Ribbon Schur Functions

Abstract We present a new determinantal expression for Schur functions. Previous expressions were due to Jacobi, Trudi, Giambelli and others (see [7]) and involved elementary symmetric functions or hook functions. We give, in Theorem 1.1, a decomposition of a Schur function into ribbon functions (also called skew hook functions, new functions by MacMahon, and MacMahon functions by others). We provide two different proofs of this result in Sections 2 and 3. In Section 2, we use Bazin's formula for the minors of a general matrix, as we already did in [6], to decompose a skew Schur function into hooks. In Section 3, we show how to pass from hooks to ribbons and conversely. In Section 4, we generalize to skew Schur functions. In Section 5, we give some applications, and show how such constructions, in the case of staircase partitions, generalize the classical continued fraction for the tangent function due to Euler.