# Shifted tableaux and products of Schur's symmetric functions

## TL;DR

This paper introduces new combinatorial objects called semistandard increasing decomposition tableaux, explores their relation to existing tableaux, and provides combinatorial formulas for generalized Littlewood--Richardson coefficients and big Schur functions.

## Contribution

It presents new combinatorial models and algorithms for understanding products of Schur's symmetric functions and their coefficients, including shifted Littlewood--Richardson coefficients.

## Key findings

- New combinatorial objects: semistandard increasing decomposition tableaux.
- Expressed big Schur functions as sums of products of Schur P-functions.
- Derived Giambelli formulas for big Schur functions as determinants and Pfaffians.

## Abstract

We introduce a new combinatorial object, semistandard increasing decomposition tableau and study its relation to a semistandard decomposition tableau introduced by Kra\'skiewicz and developed by Lam and Serrano. We also introduce generalized Littlewood--Richardson coefficients for products of Schur's symmetric functions and give combinatorial descriptions in terms of tableau words. The insertion algorithms play central roles for proofs. A new description of shifted Littlewood--Richardson coefficients is given in terms of semistandard increasing decomposition tableaux. We show that a "big" Schur function is expressed as a sum of products of two Schur $P$-functions, and vice versa. As an application, we derive two Giambelli formulae for big Schur functions: one is a determinant and the other is a Pfaffian.