h = branch(f, S1, S2)h = branch(f, m, n)Implements the classical branching rules of R.\ C.\ King, Branching rules for classical Lie groups using tensor and spinor methods, J.\ Phys.\ A 8 (1975), 429–449. For any partition $\lambda$ and a two-factor restriction of the classical group, the character decomposes by a universal formula expressed via the triple Littlewood–Richardson coefficient $c^\lambda_{\alpha,\beta,\gamma}$ (the coefficient of $s_\lambda$ in $s_\alpha s_\beta s_\gamma$).
$\bullet$ GL: $s_\lambda \mid_{GL(m)\times GL(n)} = \sum c^\lambda_{\mu,\nu}\, s_\mu \otimes s_\nu$ (the coproduct of Schur functions).
$\bullet$ Sp: $sp_\lambda \mid_{Sp(2m)\times Sp(2n)} = \sum_{\delta\text{ cols-even}} c^\lambda_{\delta,\mu,\nu}\, sp_\mu \otimes sp_\nu$, the sum running over partitions $\delta$ whose conjugate has all parts even (equivalently, the parts of $\delta$ appear with even multiplicity).
$\bullet$ O: $o_\lambda \mid_{O(a)\times O(b)} = \sum_{\delta\text{ rows-even}} c^\lambda_{\delta,\mu,\nu}\, o_\mu \otimes o_\nu$, where $\delta$ ranges over partitions with all parts even.
For finite-rank factors the output partitions are collapsed via the Sam–Snowden–Weyman modification rules, so any $(\mu,\nu)$ that is killed or re-signed by the rule is handled automatically.
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A slightly larger GL example: the restriction of $s_{3,2}$ along $GL(\cdot) \times GL(\cdot)$ produces the full coproduct of the Schur function, one term per ordered pair $(\mu,\nu)$ with $c^{(3,2)}_{\mu,\nu}$ nonzero:
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The Sp branching picks up an extra (mu, nu) = ({}, {}) term for $sp_{1,1}$ via $\delta = (1,1)$:
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For $sp_{2,1}$ the Sp branching sum runs over the two columns-even deltas $\delta = ()$ and $\delta = (1,1)$; the latter gives the correction terms supported on partitions of total weight one:
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On the orthogonal side, the branching of $o_{2,1}$ runs over rows-even deltas $\delta = ()$ and $\delta = (2)$, contributing correction terms of total weight one:
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The ZZ, ZZ form is a convenience that builds anonymous factor rings of the requested ranks (with the same GroupActing and, for O, the same OddOrEven).
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The (m,n) shortcut agrees with the result of spelling out anonymous factor rings. For the GL branching of $s_{3,2}$ along $GL(2)\times GL(2)$:
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The object branch is a method function.
The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SchurRings.m2:8542:0.