# Pieri formula

## 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]