All g ∈ {0,1} — multiplicity-free (confirms Rosas 2001). 34.6% nonzero.
- **Near-rectangular GCT triples** (11,480 triples): Max g = 105,927,325. Only 10.1% nonzero — the GCT-relevant region is sparse.
- **Random sample** (1,000 triples): 94.9% nonzero (95% CI: 93.4%–96.1%). Max sampled g = 1.3 × 10¹⁸.

### Nonzero Fraction Trend

79.5% (S₂₀) → 89.9% (S₃₀) → 94.9% (S₄₀) — approaching density 1 as n → ∞.

### 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)
  analysis_n40.json       # targeted Kronecker results + statistics
```


### Float64 Binary Character Tables (.dbin files) — April 2026

Float64 (double-precision) binary versions of the character tables, formatted for direct GPU consumption by the CUDA Kronecker kernel. Each file stores the full character table as a flat row-major array of float64 values, dimensions p(n) x p(n).

```
s30/char_table_n30.dbin   # 240 MB — S30 character table, float64 binary
s40/char_table_n40.dbin   # 11.15 GB — S40 character table, float64 binary
```

### Full S_40 Kronecker Table — Status

The full computation of all 8.68 trillion unique Kronecker triples requires a new GPU kernel with int128 or multi-precision arithmetic. The existing slab-by-slab FP64 kernel cannot handle character values up to 5.9 × 10²². This is planned for the 8×B200 cluster.

## S_30 — Largest Complete Kronecker Table Published

26,391,236,124 nonzero Kronecker coefficients out of 29,347,802,420 total triples (89.9%) for S_30, computed in 296 seconds on a single NVIDIA B200. Max |g(lambda, mu, nu)| = 51,798,395,983,223,240. To our knowledge, the largest complete Kronecker coefficient table 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) = Σ_ρ (1/z_ρ) χ^λ(ρ) χ^μ(ρ) χ^ν(ρ). GPU-parallel: one thread per (i,k) pair for fixed j, atomic reduction.
3. **Targeted computation** (S_40): Exact rational arithmetic via Python `Fraction` for selected partition families (hooks, near-rectangles, random sample).
4. **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](https://github.com/cahlen/idontknow/blob/main/scripts/experiments/kronecker-coefficients/char_table.py), [kronecker_gpu.cu](https://github.com/cahlen/idontknow/blob/main/scripts/experiments/kronecker-coefficients/kronecker_gpu.cu), [analyze_n40.py](https://github.com/cahlen/idontknow/blob/main/scripts/experiments/kronecker-coefficients/analyze_n40.py)
- **Findings**: [S_30](https://bigcompute.science/findings/kronecker-s30-largest-computation/), [S_40](https://bigcompute.science/findings/kronecker-s40-character-table/)
- **MCP Server**: `mcp.bigcompute.science` (22 tools, no auth)
- **AGENTS.md**: [Contribution guide](https://github.com/cahlen/idontknow/blob/main/AGENTS.md)

## Understanding This Data

Symmetric groups describe all the ways you can rearrange a set of objects. Kronecker coefficients answer the question: when you combine two of these rearrangement patterns, what patterns do you get?

Each character table lists the "partitions" of n (ways to write n as a sum, like 5 = 3+2 = 2+2+1) and the character values that describe each rearrangement pattern. For S20, there are 627 partitions; for S30, there are 5,604; for S40, there are 37,338. The Kronecker coefficient g(lambda, mu, nu) for three partitions tells you how many times the pattern nu appears when you combine patterns lambda and mu. A value of 0 means "never"; a value of 1 means "exactly once."

A concrete example from S20: given three specific partitions of 20, the Kronecker coefficient might be 14, meaning those two patterns combine to produce the third pattern in 14 distinct ways.

These numbers grow fast. At S30, there are 26.4 billion nonzero coefficients -- this is the largest complete Kronecker table ever published. At S40, even the character values exceed what a 64-bit integer can hold (the largest is 5.9 times 10^22), so the full Kronecker table requires a special 128-bit GPU kernel that has not been built yet.

Why does anyone outside pure math care? Geometric Complexity Theory, one approach to the famous P vs NP problem, needs exactly these coefficients. Computing them at scale is a prerequisite for that research program. Kronecker coefficients also show up in quantum information theory, where they describe entanglement structure.

## Citation

```bibtex
@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.