HW 4, Problem 2 – Details from lecture 15

I’ve been playing with Hopf algebras of symmetric functions all day, but I’m unclear on the details of the algebraic structure. To jump-start the discussion, I’ll begin with my own questions:

The notes from lectures 14-15 say that the product in this algebra is free commutative, what does that mean exactly? How do we define the unit when its input isn’t 1?

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2 comments

  1. An algebra A is said to be “free commutative” on a subset X if and only if

    1) the algebra A is generated by X (as a k-algebra);

    2) the elements of X are algebraically independent (i. e., if some polynomial (with coefficients in k) applied to the elements of X evaluates to 0, then it must have been the zero polynomial).

    In the case of the Remark on page 54 of the notes, the claim is that $ latex H\left(\mathcal P^{\ast}\right) $ is free commutative over the subset $ latex \mathcal P $. This is NOT a trivial claim; it is an algebraic rewriting of the fact that a finitely poset with 0 and 1 (or, more generally, a poset which is not a disjoint union of more than 1 nontrivial poset) can be UNIQUELY factored as a direct product of posets. I think this used to be a conjecture of Birkhoff and was subsequently proved by Hashimoto (J. Hashimoto, “On the product decomposition of partially ordered sets”, Mathematica Japonicae v.1 (1948), pp. 120-123). If you (like me) are uncomfortable with using difficult facts like this, I assume that there are variations of the definition of an incidence Hopf algebra which don’t involve factoring out isomorphisms (i. e., isomorphic posets are not being set equal per decree, although they can still end up equal), although I would like to know anything more precise about them.

    I am not sure whether the Remark in Ardila’s notes is completely reasonable. In my opinion, he should not require that $ latex \mathcal P $ is closed under taking intervals (this is unnecessarily restrictive and not satisfied, e. g., for permutahedra), but instead require that any interval of a poset in $ latex \mathcal P $ is in $ latex \mathcal P^{\ast} $ (or, what is equivalent, that $ latex \mathcal P^{\ast} $ is closed under taking intervals). Even if a poset is not a direct product of more than 1 poset, this does not have to hold for its intervals!

  2. allahandrow

    For b, it says that the coproduct is given by \sum_{i+j=n} B_k \otimes B_{n-k} where we extend multiplicatively. Is there a typo? Or is i+j=k, and k \le n?

    Thank you.

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