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In a recent paper (arXiv:2301.09744), Erickson and Hunziker considerpartitions in which the arm-leg difference is an arbitrary constant $m$. Inprevious works, these partitions are called $(-m)$-asymmetric partitions.Regarding these partitions and their conjugates as the highest weights, theyprove an identity yielding an infinite family of dimension equalities between$\mathfrak{gl}_n$ and $\mathfrak{gl}_{n+m}$ modules. Their proof proceeds bythe manipulations of the hook content formula. We give a simple bijective proofof their result.

A short bijective proof of dimension identities of Erickson and Hunziker