An exact sequence in the representation theory of SL(2)

摘要

Let V be the standard two-dimensional representation of the algebraic group G = SL(2, C), and write V-n = Sym(n) V for the irreducible (n + I)-dimensional representation of G on the nth symmetric tensor power of V. Also consider the (2(n))-dimensional space W-n = V-circle timesn, obtained as the nth tensor power of V. It is known that each V-n can be written in terms of W-0, ..., W-n as V-n = W-n - ((n-1)(1)) Wn-2 + ((n-2)(2)) Wn-4 - ..., where we view V-n and the W-i as virtual representations of G. We explain this phenomenon by writing down an exact sequence that gives a "resolution" of V-n in terms of W-0,..., W-n. References: 3