Transformations of Border Strips and Schur Function Determinants

William Y. C. Chen, Guo-Guang Yan and Arthur L. B. Yang

  Abstract:  We introduce the notion of the cutting strip of an outside decomposition of a skew shape, and show that cutting strips are in one-to-one correspondence with outside decompositions for a given skew shape. Out-side decompositions are introduced by Hamel and Goulden and are used to give an identity for the skew Schur function that unifies the determinantal expressions for the skew Schur functions that unifies the determinantal expressions for the skew Schur functions and the rim ribbon determinant due to Lascoux and Pragacz. Using cutting strips, one obtains a formula for the number of outside decompositions of a given skew shape. Moreover, one can define the basic transformations which we call the twist transformation among cutting strips,and dervive a transformation theorem for the determinantal formula of Hamel and Goulden. The special case of the transformation theorem for the Giambelli identity and the rim ribbon identity was obtained by Lascoux and Pragacz. Our transformation theorem also applies to the supersymmetric skew Schur function.

  AMS Classification:  05E05, 05A15.

  Keywords:  Young diagram, border strip, outside decomposition, Schur function determinants, Jacobi-Trudi identity, Giambelli identity, Lascoux-Pragacz identity.

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