Planar decompositions of tableaux and Schur function determinants
The authors give a general expression for the skew Schur functions as determinant of Schur functions of strip shape, obtained through a planar outside decomposition of the skew shape. This determinantal identity contains the classical Jacobi-Trudi determinant, its dual, the Giambelli determinant and more recent Lascoux and Pragacz's rim ribbon determinant as particular cases. A planar outside decomposition of the shape \(\lambda/\mu\) is a totally ordered \(m\)-tuple \((\theta_1,\dots, \theta_m)\) of strips in the diagram of \(\lambda/\mu\) such that their disjoint union is the shape \(\lambda/\mu\), with the condition that each strip starts on the left or bottom perimeter of the diagram and ends on the right or top perimeter of the diagram. Given two strips \(\theta_i\) and \(\theta_j\) of a diagram, the authors define a new strip \(\theta_i\# \theta_j\), obtained in a norm-commutative way from superimposing \(\theta_i\) and \(\theta_j\). The main result states that, for any fixed outside decomposition \((\theta_1,\dots, \theta_m)\) of \(\lambda/\mu\), the following identity holds: \(s_{\lambda/\mu}(X)= \text{det}(s_{\theta_i\# \theta_j}(X))_{m\times m}\), where \(s_{\lambda/\mu}\) denotes the Schur function associated to \(\lambda/\mu\). The proof is combinatorial and establishes a bijection between the set of all tableaux of a given shape with a fixed outside decomposition, and non-intersecting \(m\)-tuples of lattice paths. The paths used here have four types of permissible steps: up vertical, down vertical, right horizontal and down diagonal. An outside decomposition of a shape induces a decomposition of a tableau of the same shape as a union of tableaux of strip shape. Each tableau of strip shape in the original tableau is bijectively mapped to a lattice path, in which every non-vertical step ending at height \(i\) corresponds to an \(i\) in a box of the tableau of strip shape. The authors verify in detail that an \(m\)-tuple of intersecting lattice paths does not correspond to a tableau. The theorem then follows by the well-known Gessel-Viennot lattice path procedure, which defines a sign-reversing, weight-preserving involution on intersecting \(m\)-tuples of lattice paths: a determinant of symmetric functions gives the generating function for all \(m\)-tuples of lattice paths, intersecting and non-intersecting, hence the contribution of the intersecting ones to the determinantal sum is \(0\). In this paper the Gessel-Viennot lattice path argument, used in the proof of the Jacobi- Trudi identity, is adapted to paths having also diagonal steps (Gessel- Viennot paths generate only tableaux in the shape of a row).
scientific article; zbMATH DE number 798912