Pascal's triangle, the Catalan triangle, and the Motzkin triangle are classical objects in combinatorics — each row generated by a simple recursion, each entry counting something concrete (binomial coefficients, lattice paths, non-crossing partitions). They appear to be purely combinatorial, governed by counting rules with no algebraic content.
This paper reveals the algebraic content. The numbers in all three triangles — and their generalizations — are multiplicities in tensor products of sl₂ representations. The entry in row n, column k counts how many times a particular irreducible representation appears when you tensor together specific sl₂ modules. The multiplicity is expressible as a difference of generalized binomial coefficients.
The derivation is elementary: the generalized triangles satisfy the Pascal rule with appropriate initial conditions, and the representation-theoretic interpretation explains why. The “sum of squares” phenomenon observed in these triangles — where certain row sums equal sums of squares of entries from smaller rows — gets a clean explanation as a decomposition of tensor squares.
Three combinatorial objects, independently discovered and studied for their counting properties, turn out to be shadows of the same algebraic structure. The numbers were always multiplicities. The triangle was always a representation.