Kronecker S$_{40}$: Complete Character Table and Targeted Coefficients

The Finding

We computed the complete character table of the symmetric group $S_{40}$ and used it to extract targeted Kronecker coefficients via exact arithmetic:

Metric	S$_{20}$	S$_{30}$	S$_{40}$
Partitions $p(n)$	627	5,604	37,338
Character table entries	393K	31.4M	1.394 billion
Max $\|\chi\|$	fits int64	fits int64	5.90 $\times 10^{22}$ (exceeds int64)
Kronecker nonzero %	79.5% (exact)	89.9% (exact)	94.9% (sampled, 95% CI: 93.4%—96.1%)
Max $g(\lambda,\mu,\nu)$	6,408,361	24.2 trillion	$\geq 1.30 \times 10^{18}$ (sampled)
Character table time	1.7 sec	220 sec	9.5 hours

The S$_{40}$ character table has 37,338 rows and 37,338 columns (1.394 billion entries), computed via the Murnaghan-Nakayama rule. To our knowledge, this is the first publicly archived explicit $S_{40}$ character table file available as a downloadable dataset. (Note: the GAP system has provided on-demand computation of CharacterTable("Symmetric",40) since 1997, so the table has been computable for decades. Our contribution is pre-computing and publishing the full 4.6 GB file for direct access.)

Why This Matters

The int64 Barrier

At $n = 40$, character values first exceed the 64-bit integer range. The maximum absolute value $|\chi^\lambda(\rho)|$ is $5.90 \times 10^{22}$, compared to the int64 limit of $9.22 \times 10^{18}$. This makes the Kronecker triple-sum $g(\lambda,\mu,\nu) = \sum_\rho \frac{1}{z_\rho} \chi^\lambda(\rho) \chi^\mu(\rho) \chi^\nu(\rho)$ impossible with standard FP64 arithmetic on GPU. The full computation of all 8.68 trillion unique triples will require int128 or multi-precision GPU arithmetic — a significant engineering challenge.

Nonzero Fraction Approaching 1

The fraction of nonzero Kronecker coefficients grows monotonically with $n$:

$79.5\% \;\xrightarrow{n=20 \to 30}\; 89.9\% \;\xrightarrow{n=30 \to 40}\; 94.9\%$

This supports the conjecture that the Kronecker cone has density approaching 1 as $n \to \infty$. At $n = 40$, only about 1 in 20 Kronecker triples vanish. The precise rate of convergence to 1 is itself an open question related to the geometry of the Kronecker polytope.

Hooks are Multiplicity-Free

All 11,480 hook $\times$ hook $\times$ hook Kronecker triples have $g \in \{0, 1\}$. This confirms a known theorem: the tensor product of hook representations decomposes without multiplicities. We computed this exactly at unprecedented scale (40 hooks, each requiring a 37,338-term exact rational sum).

• 34.6% of hook triples are nonzero — much lower than the 94.9% overall rate
• This makes mathematical sense: hooks are “thin” representations with highly structured characters

GCT-Relevant Near-Rectangular Coefficients

For the Mulmuley-Sohoni geometric complexity theory program, the coefficients of near-rectangular partitions are most relevant. We computed all 11,480 triples from our set of 40 near-rectangular partitions of 40 (defined as partitions with at most 2 distinct part sizes differing by 1):

Triple	$g$
$(6^5 5^2, \; 6^5 5^2, \; 6^5 5^2)$	105,927,325
$(7^4 6^2, \; 6^5 5^2, \; 6^5 5^2)$	102,910,706
$(7^4 6^2, \; 7^4 6^2, \; 6^5 5^2)$	95,860,198
$(7^4 6^2, \; 7^4 6^2, \; 7^4 6^2)$	92,773,073
$(7^4 6^2, \; 6^5 5^2, \; 5^4 4^5)$	71,187,464

The near-rectangular nonzero rate is only 10.1% — far lower than the overall 94.9%. Our “near-rectangular” set consists of all 40 partitions of 40 with at most 2 distinct part sizes differing by 1 — this includes exact rectangles like $(8^5)$, $(5^8)$, $(4^{10})$, $(2^{20})$, $(20^2)$, $(40)$, $(1^{40})$ and near-rectangles like $(6^5 5^2)$, $(7^4 6^2)$, $(5^4 4^5)$, $(4^7 3^4)$, $(3^{10} 2^5)$, etc. (full list: identify_near_rectangles() in analyze_n40.py). This is broader than the strict GCT-relevant rectangular shapes, so 10.1% is likely an upper bound on GCT-relevant positivity. Checksum: SHA-256 of the near_rectangular_kronecker JSON object in analysis_n40.json is 5e9806f9...df8a1a8a. This is significant: the GCT-relevant region of the Kronecker cone is much sparser than the generic region. Positivity in this region cannot be taken for granted.

Character Table Analysis

Value Distribution

The 1.394 billion character values span 23 orders of magnitude:

log₁₀ magnitude	Count	Fraction
0 (values 1—9)	252,183,229	28.3%
1—5	580,078,760	65.1%
6—10	52,108,263	5.84%
11—15	5,131,252	0.58%
16—22	490,323	0.055%
zero	502,432,617	36.0%

The distribution is sharply concentrated: 93.4% of nonzero values have absolute value $< 10^6$. The tail is extremely thin but reaches $5.9 \times 10^{22}$.

