# The Smith normal form of a specialized Giambelli-type matrix

## TL;DR

This paper determines the Smith normal form of specialized Giambelli-type matrices, unifying previous results and providing a method to derive forms for various matrices related to Schur functions.

## Contribution

It extends Stanley's work by computing Smith normal forms for a broader class of matrices associated with Schur functions and shows how to derive these forms via stabilization operations.

## Key findings

- Smith normal form of specialized Giambelli matrices is obtained.
- Smith normal form of specialized Lascoux-Pragacz matrices is determined.
- A method to derive forms from classical Giambelli matrices is established.

## Abstract

In the study of determinant formulas for Schur functions, Hamel and Goulden introduced a class of Giambelli-type matrices with respect to outside decompositions of partition diagrams, which unify the Jacobi-Trudi matrices, the Giambelli matrices and the Lascoux-Pragacz matrices. Stanley determined the Smith normal form of a specialized Jacobi-Trudi matrix. Motivated by Stanley's work, we obtain the Smith normal form of a specialized Giambelli matrix and a specialized Lascoux-Pragacz matrix. Furthermore, we show that, for a given partition, the Smith normal form of any specialized Giambelli-type matrix can be obtained from that of the corresponding specialization of the classical Giambelli matrix by a sequence of stabilization operations.