A non-iterative straightening algorithm and orthogonality for skew Schur modules
Reuven Hodges, Hanzhang Yin
TL;DR
The paper addresses the computational bottleneck of the classical iterative straightening algorithm for skew Schur modules by introducing a non-iterative, determinantal framework that yields a canonical D-basis. It constructs skew Schur modules $E^{\lambda/\mu}$ via a universal property and a determinantal map into a polynomial ring, then defines rearrangement coefficients $\mathcal{R}_{F,S}$ to encode straightening data. The main contributions are a non-iterative straightening formula, an explicit D-basis obtained by Gram–Schmidt, and an inner-product structure that interprets straightening as an orthogonal projection, with practical implications for Young flattenings and Foulkes-type problems. This yields both a theoretical geometry of skew Schur modules and concrete computational tools for representation-theoretic and invariant-theoretic computations, including dimension analyses and efficient coefficient evaluation.
Abstract
We generalize Fulton's determinantal construction of Schur modules to the skew setting, providing an explicit and functorial presentation using only elementary linear algebra and determinantal identities, in parallel with the partition case. Building on the non-iterative straightening formula of the first author for partition shapes, we develop a non-iterative straightening algorithm for skew Schur modules that expresses arbitrary elements in a new D-basis with an explicit closed coefficient formula. We then show that this D-basis is the result of applying Gram-Schmidt orthogonalization to the semistandard tableau basis, which identifies a natural inner product on the skew Schur module and recasts straightening as an orthogonal projection.