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- W58100147 abstract "We start with the following simple observation. Let V be a representation of a reductive group G, and let f 1, f 2, . . . , f n be homogeneous invariant functions. Then the polarizations of f 1, f 2, . . . , f n define the nullcone of k ≤ m copies of V if and only if every linear subspace L of the nullcone of V of dimension ≤ m is annhilated by a one-parameter subgroup (shortly a 1-PSG). This means that there is a group homomorphism ( lambda mathbb{C} : * to {rm }G) such that ( lim _{t to 0} lambda (t)x = 0) for all x ∈ L. This is then applied to many examples. A surprising result is about the group SL2 where almost all representations V have the property that all linear subspaces of the nullcone are annihilated. Again, this has interesting applications to the invariants on several copies. Another result concerns the n-qubits which appear in quantum computing. This is the representation of a product of NumberingDepth=0 copies of SL2 on the n-fold tensor product ( mathbb{C}^2 otimes mathbb{C}^2 otimes cdot cdot cdot otimes mathbb{C}^2). Here we show just the opposite, namely that the polarizations never define the nullcone of several copies if n ≥3. (An earlier version of this paper, distributed in 2002, was split into two parts; the first part with the title “On the nullcone of representations of reductive groups” is published in Pacific J. Math. 224 (2006), 119–140)." @default.
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- W58100147 date "2009-11-14" @default.
- W58100147 modified "2023-09-26" @default.
- W58100147 title "Polarizations and Nullcone of Representations of Reductive Groups" @default.
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- W58100147 doi "https://doi.org/10.1007/978-0-8176-4875-6_8" @default.
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