divMult -- multiplication and comultiplication of divided powers

The divided-power algebra $D^*(E) = \bigoplus_{d \ge 0} D^d(E)$ on a free module $E$ of rank $n$ is a graded bi-algebra. Its basis elements are indexed by weakly-increasing tuples of integers, or equivalently by their exponent vectors. This package provides both the multiplication $D^p \otimes D^q \to D^{p+q}$ and the comultiplication $D^{p+q} \to D^p \otimes D^q$ on those bases.

Multiplication. divMult(L1, L2) takes two weakly-increasing lists L1, L2 (basis elements of $D^{|L_1|}$ and $D^{|L_2|}$) and returns the pair (c, L) where L is the sorted concatenation and c is the binomial coefficient encoding the divided-power product rule $x^{(p)} \cdot x^{(q)} = \binom{p+q}{p}\, x^{(p+q)}$.

Comultiplication. divComult(L, p) takes a basis element L of $D^{|L|}$ and returns the set of ways to split it as a pair (A, B) with |A| = p, |B| = |L| - p, and $A + B = L$ as multisets. This is the comultiplication $D^{|L|}(E) \to D^{p}(E) \otimes D^{|L| - p}(E)$.

Internally, divComult drives the Weyl-side straightening: a single Garnir shuffle across adjacent rows proceeds by comultiplying the concatenated row tails and remultiplying into the partner row; see towardWeylStandard. A basis element can be stored either as a weakly-increasing tuple or as an exponent vector recording the multiplicity of each letter; for efficiency, much of the straightening uses the exponent-vector form, and both forms are accepted as input below.

Variants. Both operations come in three forms:

• divMult(L1, L2) and divComult(L, p): tuple (weakly-increasing list) input; the alphabet size is inferred from the maximum entry appearing in the input.
• divMult(L1, L2, n) and divComult(L, p, n): tuple input with an explicit alphabet size n, useful when the tuples are padded or when the caller needs all output tuples to have a uniform length.
• divMult(S1, S2) and divComult(S, p): sequence input, interpreted as an exponent vector (that is, S_i is the multiplicity of the letter i). Working directly with exponent vectors avoids the conversion to and from tuples, and is the form used inside the straightening loop.