There once was an antigen test that could detect Delta, but that was coincidence. They used two types of antibodies in the test, and the test was positive if both of them could find binding sites. With Delta, one of them failed. That was a hint, but no proof. The only way to really distinguish the variants is to put a probe into a machine from Illumina or BMG, wait for the RNA sequence, and to compare it with the sequences in the open access databases - lab work.