# skew Schur identity

Let $$\lambda$$ be a partition of size at least 2, and let $$n>0$$ be an integer. Prove that $$s_{\lambda/(1)}.h_{n}=\sum_{\lambda^{+}\supseteq \lambda\\\lambda^{+}/\lambda\ \text{hor.} \ n\ \text{strip} }s_{\lambda^{+}/(1)}-\sum_{\lambda^{+}\supseteq \lambda\\\lambda^{+}/\lambda\ \text{hor.} \ n-1\ \text{strip} }s_{\lambda^{+}}$$

where $$s_\lambda$$ and $$s_{\lambda/(1)}$$ are Schur and skew-Schur functions and $$h_n$$ is complete homogeneous symmetric function.

My idea: I want to use following two lemmas proved in Stanley's book in chapter 7 pages 339 and 340.

(1) We have $$s_vs_n=\sum_{\lambda}s_{\lambda}$$

summed over all partitions $$\lambda$$ such that $$\lambda/v$$ is a horizontal strip of size $$n$$.

(2) We have $$s_{\lambda/n}=\sum_{v}s_{v}$$ where $$v$$ ranges over all partitions $$v\subseteq \lambda$$ for which $$\lambda/v$$ is a horizontal strip of size $$n$$.

So $$s_{\lambda/(1)}=\sum_{\lambda^{-}\subseteq \lambda\\ \lambda/\lambda^{-} \ \text{hor.} 1 \ \text{strip}}s_{\lambda^{-}}$$

Whence $$s_{\lambda/(1)} h_n=s_{\lambda/(1)} s_n=\sum_{\lambda^{-}\subseteq \lambda\\ \lambda/\lambda^{-} \ \text{hor.} 1 \ \text{strip}}s_{\lambda^{-}}s_n=\sum_{\lambda^{-}\subseteq \lambda\\ \lambda/\lambda^{-}\ \text{hor.} 1 \ \text{strip}}\left(\sum_{v/\lambda^{-} \ \text{hor.} \ n \ \text{strip}}s_v\right)=\sum_{v,\lambda^{-}\\ \lambda/\lambda^{-} \text{hor.} \ 1 \ \text{strip}\\ v/\lambda^{-} \ \text{hor.} \ n\ \text{strip}}s_v$$

On the other hand,

$$\sum_{\lambda^{+}\supseteq \lambda\\\lambda^{+}/\lambda\ \text{hor.} \ n\ \text{strip} }s_{\lambda^{+}/(1)}=\sum_{\lambda^{+}\supseteq \lambda\\\lambda^{+}/\lambda\ \text{hor.} \ n\ \text{strip} }\left(\sum_{\lambda^{+}/v' \text{hor.} \ 1 \ \text{strip}}s_{v'} \right)=\sum_{\lambda^+,v'\\\lambda^{+}/\lambda \ \text{hor.} \ n\ \text{strip}\\ \lambda^{+}/v' \ \text{hor.} \ 1\ \text{strip}}s_{v'}$$

So it suffices to show

$$\sum_{v,\lambda^{-}\\ \lambda/\lambda^{-} \text{hor.} \ 1 \ \text{strip}\\ v/\lambda^{-} \ \text{hor.} \ n\ \text{strip}}s_v+\sum_{\lambda^{+}\supseteq \lambda\\\lambda^{+}/\lambda\ \text{hor.} \ n-1\ \text{strip} }s_{\lambda^{+}}=\sum_{\lambda^+,v'\\\lambda^{+}/\lambda \ \text{hor.} \ n\ \text{strip}\\ \lambda^{+}/v' \ \text{hor.} \ 1\ \text{strip}}s_{v'}$$

The last equation is intuitive but I could not show that.