Lifting set/multiset duality

Wildon's Weblog 2024-11-24

This is a pointer to a note in the same spirit as the previous post, this time lifting the identity

$\sum_{k=0}^n (-1)^k \left(\!\binom{n}{k}\!\right) \binom{n}{k} = 0$

where $\left(\!\binom{n}{k}\!\right) = \binom{n+k-1}{k}$ is the number of $k$ -multisets of a set of size $n$ to a long exact sequence of polynomial representations of $\mathrm{GL}_n(F)$ , working over an arbitrary field $F$ . We then descend to get quick proofs of generalizations of the symmetric function identity

$\sum_{r=0}^n (-1)^r h_{n-r}(x_1,\ldots, x_D)e_r(x_1,\ldots, x_D) = 0$

for $D \ge r$ and the corresponding $q$ -binomial identity, obtained by specializing at $1,q, \ldots, q^{D-1}$ , namely

$\sum_{r=0}^n (-1)^r q^{r(r-1)2} \genfrac{[}{]}{0pt}{}{n-r+D-1}{n-r}_q \genfrac{[}{]}{0pt}{}{D}{r}_q = 0.$