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The q-deformed Fock spaces of higher levels were introduced byJimbo-Misra-Miwa-Okado. The q-decomposition matrix is a transition matrix fromthe standard basis to the canonical basis defined by Uglov in the q-deformedFock space. In this paper, we show that parts of q-decomposition matrices oflevel $\ell$ coincides with that of level $\ell - 1$ under certain conditionsof multi charge.

A comparison of q-decomposition numbers in the q-deformed Fock spaces of higher levels