shifted symmetric polynomials

Question

BACKGROUND

When defining shifted symmetric polynomials we do it in the following way: Let $\mu=(\mu_1,..., \mu_n)$ be a partition with length less or equal to $n$.

Then $$s_{\mu}^*(x_1,...,x_n)=\frac{\det[(x_i+n-i|\mu_j+n-j)]}{\det[(x_i+n-i|n-j)]}$$

MY QUESTION

1. When reading Okounkov-Olshanski's paper about shifted symmetric functions, they say that this is indeed a polynomial because the denominator is the Vandermonde determinant in the variables $$x_i'=x_i-i+c$$ where $c$ is a constant and the numerator is skew-symmetric in these variables. Why does this happen? Why a skew-symmetric determinant is divisible by Vandermonde's determinant?

2. Moreover, I would like to know if this $c$ can change on each $x_i$. I mean, can $x_1'=x_1-1+2$ and $x_2'=x_2-2+4$?

Answer 1

1. Why is every skew-symmetric polynomial in $x_1, x_2, \ldots, x_n$ divisible by the Vandermonde determinant? Let me go overkill on this question and refer to Theorem 9.6 in my Regular elements of a ring, monic polynomials and "lcm-coprimality", where I also show that the resulting quotient will be a symmetric polynomial (I think you want this, too). Note that my theorem has an extra condition, namely that the skew-symmetric polynomial becomes $0$ if you set two of the variables to be equal; but this follows from the skew-symmetry whenever $2$ is invertible in your base ring. And yes, I am working over an arbitrary commutative ring, which is the reason why it takes me so long to prove it. Proofs that work over fields are legion and should be easy to find; e.g., such an argument appears (in a particular case) in the proof of Theorem 1.0.1 in Paul Garrett's Algebra notes, Section 17. See also https://mathoverflow.net/questions/139071/every-antisymmetric-alternating-polynomial-is-divisible-by-vandermonde-product (where I gave a reference to a different overkill proof).

2. I don't really see the purpose of $c$; as far as I am concerned, you can just pick $c = 0$ or $c = n$ (to match the definition better). But you have to pick some $c$ and stick with it; the numerator won't be skew-symmetric if you pick a different $c$ for each variable.

3. When posing such questions, it is always helpful to cite the paper/text you are reading. There may be context that is missing from the question. Maybe it will explain the purpose of $c$?