Schur functions and semistandard tableaux

For shape λ, assign a variable xi to every entry i and multiply over all boxes. The Schur function sλ is the sum of these weights over all semistandard tableaux of shape λ. In N variables, entries are restricted to 1 through N. A column longer than N cannot be strictly increasing, so the specialization vanishes when λ has more than N rows. By contrast, a single row can have arbitrarily many boxes because repeated entries along a row are allowed.

Collecting equal monomials yields nonnegative integers called Kostka numbers. A coefficient counts tableaux with one specified content vector, not every rearrangement of that vector at once. Symmetry ensures that permuted contents have equal counts; proving this requires a weight-switching argument or an equivalent algebraic construction. It does not follow merely from the tableau definition looking symmetric, because the order on the entry alphabet enters the row and column conditions asymmetrically.

The Jacobi–Trudi theorem expresses sλ as the determinant with entry h at index λi−i+j, using at least as many rows as the length of λ and padding λ with zeros if extra rows are used. Negative-index complete functions are zero and h0 is one. This is an equality in the stable ring over Z. Its determinant proof can be organized by intersecting lattice paths: a sign-reversing involution cancels intersecting path families, leaving the nonintersecting ones that encode tableaux. The theorem is used here with that interpretation rather than proved in full.

The one-row determinant gives sr=hr, while the one-column version gives s at shape (1 repeated r times) equal to er. These boundary cases are valuable convention checks. More generally, Schur functions form an integral basis in every degree. Therefore a symmetric function can be Schur-positive even if a determinant used to calculate it has minus signs. Positivity concerns its final basis coordinates and the underlying combinatorial objects, not the syntax of one formula.

Setting every variable equal to one counts all admissible tableaux on that finite alphabet. In representation theory over C, this number is also the dimension of the corresponding polynomial irreducible general linear representation when the shape has at most N rows. That interpretation is an additional theorem; the tableau count is already defined without representation theory. Setting selected variables to zero instead restricts allowed entries and can reveal that a nonzero stable function has disappeared under a small-alphabet specialization.

For shape (2,1) on a two-letter alphabet, strictness forces the left column to be 1 above 2. Only the top-right box remains free, and it may be either 1 or 2. This makes the stable distinctions tangible: the two monomials have different contents even though their all-ones values coincide. Standard tableaux are a separate specialization of the combinatorics, using each label exactly once; they are not obtained simply by setting every xi to one.