toSymm -- Convert a Schur ring element to an element of the associated symmetric ring

Every SchurRing has an associated symmetricRing with variables $e_i$, $h_i$, $p_i$. toSymm rewrites a Schur-basis element in that symmetric ring via the Jacobi-Trudi determinant $s_\lambda = \det(h_{\lambda_i - i + j})$. Scalars are returned unchanged.

This is the dual of toS, and the two together let you move freely between Schur-basis and $e$/$h$/$p$-basis representations.

A larger partition produces a bigger Jacobi-Trudi expansion in the $e$-basis; here is $s_{3,2,1}$ written in the symmetric ring:

toSymm is additive and is a left-inverse of toS, so toS toSymm is the identity on Schur-basis elements. It distributes over sums and scalars just like any ring map:

In a tensor-product (two-layer) Schur ring, toSymm is applied to the outermost layer only, so a product like $s_{2,1}\otimes t_{1,1}$ returns an element of the symmetric ring of S times the original t factor: