An Exact Sequence in the Representation Theory of SL(2)

Abstract

Let Vbe the standard two-dimensional representation of the algebraic group G = SL(2, C), and write V _n  = Sym ⁿ Vfor the irreducible (n + 1)-dimensional representation of Gon the nth symmetric tensor power of V. Also consider the (2 ⁿ )-dimensional space W _n  = V ^⊗n , 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  −  W _n−2 +  W _n−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 .

Dedicated in memory of my colleague Ahmad Shamsuddin.

Key Words:

• Representation theory
• Complexes
• SL(2)
• Resolutions

AMS 1991 Subject Classification:

• 20G05
• 15A72
• 16E05

Acknowledgments

Notes

Dedicated in memory of my colleague Ahmad Shamsuddin.