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5144 bytes, as of the version from 2026-10-28 08:03 (current). Recorded here, not run.
# Argument selection for Jack polynomials (T575)
This is a build rationale, not an independent critique or acceptance.
The authoritative instructions are the archived local/live skills listed in
`agents/runs/20261028T074202Z-build-skills/manifest.json`.
Both files have SHA-256
`e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`.
The run read both byte-identical versions before computation.
## Quantity and conventions
One object is stored: the integral Jack polynomial J_lambda^(alpha), expanded
in Z[x1,...,xn,a], where the written letter a denotes alpha. No evaluations at
selected numerical alpha are stored. The x variables and alpha are formal
indeterminates, not free choices of evaluation points or entry parameters.
The P normalization is monic in m_lambda and orthogonal for
<p_mu,p_nu> = delta_mu,nu*z_mu*alpha^length(mu). J equals P times
c_lambda(alpha) = product_s (alpha*arm(s)+leg(s)+1).
The symmetric function is formed before all alphabet variables beyond n are
set to zero. The domain is positive n and partitions with at most n positive
parts. A partition key has comma-separated positive parts, largest first,
without zero padding. Entry parameter order is n, lambda.
Sources checked: Nakajima, *Jack polynomials and Hilbert schemes of points on
surfaces*, section 2(i), Theorem 2.3 and equation (2.4),
https://arxiv.org/html/alg-geom/9610021v1#S2.SS1;
Stanley, *Some combinatorial properties of Jack symmetric functions* (1989),
Theorem 1.1 and Proposition 1.2,
https://math.mit.edu/~rstan/pubs/pubfiles/73.pdf;
and SageMath's Jack symmetric functions documentation (J/P and c1),
https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/sf/jack.html.
## Every independently chosen parameter
- **n**, category `discrete_index`, is the number of alphabet variables, a
structural dimension of the polynomial. Choose n=2,3,4. This includes
binary through four-variable symmetric polynomials and allows every shape
with up to four rows in the chosen degrees. n=1 is an explicit product
times a monomial, given in the table's boundary formula. There is no
arithmetic or continuous input mesh. A fifth alphabet variable would add
another dimension but its first five-row slice requires degree 5, where
expansions already exceed the readability choice at n=4. We therefore
keep breadth through four rows rather than a thin n=5 slice or much deeper
binary-only coverage. The source's finite-alphabet specialization and
length restriction justify this index, not its numerical cutoff.
- **lambda**, category `arithmetic_structure`, is the partition shape that
indexes the Jack basis. For n=2 and n=3 include EVERY shape of weights
2 through 5 with length at most n. For n=4 include EVERY shape of weights
2 through 4. Rows, columns and interior shapes are all retained, including
simple columns. Shape weights 0 and 1 are given by the boundary formulas.
Partition structure, dominance and the Jack basis are established by the
sources cited for the definition. No separate fame or application is
asserted for every shape. These low-degree complete slices are useful
explicit reference expansions of a named symmetric-function basis.
The finite cutoffs are noncanonical readability decisions, independent of
numeral base. The selection is discrete/arithmetic, not `reference_sample`:
that category would apply to numerical specializations of alpha, which this
table does not make. There is no hidden coupling grid beside the shape index.
## Extent and measurement
The queued proposal had n=2,3,4 and degrees 2..5 throughout: 40 entries.
Exact computation measured its maximum as 3341 characters at n=4, lambda=5.
The whole n=4, degree-5 slice is omitted, not just its difficult shapes.
The final extent is 34 entries: 10 for n=2, 14 for n=3, 10 for n=4.
The longest polynomial is J_(4) in four variables, 1412 characters; the
three-variable degree-5 maximum is 1308. Retaining the four-variable degree-4
slice admits the first four-row shape (1,1,1,1) and all shapes of that degree.
This modest length above the usual 1100--1300-character guideline is accepted
for that complete slice, while 3341 is too unwieldy. The final block is around
13 KB, well below 160 KB. Length, not a desired entry count, sets the boundary.
Enumeration is increasing n, increasing weight, then descending lexicographic
partition order. `generate.py`, `checks.py`, and the completeness note agree.
## Proposal adjustments
1. The n=4 upper degree was reduced from 5 to 4 after measuring the expansions.
2. NumberDB's polynomial parser accepts a letter optionally followed by digits
and rejects the name `alpha`. Stored `a` is explicitly defined as alpha;
x1,...,xn, the normalization and entry identities are unchanged.
No source disagreed about the chosen integral normalization. No numerical
precision is chosen: all coefficients are exact integers, so 100-digit
padding would misrepresent the data. A separate actor must assess this
selection, inspect all generator dependencies, recompute values, and produce
the critique sidecar; this build does not supply that verdict.