OddOrEven -- Select type $B_n$ or type $D_n$ for an orthogonal Schur ring

This is an optional argument for the schurRing constructor (and, by extension, the symmetricRing and specialize methods), relevant only when GroupActing => "O". It distinguishes the two flavors of orthogonal group: odd (O(2n+1), type $B_n$) and even (O(2n), type $D_n$). Its possible values are "Odd" (the default) and "Even"; any other value raises an error.

For stable orthogonal rings (rank infinity) the distinction is invisible at the level of multiplication, but it affects the dimension formula and the modification rule applied when specialize is called.

At rank 3 and $\lambda = (2,1)$, the difference is much more pronounced: O(7) (type $B_3$) gives a 105-dimensional irreducible, whereas O(6) (type $D_3$) gives a 64-dimensional one:

The same partition of larger weight separates the two flavors by an even bigger factor. For $\lambda = (3,2)$ the $O(7)$ Weyl dimension is 693 while the $O(6)$ Weyl dimension is 300:

Supplying OddOrEven for a non-orthogonal ring (for example GroupActing => "Sp" or "GL") is an error.

Functions with optional argument named OddOrEven:

• schurRing(...,OddOrEven=>...)
• specialize(...,OddOrEven=>...)
• symmetricRing(...,OddOrEven=>...)