The Monomial Lattice in Modular Symmetric Power Representations

Abstract

Let p be a prime. We study the structure of and the inclusion relations among the terms in the monomial lattice in the modular symmetric power representations of $\text {GL}_{2}(\mathbb {F}_{p})$ . We also determine the structure of certain related quotients of the symmetric power representations which arise when studying the reductions of local Galois representations of slope at most p. In particular, we show that these quotients are periodic and depend only on the congruence class modulo p(p − 1). Many of our results are stated in terms of the sizes of various sums of digits in base p-expansions and in terms of the vanishing or non-vanishing of certain binomial coefficients modulo p.