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Math Notes

University Mathematics

ConceptComplete

Branching Rules and Restrictions

The branching rules describe how representations of SnS_n decompose when restricted to Sn−1S_{n-1}, revealing beautiful recursive structure.

TheoremBranching Rule for $S_n \downarrow S_{n-1}$

Let VλV^\lambda be the irreducible representation of SnS_n corresponding to partition λ⊢n\lambda \vdash n. Then: ResSn−1Sn(Vλ)≅⨁μVμ\text{Res}_{S_{n-1}}^{S_n}(V^\lambda) \cong \bigoplus_{\mu} V^\mu where the sum is over all partitions μ⊢(n−1)\mu \vdash (n-1) obtained by removing one box from the Young diagram of λ\lambda.

Conversely (by Frobenius reciprocity): IndSn−1Sn(Vμ)≅⨁λVλ\text{Ind}_{S_{n-1}}^{S_n}(V^\mu) \cong \bigoplus_{\lambda} V^\lambda where the sum is over all partitions λ⊢n\lambda \vdash n obtained by adding one box to μ\mu.

ExampleBranching from $S_4$ to $S_3$

Consider λ=(2,2)⊢4\lambda = (2,2) \vdash 4. Removing one box gives partitions:

  • Remove from first row: (1,2)=(2,1)⊢3(1,2) = (2,1) \vdash 3
  • Remove from second row: (2,1)⊢3(2,1) \vdash 3

Thus V^{(2,2)}|_{S_3} \cong V^{(2,1)} \oplus V^{(2,1)} \cong V^{(2,1)}^{\oplus 2}.

The dimension check: dim⁡V(2,2)=2\dim V^{(2,2)} = 2 (from hook formula), and dim⁡V(2,1)=2\dim V^{(2,1)} = 2, so 2=2×12 = 2 \times 1 suggests multiplicity 1, but actually both removal positions give the same μ=(2,1)\mu = (2,1), so multiplicity is 2. Wait, that doesn't match dimensions...

Actually, removing boxes from (2,2)(2,2): either (2,1)(2,1) or (1,2)=(2,1)(1,2) = (2,1) (same partition). So there are two ways, giving V^{(2,1)}^{\oplus 2}. But dim⁡V(2,2)=2\dim V^{(2,2)} = 2 and dim⁡V(2,1)=2\dim V^{(2,1)} = 2, so we need 2=m×22 = m \times 2, thus m=1m=1. Let me reconsider.

The removable boxes from (2,2)(2,2) are: one from (2,0)(2,0) position giving (2,1)(2,1), or one from (1,1)(1,1) position giving (2,1)(2,1). Both give the same partition, so it appears once. Thus V(2,2)∣S3≅V(2,1)V^{(2,2)}|_{S_3} \cong V^{(2,1)}.

DefinitionYoung's Lattice

The branching rules define a partial order on partitions: μ≺λ\mu \prec \lambda if μ\mu is obtained from λ\lambda by removing one box. This structure is called Young's lattice, and irreducible representations form a graded poset under restriction and induction.

TheoremPieri Rule

For the hook (n)(n) and vertical partition (1n)(1^n) (trivial and sign representations), the tensor products with irreducibles satisfy: Vλ⊗V(n)≅⨁VμV^\lambda \otimes V^{(n)} \cong \bigoplus V^\mu where μ\mu ranges over partitions obtained from λ\lambda by adding nn boxes, no two in the same column.

Similarly for the sign representation (1n)(1^n).

Remark

Branching rules are fundamental for:

  • Recursive construction: Building SnS_n representations from Sn−1S_{n-1}
  • Representation stability: Understanding asymptotic behavior as n→∞n \to \infty
  • Quantum groups: q-deformed versions yield deep connections to knot theory
  • Symmetric functions: The branching corresponds to multiplication by elementary symmetric functions

The branching structure connects representation theory to algebraic combinatorics, probability (random matrices), and mathematical physics (Yang-Baxter equations).