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.

令p为质数。我们研究了\（\ text {GL} _ {2}（\ mathbb {F} _ {p}）\）的模对称幂表示中单项格中项的结构和包含关系。我们还确定了对称幂表示的某些相关商的结构，这些相关商在研究最多p的边坡的局部Galois表示的减少时出现。特别是，我们表明这些商是周期性的，并且仅取决于模p（p − 1）的同余类。我们的许多结果都以基数p中各种数字之和的大小表示-展开，并且根据模p的某些二项式系数的消失或不消失。