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On tensor products of modular representations of symmetric groups. (English) Zbl 1021.20010

Main Theorem: Let \(D^\lambda\) and \(D^\mu\) be two irreducible \(F\Sigma_n\)-modules of dimensions greater than 1. Assume that \(D^\lambda\otimes D^\mu\) is irreducible. Then \(p=2\), \(n\) is even, and if \[ \lambda=(\lambda_1>\lambda_2>\cdots>\lambda_r>0)\quad\text{and}\quad\mu=(\mu_1>\mu_2>\cdots>\mu_s>0), \] then \(\lambda_1\equiv\lambda_2\equiv\cdots\equiv\lambda_r\pmod 2\) or \(\mu_1\equiv\mu_2\equiv\cdots\equiv\mu_s\pmod 2\) (or both).

MSC:

20C30 Representations of finite symmetric groups
20C20 Modular representations and characters
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