Largest Irreducible Representations

The maximum dimension of an irreducible representation of $S_{40}$ is:

$\dim(\lambda_{\max}) = 58{,}965{,}081{,}685{,}061{,}803{,}130{,}880 \approx 5.9 \times 10^{22}$

achieved by the partition $\lambda = (10, 8, 6, 5, 4, 3, 2, 1, 1)$. The sum of squared dimensions equals $40!$ exactly:

$\sum_{\lambda \vdash 40} \dim(\lambda)^2 = 815{,}915{,}283{,}247{,}897{,}734{,}345{,}611{,}269{,}596{,}115{,}894{,}272{,}000{,}000{,}000 = 40!$

confirming the character table’s correctness.

Positive/Negative Balance

Of the 891.7 million nonzero entries:

• 447.0 million (50.1%) are positive
• 444.7 million (49.9%) are negative

The near-perfect symmetry reflects the interplay between even and odd border strips in the Murnaghan-Nakayama recursion.

Cross-$n$ Trends

$n$	$p(n)$	Char entries	Kronecker triples	Nonzero %	Max $g$	Char time
5	7	49	84	—	—	< 0.01s
20	627	393K	41.1M	79.5%	$6.4 \times 10^6$	1.7s
30	5,604	31.4M	29.3B	89.9%	$2.4 \times 10^{13}$	220s
40	37,338	1.394B	8.68T	94.9%	$\geq 1.3 \times 10^{18}$	9.5 hr

The growth rates suggest:

• Nonzero fraction: fits $1 - c/n^\alpha$ for some $\alpha > 0$, approaching 1
• Max coefficient: grows super-exponentially with $n$
• Character table time: roughly $O(p(n)^2 \cdot n)$, dominated by MN recursion depth

Method

Character Table (CPU, 9.5 hours)

The character table is computed via the Murnaghan-Nakayama rule: for each partition $\lambda \vdash 40$ and cycle type $\rho \vdash 40$, recursively remove border strips of sizes given by the parts of $\rho$. Each strip contributes $(-1)^{\text{height}}$ to the character value.

• $37{,}338 \times 37{,}338 = 1{,}394{,}126{,}244$ entries
• Hardware: 64-core CPU (2x Intel Xeon Platinum 8570), Python + memoized MN recursion
• Validated: $\sum \dim^2 = 40!$ (exact), row orthogonality, column orthogonality. All three validation checks were performed on the full $n = 40$ table (not just smaller $n$). Maximum orthogonality residual: exactly 0 (exact integer arithmetic throughout).
• Saved as text (4.6 GB) because values exceed int64

Targeted Kronecker Coefficients (CPU, exact arithmetic)

For selected partitions, we compute $g(\lambda, \mu, \nu)$ exactly using Python’s arbitrary-precision Fraction type:

$g(\lambda, \mu, \nu) = \sum_{\rho \vdash 40} \frac{1}{z_\rho}\, \chi^\lambda(\rho)\, \chi^\mu(\rho)\, \chi^\nu(\rho)$

Each sum has 37,338 terms with integer numerators up to $\sim 10^{68}$. The result is guaranteed to be an exact integer.

• Hook triples: 11,480 triples, 368 seconds
• Near-rectangular triples: 11,480 triples, 304 seconds
• Random sample: 1,000 triples (uniform over unordered triples $i \leq j \leq k$ drawn without replacement via random.seed(42) then sorted(random.sample(range(37338), 3)); see analyze_n40.py lines 343—353), 17 seconds

Full Computation Status

The full Kronecker table (8.68 trillion triples) requires a new GPU kernel with int128 arithmetic. The slab-by-slab approach from S$_{30}$ cannot be directly reused. This is planned for the 8$\times$B200 cluster with an estimated runtime of several days.

Reproduce

git clone https://github.com/cahlen/idontknow
cd idontknow

# Step 1: Compute character table (CPU, ~9.5 hours for n=40)
python3 scripts/experiments/kronecker-coefficients/char_table.py 40

# Step 2: Analyze and compute targeted Kronecker coefficients
python3 scripts/experiments/kronecker-coefficients/analyze_n40.py

Data

• Hugging Face: cahlen/kronecker-coefficients
• Source code: github.com/cahlen/idontknow

References

1. Murnaghan, F.D. (1938). “The Analysis of the Kronecker Product of Irreducible Representations of the Symmetric Group.” American Journal of Mathematics, 60(3), pp. 761—784.
2. Rosas, M.H. (2001). “The Kronecker Product of Schur Functions Indexed by Two-Row Shapes or Hook Shapes.” Journal of Algebraic Combinatorics, 14(2), pp. 153—173.
3. Ikenmeyer, C., Mulmuley, K., and Walter, M. (2017). “On vanishing of Kronecker coefficients.” Computational Complexity, 26(4), pp. 949—992.
4. Pak, I. and Panova, G. (2017). “On the complexity of computing Kronecker coefficients.” Computational Complexity, 26(1), pp. 1—36.
5. Mulmuley, K.D. and Sohoni, M.A. (2001). “Geometric complexity theory I: An approach to the P vs. NP and related problems.” SIAM Journal on Computing, 31(2), pp. 496—526.

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Computed 2026-04-03 (character table on CPU, targeted Kronecker coefficients via exact arithmetic). This work was produced through human—AI collaboration (Cahlen Humphreys + Claude). Not independently peer-reviewed. All code and data open for verification at github.com/cahlen/idontknow.