For any commutative ring $R$, let $\mathbf{Symm}_R$ be the ring of symmetric functions in countably many variables $x_1$, $x_2$, $x_3$, ... over $R$. ("Symmetric functions" really means symmetric power series of bounded degree.) It is known that $\mathbf{Symm}_R$ is generated by the elementary symmetric polynomials $e_1$, $e_2$, $e_3$, ... as an $R$-algebra. If $R$ is a $\mathbb Q$-algebra, then $\mathbf{Symm}_R$ is also generated by the power sum polynomials $p_1$, $p_2$, $p_3$, ... as an $R$-algebra. Note that $\mathbf{Symm}_{\mathbb Z} \subseteq \mathbf{Symm}_{\mathbb Q}$.

There are two ways to define a comultiplication (in the sense of coalgebras) on $R$:

The first comultiplication, called $\Delta_1$, is defined by $\Delta_1\left(e_n\right) = \sum\limits_{i+j=n} e_i\otimes e_j$ for all $n\in\mathbb N$, where the sum allows $i$ and $j$ to be zero (and $e_0$ has to be understood as $1$). This comultiplication satisfies $\Delta_1\left(f\left(x_1,x_2,x_3,...\right)\right) = f\left(x_1,x_2,x_3,...,y_1,y_2,y_3,...\right)$, where we identify the tensor product $\mathbf{Symm}_R\otimes_R\mathbf{Symm}_R$ as a ring of certain power series in "$2$ times countably many" indeterminates $x_1$, $x_2$, $x_3$, ..., $y_1$, $y_2$, $y_3$, .... In order to make sense of the term $f\left(x_1,x_2,x_3,...,y_1,y_2,y_3,...\right)$, one has to recall that $f$ is a symmetric polynomial, so that one can reorder its arguments in any way, for example as $x_1,y_1,x_2,y_2,x_3,y_3,...$.

The second comultiplication, denoted by $\Delta_2$, satisfies $\Delta_2\left(p_n\right) = p_n\otimes p_n$ for all positive integers $n$. This is not enough to define it because $p_1$, $p_2$, $p_3$, ... don't always generate the $R$-algebra $\mathbf{Symm}_R$, but at least they generate it when $R$ is a $\mathbb Q$-algebra, so one can use this definition for $R=\mathbb Q$, then show that $\Delta_2\left(\mathbf{Symm}_{\mathbb Z}\right) \subseteq \mathbf{Symm}_{\mathbb Z}\otimes_{\mathbb Z}\mathbf{Symm}_{\mathbb Z}$, so that $\Delta_2$ is also defined for $R=\mathbb Z$, and consequently (since $\mathbb Z$ is the initial object in the category of rings) also defined for any $R$. Of course, one could just as well give a more direct definition of $\Delta_2$, by setting

$\Delta_2\left(f\left(x_1,x_2,x_3,...\right)\right) = f\left(x_1y_1,x_1y_2,x_1y_3,...,x_2y_1,x_2y_2,x_2y_3,...,x_3y_1,x_3y_2,x_3y_3,...\right)$.

Again, one has to invoke symmetry of $f$ for the right hand side to make sense here. This time one also needs to check that the right hand side is well-defined at all, what with the infinitely many terms involving the same $x_i$. Yet another way to define $\Delta_2$ is by the identity $\Delta_2\left(e_n\right) = \sum\limits_{\lambda\text{ is a partition of }n} s_{\lambda}\otimes s_{\lambda^t}$, where $s_{\mu}$ are the Schur polynomials, and $\lambda^t$ is the conjugate partition of $\lambda$. (A categorification of this identity is the isomorphism from MathOverflow question #120873 posted earlier today.)

This suggests defining a third comultiplication $\Delta_3$ by

$\Delta_3\left(f\left(x_1,x_2,x_3,...\right)\right) = f\left(x_1+y_1,x_1+y_2,x_1+y_3,...,x_2+y_1,x_2+y_2,x_2+y_3,...,x_3+y_1,x_3+y_2,x_3+y_3,...\right)$.

