# Why is $\sum_\mu [s_\lambda](s_\lambda s_\mu)_{/\mu} = \sum_\rho [s_\rho](s_{\rho /\mu}) s_\mu$ for any partition $\lambda$?

Say $$\lambda \vdash n$$ and $$\mu \vdash m$$ and $$|\rho|=|\lambda|+|\mu|$$ then for all $$\lambda$$ and $$m$$ it appears to hold that

$$\sum_\mu [s_\lambda](s_\lambda s_\mu)_{/\mu} = \sum_\rho [s_\rho](s_{\rho /\mu}) s_\mu$$ with the (abusive?) shorthand $$(s_\lambda s_\mu)_{/\mu}$$ standing for the expansion of the product $$s_\lambda s_\mu$$ and taking the skew Schur function $$s_{\xi/\mu}$$ of each term $$s_\xi$$, and analogously $$(s_{\rho /\mu}) s_\mu$$ standing for the expansion of $$s_{\rho/\mu}$$ and multiplying each term with $$s_\mu$$ and expanding again. The $$[s_\lambda ]$$ denotes the coefficient of $$s_\lambda$$ in the polynome.

Example: take $$\lambda = \{3, 1\}$$ and $$m=3$$

then the left hand side evaluates to $$6+11+4 = 21$$ (running over the 3 partitions of 3) and the right hand side gives $$0+1+2+2+2+6+2+2+1+2+1+0+0+0+0 = 21$$ over the 15 partitions of 7.

Does this hold generally? Is this well known? Is it in EC2? Is it easy to prove?

• You should recall the meaning of "majoring" notation $\lambda \vdash n$ and/or give a reference. Mar 19 at 17:18
• @JeanMarie, the condition that $\lambda$ majors $\mu$ is not needed. A propos, "majoring" is equivalent to "at least covering" when seen as Young diagrams. Mar 20 at 18:38