Generalized Knuth Equivalence for Schur Positivity

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" summary

Generalizes Knuth equivalence inside the plactic algebra to larger quotients and shows that the noncommutative Jacobi–Trudi element J_lambda equals a positive sum of column‑reading words under Fomin–Greene relations, so the generating function D(f) becomes a Schur expansion. Uses RSK, Gessel quasisymmetric functions, descent sets and an inner‑product orthogonal complement; recovers Stanley symmetric functions and gives new Schur‑positivity cases (notably certain three‑column Macdonald/LLT instances).

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" chapters (10)

1. 0:00Positivity in algebraic combinatorics
2. 2:09Approaches to positivity problems
3. 5:21Plactic algebra equivalence
4. 16:12Non-commutative symmetric functions
5. 21:18Descent sets and quasi-symmetric
6. 33:44Non-commutative elementary symmetric
7. 43:04Theorem on Schur functions
8. 46:17Schur positivity of plactic classes
9. 51:03Stanley symmetric functions
10. 52:54Q&A