An exact sequence in the representation theory of SL(2)
نویسندگان
چکیده
Let V be the standard two-dimensional representation of the algebraic group G = SL(2,C), and write Vn = SymnV for the irreducible (n+1)-dimensional representation of G on the nth symmetric tensor power of V . Also write Wn = V ⊗n for the nth tensor power of V (which has dimension 2n). Then, as virtual representations, it is known that each Vn can be written in terms of W0, . . . , Wn as Vn = Wn − `n−1 1 ́ Wn−2 + `n−2 2 ́ Wn−4 − . . . . We explain this phenomenon by writing down an exact sequence that gives a “resolution” of Vn in terms of W0, . . . , Wn.
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