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Let $p$ be a prime. We study the structure of and the inclusion relationsamong the terms in the monomial lattice in the modular symmetric powerrepresentations of $\mathrm{GL}_2(\mathbb{F}_p)$. We also determine thestructure of certain related quotients of the symmetric power representationswhich arise when studying the reductions of local Galois representations ofslope at most $p$. In particular, we show that these quotients are periodic anddepend only on the congruence class modulo $p(p-1)$. Many of our results arestated in terms of the sizes of various sums of digits in base $p$-expansionsand in terms of the vanishing or non-vanishing of certain binomial coefficientsmodulo $p$.

The Monomial Lattice in Modular Symmetric Power Representations