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.