This, however, doesn't go well: Setting $f=p_n=x_1^n+x_2^n+x_3^n+...$, the right hand side becomes $\sum\limits_{i\geq 1,\ j\geq 1} \left(x_i+y_j\right)^n$, which involves summing infinitely many $x_i^n$'s for every $i\geq 1$, and summing infinitely many $y_i^n$'s for every $i\geq 1$.

Fortunately, these are the only undefined terms on the right hand side; all the other infinite sums do make sense. If we replace every undefined sum of infinitely many $x_i^n$'s by $rx_i^n$ for some fixed integer $r$, then we obtain

$\Delta_3\left(p_n\right) = \sum\limits_{i=1}^{n-1} \dbinom{n}{i} p_i \otimes p_{n-i} + r \otimes p_n + p_n \otimes r$.


So fix an integer $r$, and let us define a map $\Delta_3 : \mathbf{Symm}_{\mathbb Q} \to \mathbf{Symm}_{\mathbb Q} \otimes_{\mathbb Q} \mathbf{Symm}_{\mathbb Q}$ by

$\Delta_3\left(p_n\right) = \sum\limits_{i=1}^{n-1} \dbinom{n}{i} p_i \otimes p_{n-i} + r \otimes p_n + p_n \otimes r$ for every positive integer $n$.

This $\Delta_3$ is easily seen to be coassociative, and for $r=1$ even counital (with respect to the standard counity of $\mathbf{Symm}_{\mathbb Q}$).

Do we have $\Delta_3\left(\mathbf{Symm}_{\mathbb Z}\right) \subseteq \mathbf{Symm}_{\mathbb Z}\otimes_{\mathbb Z}\mathbf{Symm}_{\mathbb Z}$ for every $r$ ? In other words, can this $\Delta_3$ be defined over any commutative ring $R$ ? What is the combinatorial meaning of this $\Delta_3$ ?


We need to prove that $\Delta_3\left(e_n\right)$ has integer coefficients for all $n$ and $r$. Sage code and some output (please don't imitate my code) verifies my suspicion for $r=0,1$ and $n=1,2,3,...,9$. I also have some not very tangible semi-proof arguments.


1 Answer 1


I have a proof of the integrality of $\Delta_3$ using Dwork's lemma (which tells when a given vector $\left(v_1,v_2,v_3,...\right)\in A^{\left\lbrace 1,2,3,...\right\rbrace}$ over some commutative ring $A$ is the vector of ghost components of a big Witt vector -- note that this is equivalent to the existence of a ring homomorphism $f:\mathbf{Symm}_{\mathbb Z} \to A$ which sends each power sum $p_n$ to $v_n$) and some binomial coefficient congruences (I am eventually going to write this up, though I cannot give a good upper bound on the "eventually").

Meanwhile this stuff doesn't seem so new. On page 14 of Andrew Baker, Birgit Richter, Quasisymmetric functions from a topological point of view, arXiv:math/0605743v4, I see a coproduct $\psi_{\otimes}$ which is exactly mine if I get it correctly that $q_0$ is an arbitrary integer. They seem to have a proof based on topology. They don't seem to notice the need for renormalization, though.

On the other hand, on the level of "algebraic rings" (commutative algebraic groups with an additional algebraic monoid structure that distributes over the algebraic group structure), the Hopf algebra $\mathbf{Symm}_{\mathbb Z}$ equipped with the comultiplication $\Delta_3$ seems to more-or-less represent the ring $\widetilde{A}$ defined in §3 of Berthelot, Grothendieck, Illusie, SGA 6, exposé 1. I am sorry for the weasel words, but I don't have the time to make more concrete assertions these days.

EDIT: Two (or three, depending on how you count) proofs of the integrality of $\Delta_3$ are now in Vic Reiner's and my notes on Hopf algebras in Combinatorics (the version with solutions). See Exercise 2.75(f) and Exercise 2.80(e). (The numbering is volatile, but the first exercise has the words "Define a $\mathbb Q$-algebra homomorphism", and the second exercise talks about "new solutions to parts (b), (c), (d), (e) and (f)".)


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