Littlewood–Richardson coefficients and products

If the diagram of λ is contained in ν, the skew shape ν/λ consists of boxes in ν that are outside λ. Fill only those boxes, comparing entries in the usual skew tableau row and column positions; deleted boxes receive no entries and impose no artificial inequalities. Content μ means that exactly μi entries equal i. Necessarily the number of skew boxes equals the size of μ. These elementary support conditions eliminate many candidate coefficients before any word is tested.

Read the filled skew diagram in the declared order. The word is lattice, or ballot, if every initial segment contains at least as many 1s as 2s, at least as many 2s as 3s, and so on. Checking only the whole word is insufficient: its total content is already a partition. The first symbol must therefore be 1. The LR theorem states that the coefficient cλμν counts exactly the semistandard skew fillings satisfying this prefix condition. We state the theorem and verify small cases rather than derive its general proof.

When μ is a single row of length r, every skew entry is 1. Column strictness then allows at most one added box in each column, while the word is automatically ballot. Thus multiplication by hr adds a horizontal r-strip, each allowable shape with coefficient one. Horizontal refers to the no-two-in-one-column condition; the added boxes need not lie in a single row or form a connected region. For er, the corresponding theorem adds a vertical strip, with at most one new box in each row.

General LR products are not multiplicity-free. Distinct ballot fillings of the same skew shape and content can yield coefficients two or larger. Commutativity of symmetric functions gives cλμν=cμλν even though the chosen tableau description treats λ as removed boxes and μ as content. This equality is not visually automatic from the definition. A bijection or algebraic argument must explain it; exchanging shape and content in a single filling without changing anything else is not a valid proof.

The product identity lives in the stable ring over Z. After specialization to N variables, terms whose shapes have more than N rows vanish. Therefore one cannot infer that their stable coefficient was zero by observing a small-variable polynomial product. In complex representation theory the surviving coefficients describe polynomial GLN tensor products when both input shapes fit in N rows; the stable rule and the finite-dimensional interpretation remain consistent through this vanishing mechanism.

A useful verification workflow combines three independent constraints. First check sizes and containment. Then enumerate row- and column-admissible fillings for the required content. Finally test every word prefix, not just final totals. All-ones specialization provides a separate dimension check on a full product, but it does not isolate individual coefficients. This distinction becomes important for positivity conjectures: an integer can be nonnegative for representation-theoretic reasons even when no efficient positive counting formula has yet been found.