A basis of bideterminants for the coordinate ring of the orthogonal group

TL;DR

This paper constructs a basis of bideterminants for the coordinate ring of the orthogonal group over an infinite field, using Young tableaux, and describes its module structure explicitly.

Contribution

It introduces a new basis for the coordinate ring of O(n) using O(n)-standard Young tableaux and provides an explicit bimodule filtration.

Findings

Basis of bideterminants indexed by O(n)-standard Young tableaux

Explicit bimodule filtration of the coordinate ring

Isomorphism of factors to orthogonal Schur modules

Abstract

We give a basis of bideterminants for the coordinate ring K[O(n)] of the orthogonal group O(n,K), where K is an infinite field of characteristic not 2. The bideterminants are indexed by pairs of Young tableaux which are O(n)-standard in the sense of King-Welsh. We also give an explicit filtration of K[O(n)] as an O(n,K)-bimodule, whose factors are isormorphic to the tensor product of orthogonal analogues of left and right Schur modules.