No finitely generated symmetric operad has GK-dimension strictly between 1 and 2; finitely generated prime symmetric operads of GK-dimension 1 are classified.
Wagner, The faithful linear representation of least degree of S_n and A_n over a field of characteristic 2, Math
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Symmetric operads of GK-dimension one
No finitely generated symmetric operad has GK-dimension strictly between 1 and 2; finitely generated prime symmetric operads of GK-dimension 1 are classified.