Enumerative Geometry Littlewood Richardson

Review 01

  • Structure of cohomology ring.
    • Multiplying, configuration by adding horizontal strip of length \(b\).
  • Given partitions, how to determine the structure coefficitents?
    • Language of Young Tableaux for Littlewood-Richardson rule.

Young Tableaux

  • Semi-Standard Young Tableaux for given a partition is
  • Entries in rows are non-decreasing,
  • Entries in column are strictly increasing.
  • Z-module Multiplication
  • Row insertion
  • Reading word: read rows from bottom to top.
  • Jeu-de-taquin (Jdt)
    • For partition inclusion: Skew shape, inner corner, outer corner
    • Skew Semi-Standard Young Tableaux (SSYT)
    • To make a semi-standard T from a skew T, slide the entries into the empty space.
    • Jdt slide for inner corner makes it an outer corner. Rectification: when Jdt done for all inner corners.
    • Define Jdt product and show it is equivalent to row insertion.

Symmetric Polynomials

  • Elements of invariant ring \(\mathbb{Z}[x_1,\ldots,x_n]^{S_n}= \wedge_n\)where \(S_n\) acts by permuting elements.
  • Monomial. Elementary (biject column). power-sum. complete homogeneous (biject row).
  • Schur Polynomial interpolate \(h_k\) to \(e_k\)
    • \(\mathbb{Z}\) basis
    • Expand positively in monomial basis
    • Symmetric polynomial
  • Define similar family in Tableaux ring giving a ring homomorphism (no.of celss stays the same).
    • Pieri’s Rule for Tableaux homomorphs to Schur polynomial.
    • Product Commutative specifically for the \(S_\lambda\).
  • Connection to Schubert Classes
    • Both satisfying Pieri’s Rule in respective rings gives a homomorphism to cohomology ring.

Littlewood Richardson Rule

  • Enumerate by matching cardinality of two sets to write a given partition as a product of the two given partitions.