Symmetric functions and standard bases

Define the ring Λ over Z as symmetric formal series in countably many variables with a finite bound on total degree. Each monomial has finite support; coefficients are invariant under every finite permutation of variables. Infinite sums of monomials within one degree are allowed, but an element of this ring has only finitely many homogeneous degrees. Thus the sum of all variables belongs to Λ, whereas the sum of one complete homogeneous function in every degree belongs only to a degree completion.

Setting all variables beyond the first N equal to zero gives a symmetric polynomial in N variables. In degree d, this specialization is injective once N is at least d: every partition of d has at most d nonzero parts and its distinct monomial orbit survives. This is a degreewise statement, not injectivity on the entire ring for one fixed N. For example, the elementary function of degree N plus one vanishes on N variables while remaining a nonzero symmetric function.

A partition λ is a weakly decreasing finite list of positive integers; its size is their sum. The monomial function mλ sums every distinct monomial whose nonzero exponents rearrange to λ, once each. Elementary er sums squarefree degree-r monomials. Complete homogeneous hr sums all degree-r monomials. Their products eλ and hλ, indexed by partitions, are integral bases, as are the monomial functions. We set e0=h0=1 and use zero for negative indices when determinant identities require them.

The power sum pr is the sum of the rth powers of the variables, and pλ is the product of its parts. These products are a basis over Q, but generally not over Z. Already e2=(p1 squared minus p2)/2 requires a denominator. No contradiction arises: the numerator has even monomial coefficients, yet its coordinates in the power-sum basis involve halves. A basis is always a basis over a declared coefficient ring; integrality cannot be inferred merely from a familiar name.

Compare the coefficient of t squared in the product identity. It gives h2=e1 squared minus e2. A repeated variable contributes once to h2 and never to e2, while a product of two distinct variables appears twice in e1 squared and once in each of h2 and e2. This direct coefficient check is useful because it explains the algebraic cancellation rather than treating a basis conversion as an unexplained matrix calculation.

Specializing all N variables to one turns elementary functions into subset counts and complete functions into multiset counts. Their values are binomial coefficients with different upper parameters. Such numerical tests can reject a false identity, but a single all-ones specialization cannot establish equality of symmetric functions: many independent degree-d functions have the same evaluation. Stable symbolic identities retain the individual monomial weights, which later allow tableau contents, representation weights and tensor decompositions to be recovered.