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THE JACOBI-TRUDI
a Schur polynomial as a determinant of complete symmetrics
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Jacobi–Trudi identity writes a Schur polynomial — the fundamental building block of symmetric-function theory — as a determinant. The Schur polynomial sλ is defined combinatorially as a sum over all semistandard Young tableaux of shape λ (fillings that weakly increase along rows and strictly increase down columns). Jacobi and Trudi proved it also equals a clean determinant of complete homogeneous symmetric polynomials: sλ = det(hλi-i+j). A messy sum over combinatorial objects becomes one determinant of simple pieces — the bridge that connects representation theory, symmetric functions, and algebraic combinatorics.
LIT verified live: for several partitions λ and random values of the variables, the Jacobi–Trudi determinant det(hλi-i+j) equals the direct sum over all semistandard Young tableaux of shape λ, to floating precision (window.__jacobitrudi). FIG no framing; the determinant of complete-homogeneous polynomials and the tableau sum are computed by different routes and agree.
LIT verified live: for several partitions λ and random values of the variables, the Jacobi–Trudi determinant det(hλi-i+j) equals the direct sum over all semistandard Young tableaux of shape λ, to floating precision (window.__jacobitrudi). FIG no framing; the determinant of complete-homogeneous polynomials and the tableau sum are computed by different routes and agree.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at split-screen — two definitions of the same Schur polynomial side by side: a sum over tableaux on one panel, a determinant of h’s on the other, landing on the identical polynomial. AVAN (AI) built the instrument: the complete-homogeneous polynomials, the Jacobi–Trudi determinant, and the tableau enumeration.
Credit as content: Carl Gustav Jacob Jacobi and Nicola Trudi (19th c.). The weave: David names the split screen; I confirm the tableau sum equals the determinant.
Credit as content: Carl Gustav Jacob Jacobi and Nicola Trudi (19th c.). The weave: David names the split screen; I confirm the tableau sum equals the determinant.
3 ONE DIMENSION
A Young diagram of shape λ with one semistandard filling; s_λ sums over every such tableau.
4 TWO DIMENSIONS · INTERACTIVE
Cycle shapes λ; the Jacobi–Trudi determinant of h's is compared to the tableau sum at a test point.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the Schur polynomial s_λ evaluated at a point.
AVAN’s addition (the inverse-companion): don’t enumerate the tableaux — take a determinant. The inverse of ‘the sum over semistandard Young tableaux’ is ‘the determinant det(hλi-i+j) of complete-homogeneous polynomials’. Magenta are the tableaux being summed; green is the Schur value the determinant computes. A combinatorial sum folded into a determinant.
LIT Genuine Jacobi–Trudi identity (Carl Gustav Jacob Jacobi & Nicola Trudi, 19th c.). Verified live: for 5 partitions λ and random variable values, the determinant det(h_{λ_i−i+j}) of complete-homogeneous polynomials equals the direct sum over all semistandard Young tableaux of shape λ, to floating precision (window.__jacobitrudi.ok, .rows).
FIG No framing; the determinant of complete-homogeneous polynomials and the tableau sum are computed by different routes and agree. The AVAN inverse is honest — instead of enumerating the tableaux, take a determinant: the inverse of 'the sum over semistandard Young tableaux' is 'the determinant det(h_{λ_i−i+j}) of complete-homogeneous polynomials'. Magenta are the tableaux being summed; green is the Schur value the determinant computes. A combinatorial sum folded into a determinant.
FIG No framing; the determinant of complete-homogeneous polynomials and the tableau sum are computed by different routes and agree. The AVAN inverse is honest — instead of enumerating the tableaux, take a determinant: the inverse of 'the sum over semistandard Young tableaux' is 'the determinant det(h_{λ_i−i+j}) of complete-homogeneous polynomials'. Magenta are the tableaux being summed; green is the Schur value the determinant computes. A combinatorial sum folded into a determinant.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SPLIT SCREEN · David Lee Wise (ROOT0), with AVAN