specialize -- Specialize a stable character to a finite rank

Every ring produced by schurRing comes in two sizes: the stable ring (rank infinity), which is a universal object admitting arbitrarily many row labels, and the finite-rank ring (rank n), on which the relevant representation-theoretic modification rule is enforced (see modificationRule). The function specialize bridges the two: it maps every partition-indexed basis element of the stable ring to its image in the finite-rank ring, collapsing or re-signing partitions that are ``too long'' via the modification rule of Sam-Snowden-Weyman (for Sp/O/RatGL) or simply truncating (for GL/SL/Sn).

GL specialization drops every Schur label with more than n parts:

Sp specialization applies the type-C modification rule: characters $sp_\lambda$ with $\ell(\lambda) > n$ are re-expressed in the finite-rank ring (possibly with a sign, or as zero).

Concrete low-rank $Sp$ specializations show the modification in action. At rank 1 ($Sp(2)$), $sp_{1,1,1}$ is modified via "C" to $-sp_{1}$, and at rank 2 ($Sp(4)$), $sp_{2,1,1}$ has $\ell(\lambda) = 3 > 2$ and is also modified to a signed lower-rank character:

O specialization distinguishes type $B_n$ (O(2n+1)) from type $D_n$ (O(2n)) via the OddOrEven option. If the stable ring has a stored OddOrEven attribute, that value is used; otherwise the option must be supplied at the call site.

The two $O$ flavors give genuinely different images of the same partition on the same target rank. For $\lambda = (2,1)$ at rank 3 we compare $O(7)$ with $O(6)$:

SL specialization drops rows of length equal to the full rank, i.e. "columns" of height $n$, because the determinant representation is trivial in $SL(n)$. Here the stable $s_{3,2}$ collapses to $s_1$ in $SL(2)$, and $s_{3,3,1}$ collapses to $s_{2,2}$ in $SL(3)$:

Tower specialization. For a SchurRing obtained by iterating the schurRing constructor over a coefficient ring that is itself a SchurRing, one can specialize several layers at once. The layers are listed from outermost to innermost, and an entry equal to infinity leaves that layer stable.

In a two-layer tower of GL and Sp flavors, the outer and inner ranks can be adjusted independently; here is the same element specialized to a second choice of ranks: