A curated record of the collaborative design, debugging, and mathematical implementation process behind the Young Diagram² Calculator website.
The project began as a request for a second webpage, parallel to an existing single Young diagram calculator, but with two drawable Young diagrams. The goal grew into a compact research-oriented calculator for operations involving two partitions: tensor products, Littlewood–Richardson style decompositions, finite-rank Lie algebra representations, Kostka tableaux, Kronecker coefficients, plethysm, Schur functors, and Grassmannian cup products.
The development style was iterative. The user repeatedly tested the generated webpage, pointed out mathematical or interface issues, and asked for targeted changes. The assistant generated updated versions of the HTML while preserving the shared stylesheet css/site.css whenever possible.
The user asked for a new page named double_young_diagram with two Young diagram canvases, basic statistics, export, and additional mathematical cards. The first implementation reused css/site.css and created a standalone two-diagram webpage.
The two diagrams were moved into one canvas panel with two grids separated by small padding. Rows and columns were unified, defaulting to a 5×5 grid. The Lie type/rank selector was moved into the canvas heading.
Unfinished cards such as the initial fixed-rank coefficient and weight-space cards were removed. The Schur card was renamed and restyled as a Littlewood–Richardson decomposition card in the compact output style of the existing website.
The user wanted diagram λ and diagram μ to remain horizontally aligned even on small screens. The layout was changed so the two grids shrink rather than stack. The diagram input card also became draggable in the small-screen card stack.
The fixed-rank Lie algebra tensor product went through several revisions. A naive truncation idea was rejected. The user emphasized that SageMath computes representation tensor products using actual Lie algebra representation algorithms, not Young diagram truncation.
A temporary version generated SageMath code, but the user wanted the answer directly in the page. The implementation moved toward browser-side Weyl-character-style computations, then away from overly expensive Kostant partition routines.
The user noticed timeouts even for small examples such as D₆ with λ=μ=(1). The discussion identified the problem: candidate loops and coefficient subroutines were still doing unnecessary weight support or Weyl alternation work.
The fixed-rank tensor product engine was rebuilt around Freudenthal character generation for the smaller factor and Brauer–Klimyk / Racah–Speiser straightening. This removed the old Kostant partition recursion from the tensor product path.
The user proposed enumerating dominant weights γ satisfying ‖γ‖≤‖λ‖+‖μ‖. This became the fixed-rank candidate source, with a dimension check used as a completeness guard.
All cards were made collapsed by default, and calculations were changed so they do not run while the corresponding card is collapsed. Export buttons write results to the single Export card textarea instead of forcing a download.
A Lie algebra invariants card was added, then a separate Kostka tableaux card was created to list all semistandard Young tableaux of shape λ and weight μ.
Cards for Kronecker decomposition and plethysm decomposition were added. A Schur functor card was then refined so that it respects the selected Dynkin type and decomposes the finite-type result as a sum of irreducibles V^γ.
A Grassmannian cup product card was inserted before Export. It computes σλ∪σμ in Gr(r,n), with r equal to the grid rows and n−r equal to the grid columns. It supports Young diagram output and a Chern-class polynomial mode.
The grids were softened: heavy black borders were removed, cells became slightly smaller, padding was reduced for large grids, and the click behavior was restored so clicking a filled square removes the southeast rectangle.
The title became Young Diagram² Calculator, the subtitle became “double diagrams · double interactions,” the computation cap increased to 25 boxes, and the default Export text became a BibTeX template.
Every computational card follows the same interaction pattern: the card is collapsed by default, the user opens it, presses compute, sees a scrollable decomposition-style output, and may press export to place a text version into the Export card.
| Feature | Algorithmic idea | Implementation note |
|---|---|---|
| Stable type A LR | Littlewood–Richardson tableaux / coefficients. | Enumerate target partitions ν and compute cνλμ. |
| Stable B/C/D | Stable Newell–Littlewood-style coefficient. | Uses auxiliary partitions α,β,γ and LR coefficients. |
| Fixed-rank tensor products | Dominant norm-bound candidates plus Freudenthal characters and Brauer–Klimyk straightening. | Dimension check guards completeness. |
| Weyl dimensions | Weyl dimension formula using positive coroots. | Fractions are reduced to integers before display. |
| Kostka tableaux | Backtracking enumeration of semistandard Young tableaux. | Lists the tableaux, not only the count. |
| Kronecker coefficients | Symmetric group characters via Murnaghan–Nakayama. | Uses character inner products over conjugacy classes. |
| Plethysm | Power-sum expansion of Schur plethysm. | Browser cap prevents runaway expansions. |
| Schur functor | Finite-type character of Vλ, Adams operations, then highest-weight peeling. | Output is expressed as irreducibles Vγ for the selected Dynkin type. |
| Grassmannian cup product | Schubert product by LR coefficients inside an r×(n−r) rectangle. | Presentation can be Young diagrams or Chern-class polynomial via Giambelli determinants. |
The most important algorithmic improvement was to avoid repeated Weyl-group alternation and instead follow the character approach used in computer algebra systems:
character = irreducibleCharacterFreudenthal(smallerFactor)
for each weight η with multiplicity m in character:
γ = shiftedWeylStraighten(largerFactor + η)
if γ is dominant and nonsingular:
coefficient[γ] += sign * mThis is paired with a candidate enumeration by invariant norm:
enumerate dominant γ with ||γ|| ≤ ||λ|| + ||μ||
compute coefficients
check: Σ coefficient(γ) dim(V^γ) = dim(V^λ) dim(V^μ)css/site.css to stay visually aligned with the existing site.[3,2,1].<section class="card collapsed lr-card" draggable="true">
<div class="card-head">
<span class="drag-handle">⋮⋮</span>
<span class="card-head-label">new computation</span>
<i class="toggle-icon">▾</i>
</div>
<div class="card-body">
<div class="lr-action-row">
<button id="compute-new" class="btn">compute</button>
<button id="export-new" class="btn btn-ghost">export</button>
</div>
<div id="new-out" class="decomposition-output">
<span class="hint">Open this card and compute.</span>
</div>
</div>
</section>This website was developed through a conversation between the user and ChatGPT. The assistant version recorded for this production document is GPT-5.5 Thinking.
The final site links back to the single Young diagram calculator:
<footer>
One applet built with p5.js, with help from ChatGPT.
The single Young diagram calculator is
<a href="https://ramified.github.io/web/database/young_diagrams.html">here</a>.
</footer>The default export textbox in the final website contains a BibTeX template so users can cite the applet. The author name, canonical URL, release date, and version number can be edited later.