# Pfaffian Formulas and Schur Q-Function Identities

## TL;DR

This paper develops Pfaffian analogues of classical matrix formulas and uses them to provide new, clearer proofs of identities related to Schur Q-functions, advancing algebraic combinatorics.

## Contribution

It introduces Pfaffian versions of the Cauchy--Binet and minor-summation formulas, offering novel tools for studying Schur Q-functions.

## Key findings

- Pfaffian analogues of classical formulas established
- New proofs for identities of Schur Q-functions provided
- Enhanced understanding of Pfaffian structures in algebraic combinatorics

## Abstract

We establish Pfaffian analogues of the Cauchy--Binet formula and the Ishikawa--Wakayama minor-summation formula. Each of these Pfaffian analogues expresses a sum of products of subpfaffians of two skew-symmetric matrices in terms of a single Pfaffian. By using these Pfaffian formulas we give new transparent proofs to several identities for Schur Q23 pa-functions.