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Mathematics
General Math
Can chatgpt accurately calculate expected lengths in Pascal's triangle?
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[QUOTE="Infrared, post: 6847283, member: 467682"] I think representations of topological groups are by definition continuous. If you don't require continuity, this looks very false. For example, consider the group ##GL(n,\mathbb{C})## equipped with the indiscrete topology, which trivially makes it compact. Then the identity map (which is not continuous) ##GL(n,\mathbb{C})\to GL(n,\mathbb{C})## can be considered as a representation, which is certainly not unitary. The righthand side of course has the standard topology. Assuming continuity, if the group is also Hausdorff, then it possesses a Haar measure ##\mu## and any representation is unitary, using the inner product ##\langle u,v\rangle=\frac{1}{\mu(G)}\int_G \langle gu,gv\rangle_0 d\mu(g)## where ##\langle \rangle_0## is any fixed inner product. Even if the group ##G## is not Hausdorff, the image of the representation ##\rho:G\to GL(V)## is a compact subgroup ##\rho(G)\subset GL(V)## which is Hausdorff (since ##GL(V)##) is. So we just apply the above argument to the group ##\rho(G)## instead, thinking of ##\rho(G)\hookrightarrow GL(V)## as a representation. I don't see where irreducibility should factor in. [/QUOTE]
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Can chatgpt accurately calculate expected lengths in Pascal's triangle?
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