Logarithmic Concavity and SI2(C)

We observe that for any logarithmically concave nite sequence a 0 , a 1 , : : : , a n of positive integers there is a representation of the Lie algebra sl 2 (C) from which this logarithmic concav-ity follows. Thus, in applying this strategy to prove logarithmic concavity, the only issue is to construct such a representation naturally from given combinatorial data. As an example, we do this when a j is the number of j-element stable sets in a claw-free graph, reproving a theorem of Hamidoune. Let a 0 , a 1 , : : : , a n be a sequence of positive integers which is logarithmically concave: that is, a 2 j a j?1 a j+1 for all 1 j n ? 1. (Such sequences occur frequently in combinatorics; see Stanley 5] and Brenti 1, 2].) As is easily seen, this implies that if 0 h i j k n and h + k = i + j then a i a j a h a k. Thus, the inequalities in the following display are eqivalent to logarithmic concavity of the sequence (a j) n j=0 Symmetric unimodal sequences of positive integers appear naturally in the representation theory of the Lie algebra sl 2 (C), as explained in Section 7 of Humphreys 4]. In particular, to prove logarithmic con-cavity of (a j) n j=0 the following construction suuces.