Calculations for the Appendix: A combinatorial formula for some \(\mathrm{SL}_n\)-invariants

One step in constructing the polynomial \(F_{8,2}\) is to find the invariant in \( V \otimes V^{*} \), where \(V\) is an irreducible representation of \(\mathrm{SL}_n\). We give a combinatorial formula for this invariant in terms of the Gelfand-Tsetlin basis of \(V\).

Remark.The LieAlgebraRepresentations package was first released in Macaulay2 version 1.25.11 in November 2025, and the documentation cited the following file: A conjectural combinatorial formula for some \(\mathrm{SL}_n\)-invariant polynomials . At that time, the main formula in these notes was only a conjecture. However, in March 2026, we proved that this formula is correct, and these old notes are now superseded by the appendix.

• Verifying Lemma A.3
• Verifying the statement of Proposition A.4
• Verifying the statement of Proposition A.6
• Verifying claims appearing in the proof of Proposition A.6
• Verifying Proposition A.7
• Verifying formulas used in the proof of Proposition A.4
• Verifying claims about \(\mathscr{B}^{*}_{+}\) in Proposition A.4
• Verifying claims appearing in the proof of Proposition A.4
• Verifying the statement of Proposition A.8
• Verifying claims appearing in the proof of Proposition A.8