On the conjectural decomposition of symmetric powers of automorphic representations for GL(3) and GL(4)

Given a cuspidal automorphic representation π for GL(3) over a number field and a positive integer k, assume that the symmetric mth power lifts of π are isobaric automorphic for m ≤ k, cuspidal for m ≤ k− 1, and that certain associated Rankin–Selberg products are isobaric automorphic. Then the number of cuspidal isobaric summands in the kth symmetric power lift is bounded above by 3 when k ≥ 7, and bounded above by 2 when k ≥ 19 with k ≡ 1 (mod 3). We then investigate the analogous problem for GL(4).

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