The Pascal Multiplicity

Pascal’s rule — each entry is the sum of the two above it — generates the binomial coefficients. Change the initial condition (the top of the triangle) and the same rule generates different sequences: Catalan numbers, Motzkin numbers, and their generalizations. The rule is identical; only the boundary changes.

These numbers are tensor product multiplicities for sl₂ representations.

When you take two representations of sl₂ (the simplest non-abelian Lie algebra) and form their tensor product, the result decomposes into irreducible representations. The multiplicity of each irreducible summand — how many times it appears — is a number sitting in one of these triangles. The Catalan triangle gives multiplicities for one class of tensor products; the Motzkin triangle gives them for another.

The proof is elementary: no heavy machinery, just direct computation showing that the Pascal recurrence applied to the right initial conditions reproduces the tensor product decomposition formulas. Each multiplicity equals a difference of generalized binomial coefficients — a closed form that makes computation trivial.

The “sum of squares” phenomenon follows from the same structure. Certain row sums in these triangles are perfect squares, and representation theory explains why: they count dimensions of invariant subspaces, which factor as products of dimensions of the constituent representations. The algebraic structure guarantees the combinatorial identity.

The representation-theoretic perspective doesn’t make the triangles deeper — Pascal’s rule is already completely understood. But it does make the triangles inevitable. The multiplicities have to satisfy Pascal’s rule because the tensor product operation has to satisfy certain associativity and branching properties. The simplest recursive rule in mathematics is the simplest because the simplest Lie algebra forces it to be.