Pieri formula

From Groupprops
Jump to: navigation, search

Statement

The Pieri formula is a formula about the Schur elements in the tableau ring. It is a particular case of the Littlewood-Richardson formula, and can also be thought of as an assertion about the values of Littlewood-Richardson numbers.

Statement in the tableau ring

  • S_\lambda.S_{(p)} = \sum_{\mu}S_\mu where \mu ranges over all partitions obtained by adding p boxes to \lambda, no two in the same column
  • S_\lambda.S_{(1^p)} = \sum_{\mu} S_\mu where \mu ranges over all partitions obtained by adding p boxes to \lambda, no two in the same row

Statement in terms of Littlewood-Richardsom numbers

The Littlewood-Richardson formula gives a general way of multiplying any two Schur elements in the tableau ring:

S_\lambda.S_\mu = \sum_\nu c_{\lambda\mu}^\nu S_\nu

where c_{\lambda\mu}^\nu are certain combinatorial quantities associated with the triple of partitions \lambda, \mu, \nu. The Pieri formula gives the special cases of this where \mu = (p) (the partition into one part) and \mu = (1^p) (the partition into p parts). What it tells us is that:

  • c_{\lambda(p)}^{\nu} = 1 if and only if \nu is obtained by adding p boxes to \lambda, no two in the same column. It is zero otherwise.
  • c_{\lambda(1^p)}^{\nu} = 1 if and only if \nu is obtained by adding p boxes to \lambda, no two in the same row. It is zero otherwise.

Statement in terms of Schur polynomials

By the canonical homomorphism from the tableau ring to the polynomial ring, which sends the Schur element S_\lambda to the polynomial s_\lambda, we get:

  • s_\lambda.s_{(p)} = \sum_\mu s_\mu where \mu ranges over all partitions obtained by adding p boxes to \lambda, no two in the same column
  • s_\lambda.s_{(1^p)} = \sum_{\mu} s_\mu where \mu ranges over all partitions obtained by adding p boxes to \lambda, no two in the same row

Notice that s_{(p)} is the same as h_p (the complete symmetric polynomial of degree p). Similarly, s_{(1^p)} is the same as e_p (the elementary symmetric polynomial of degree p). This gives us:

  • s_\lambda.h_p = \sum_\mu s_\mu where \mu ranges over all partitions obtained by adding p boxes to \lambda, no two in the same column
  • s_\lambda.e_p = \sum_{\mu} s_\mu where \mu ranges over all partitions obtained by adding p boxes to \lambda, no two in the same row

Corresponding statement for cohomology classes

The Pieri formula also holds when we interpret S_\lambda, not as the Schur element for the partition \lambda, but rather, as the cohomology class of the Schubert variety corresponding to the partition \lambda. PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]