Mathematics / research
Algebraic combinatorics
Reveal algebraic structure in finite objects through symmetric functions, tableaux, posets, matroids, and representation theory.
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Module 01 research
Algebraic combinatorics
Reveal algebraic structure in finite objects through symmetric functions, tableaux, posets, matroids, and representation theory.
- 01
Symmetric functions and standard bases
Work in the graded ring of stable symmetric functions and compare monomial, elementary, complete and power-sum coordinates.
33 min - 02
Schur functions and semistandard tableaux
Connect tableau weights, Jacobi–Trudi determinants and stable Schur bases with explicit row and column conventions.
33 min - 03
The Robinson–Schensted–Knuth correspondence
Track row insertion and recording tableaux, preserving multiplicities and recovering subsequence information.
33 min - 04
Littlewood–Richardson coefficients and products
Count Schur-product multiplicities using skew tableaux with a fully specified ballot reading convention.
33 min - 05
Symmetric-group characters and the Frobenius map
Normalize power-sum coordinates by conjugacy-class centralizers and connect induction to Schur multiplication.
33 min - 06
Incidence algebras and Möbius functions
Invert interval convolution on locally finite posets and separate local finiteness from global summability.
33 min - 07
Algebraic matroids and representability
Compare polynomial independence with vector independence, tracking field extensions and characteristic in differential linearization.
33 min - 08
Stanley–Reisner rings and simplicial complexes
Encode nonfaces by squarefree monomials and recover face counts through a graded Hilbert series.
33 min - 09
Hopf algebras in combinatorics
Orient multiplication and comultiplication correctly and calculate antipodes through symmetric-function alphabets.
33 min - 10
Positivity, real-rootedness and log-concavity
Separate basis-dependent positivity from root geometry and derive coefficient-shape consequences under sufficient hypotheses.
33 min