1. If V,W are two reps of G, show that ChiV @ W = ChiV ChiW.
2. If V,W are two reps of G, and |ChiV(g)| < dim V except for g=1, show that W is isomorphic to a subrepresentation of V@V@....@V for some number of factors.
3. Show that the condition in #2 holds for G simple, V not
4. If V is a rep, compute the character of G on the symmetric tensors Sym2V and the alternating tensors Alt2V, in terms of the character of V. For each irrep W of S4, confirm that the characters of Sym2W and Alt2W are nonnegative integer combinations of the irreducible characters.
2 comments:
nr. 2 nice--and with a bit more work
for any V one "describes asymptotically" the decomposition of V⊗V...⊗V , n large
for nr. 3, V not 0 and not (multiple of) 1.
Right you are. Corrected.
Rather than "1", which is the character, I fixed it to "a trivial representation".
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