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arXiv · 1703.01572

The Smith normal form of a specialized Giambelli-type matrix

Alice L. L. Gao·Matthew H. Y. Xie·Arthur L. B. Yang
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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.

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Alice L. L. Gao, Matthew H. Y. Xie, Arthur L. B. Yang. 2017-03-05. The Smith normal form of a specialized Giambelli-type matrix. https://arxiv.org/abs/1703.01572

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