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.
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Acknowledgements
We would like to thank Prof. Doty for useful conversations. The second author wishes to thank his advisor (the first author) for his constant encouragement and the School of Mathematics, TIFR for its support. We acknowledge support of the Department of Atomic Energy under project number 12-R&D-TFR-5.01-0500.
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Presented by: Vyjayanthi Chari
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Ghate, E., Vangala, R. The Monomial Lattice in Modular Symmetric Power Representations. Algebr Represent Theor 25, 121–185 (2022). https://doi.org/10.1007/s10468-020-10013-x
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DOI: https://doi.org/10.1007/s10468-020-10013-x
Keywords
- Modular representations of \(\text {GL}_{2}(\mathbb {F}_{p}) \)
- Structure of monomial submodules
- Reductions of crystalline representations