toO -- Expansion in the basis of orthogonal characters

Given a symmetric function f, the function toO returns the expression of f in the basis of irreducible orthogonal characters. The output is a RingElement in a SchurRing with GroupActing => "O", so that the basis element o_\lambda stands for the irreducible O-representation associated to the partition \lambda.

The conversion implements the inverse Koike branching formula s_\lambda = \sum_\delta o_{\lambda/\delta}, where \delta ranges over partitions \delta \subseteq \lambda with all parts even, and skew characters are expanded using Littlewood-Richardson coefficients.

If the target orthogonal ring is not specified, the first call to toO on an element of S creates and caches an associated stable orthogonal ring; alternatively one can supply the target ring explicitly.

The conversion is inverse to toS on the orthogonal ring. At stable rank, toS(o_{(2)}) = s_{(2)} - 1 and toS(o_{(1,1)}) = s_{(1,1)}; the product o_{(1)}^2 equals the Newell-Littlewood product o_{(2)} + o_{(1,1)} + 1.

At finite rank, the modification rule for types B_n / D_n is applied automatically. The tag OddOrEven (default "Odd") distinguishes O(2n+1) (type B_n) from O(2n) (type D_n) and is used by dim to pick the right Weyl dimension formula. Plethysm routes through the GL Schur ring.

The simplest nontrivial example of the inverse Koike branching is the second symmetric power: s_{(2)} decomposes as o_{(2)} + 1 (the traceless part plus the invariant form):

The conversion composes with the other symmetric-function transitions. One can start from an element of a Symmetric ring expressed in e- or h-variables, and let toO route it through the Schur basis:

A rank comparison between odd and even orthogonal groups shows that the Weyl dimension truly depends on the value of OddOrEven: the partition (2,1) indexes a 105-dimensional irreducible of O(2\cdot 3+1) = O(7) and a 64-dimensional irreducible of O(2\cdot 3) = O(6):