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# Argument selection for T661
## Quantity and normalization
For q>=1, n>=2 and rational s, the table holds the unnormalized finite sum
D(q,n,s)=sum chi(1)^s over all irreducible complex characters of C_q wreath S_n.
Every linear character and every multiplicity of a repeated degree is included.
Positive degrees make this definition valid for every rational s, including
zero and negative exponents. Any abelian base of order q has the same spectrum.
## Group families retained
| Base order q | Permutation degree n | Pairs | Selected s |
|---|---|---:|---|
| 1 | 3 through 20 | 18 | 0, 1, 2, 9/4, 3, 4, -2, -4 |
| 2 | 2 through 12 | 11 | 0, 1, 2, 9/4, 3, 4, -2, -4 |
| 3 | 2 through 10 | 9 | 0, 1, 2, 9/4, 3, 4, -2, -4 |
| m^3, 2<=m<=40 | 2 | 39 | 2, 9/4, 3 |
These are the same 77 group pairs as the earlier revision, now with 421 values.
All 154 earlier values and entry identities are retained. The bounds in q and n
are unchanged. The symmetric and hyperoctahedral specializations are intrinsic.
The three-colour case is an explicitly exploratory reference beyond those two
families, not a claim of an application for every group. Its cutoff and the
other finite cutoffs are practical, not definitions of the families.
The cube base orders arise from the abelian group (C_m)^3 in the two-factor
construction of Cohn, Kleinberg, Szegedy and Umans, Section 2. Here q=m^3 is
the base order and n=2 is the number of permuted copies. Their existing range
includes small cases and both sides of the coarse optimum near m=17.
## Why these exponents
- s=0 counts irreducible characters, equivalently conjugacy classes. For this
family it counts ordered q-tuples of partitions of total size n.
- s=1 is the dimension of a Gelfand model. For the cyclic-base realization,
it counts symmetric monomial matrices with nonzero entries q-th roots of
unity. These are ordinary involutions for q=1,2, but not generally for q>2.
- s=2 is the group order q^n n!, the regular-representation identity and the
natural quadratic benchmark in the matrix-multiplication application.
- s=9/4 is the literature-selected comparison exponent in Amanov, van den
Berg and Zuiddam, Theorem 1.1. It is an upper bound, not an assertion of
equality with the unknown matrix-multiplication exponent.
- s=3 is the classical sum-of-cubes matrix-block cost in CKSU, Section 2.
- s=4 determines the collision probability for Plancherel-distributed
irreducibles: if p_chi=chi(1)^2/|G|, then sum p_chi^2=D(4)/|G|^2.
- s=-2,-4 are the first two negative even exponents 2-2g, for orientable
surfaces of genus g=2,3. The exact rational sum becomes the number of
homomorphisms from the surface group after multiplication by |G|^(2g-1).
The extra exponents are supplied on the small-base reference families, rather
than indiscriminately crossing every exponent with the large cube-base slice.
For that application-specific slice, only the essential s=2 benchmark is
added. No decimal, half-integer or denominator-bounded interpolation grid is
introduced. This broadens mathematical coverage instead of densifying one
numerical interval. Higher genera and higher moments remain possible future
extensions; this selection does not claim completeness in s.
## Reproducibility and historical attachments
generate.py recomputes the degree spectrum with exact integer hook products.
Nonnegative integral exponents give exact integers; negative integral
exponents give exact rationals. Only the 9/4 entries are approximations, with
100 significant digits certified by Arb. No precomputed answer file is an
input to the generator.
checks.py independently reconstructs the small-base spectra by Young-graph
path counts, compares every entry, checks the Gelfand dimension and class
count identities, and counts products of commutators directly in four small
groups. Integer fourth-root isolation checks every 9/4 interval without Arb.
coverage-evidence.json records this revision's checks. Older numerical and
repair reports remain attached as historical evidence for their stated
154-entry scope; they do not certify the new entries. None of these files
constitutes a human acceptance of the table.
The governing numberdb-table skill was read locally and through /skill.
Both copies have SHA-256
a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278.
## Primary sources
- Gao and Jing, Irreducible Characters of the Generalized Symmetric Group,
Journal of Lie Theory 35 (2025), 593–616, Theorem 4.1:
https://doi.org/10.5802/jolt.1399
- Cohn, Kleinberg, Szegedy and Umans, Group-theoretic Algorithms for Matrix
Multiplication, Theorem 1.8 and Section 2:
https://arxiv.org/html/math/0511460v1
- Amanov, van den Berg and Zuiddam, Multiplying matrices via polynomials:
exposition and barriers, Theorem 1.1:
https://arxiv.org/html/2610.30054v1
- Marberg, Generalized Involution Models for Wreath Products, Introduction
and Section 5: https://arxiv.org/html/1007.5078v2
- Klug, Counting homomorphisms from surface groups to finite groups,
Corollary 1 and Section 3: https://arxiv.org/html/2106.11089v2