1. Again, please note the mistake I made in class and on the lecture notes. I am about to correct the notes and reupload them:

https://hopfcombinatorics.wordpress.com/2012/04/26/latex-se_lambda-1n-h_lambda-and-a-mistake-in-lecture-3/

2. The inner product on Sym has the very nice property that it is respected by the antipode; that is, . You may find this useful in the last question of the Short Homework. **If** one of you posts a proof of this fact here, then everyone will be allowed to use it in the homework (and will be very thankful to that person for their effort).

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Here’s my stab at it: Clearly, the inner product induces a norm. Let’s define it as follows: . Then we have, for any given

Hence, which implies the claim.

could someone direct me to the definition of the inner product on symmetric functions.

p.106 of the notes.

duquec, I don’t think that’s how you want to define a norm. Also, is our inner product positive definite? (It is, but why?)

Miriam, please note that I haven’t fixed the notes on the website yet – the definition there is wrong at the moment. You want to use the definition here:

https://hopfcombinatorics.wordpress.com/2012/04/26/latex-se_lambda-1n-h_lambda-and-a-mistake-in-lecture-3/

where we define the inner product between the m and the h, and then extend bilinearly.

Here’s my try at a proof:

It is enough to show for basis elements. In particular, I show

In his book Enumerative Combinatorics, Stanley provides a way to write the basis in terms of as

where (proposition 7.4.1). For a description of what is, I direct you to Stanley’s book – it has something to do with the number of matrices satisfying certain conditions. The important property of these numbers is that

(corollary 7.4.2). We know from class that

Suppose is a partition of and is a partition of . Then the left hand side is

and the right hand side is

Notice that if (or if ) then it must be that (otherwise they would be partitions of different numbers and not equal). And the right hand side is equal to the left hand side.

Thank you, Alyssa, especially from your classmates. 🙂

You should all feel free to use it in your solution to the homework.

I have one objection to your argument: I did not prove that this inner product is symmetric ( = ), which you are using in the proof. Can anyone fill in the details to that?

Thanks, Alyssa! Maybe we can work out Federico’s objection tomorrow in the office.