Department of Mathematics, La Trobe University, Melbourne, VIC 3086, Australia. E-mail: [email protected]
Received Date: April 18, 2009; Revised Date: July 24, 2009
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It is known that the Betti numbers of the Heisenberg Lie algebras are unimodal over elds of characteristic two. This note observes that they are log-concave. An example is given of a nilpotent Lie algebra in characteristic two for which the Betti numbers are unimodal but not log-concave.
The Heisenberg Lie algebra of dimension 2m + 1 is the Lie algebra having the basis and nonzero relations . For the cohomology with trivial coecients, the Betti numbers have been explicitly computed in all characteristics [1,3,4]. Recall that the Betti numbers are unimodal if bi ≤ bj for all 0 ≤ i ≤ j ≤ m and bi ≥ bj for all m ≤ i ≤ j ≤ 2m + 1, and they are concave (resp., log-concave) if bi is at least as great as the arithmetic (resp., geometric) mean of the pair bi–1, bi+1 for all 1 ≤ i ≤ 2m. So concave implies log-concave which implies unimodal. In characteristic zero, unimodality is quite common. The Heisenberg Lie algebras play a key role in the construction of all known examples of Lie algebras in characteristic zero where the Betti numbers are not unimodal . In fact, the Betti numbers of are unimodal only in characteristic two . On the other hand, we know of no nilpotent Lie algebra in characteristic two whose Betti numbers fail to be unimodal. In , the question was posed: in characteristic two, do all nilpotent Lie algebras have unimodal Betti numbers? Since logconcavity is a common route taken to prove unimodality, it is natural to ask whether the Betti numbers of the Heisenberg algebras are unimodal in characteristic two. We record the following observation as a theorem, though it is really just a corollary of the works [1,4].
Theorem 1. Over elds of characteristic two, the Betti numbers of are log-concave; i.e., for all n.
Proof. For the rest of this note we x the characteristic to be two. Emil Skoldberg showed that the Poincare polynomial is 
Though we will not need them, we mention that the individual Betti numbers are given in ; for all i ≤ m,
To establish the log-concavity, we observe that the Betti numbers of are essentially determined by those of , with a curious correction for the middle two terms. Explicitly,
This relation is easily deduced from (1). Using induction, we assume that Sm is log-concave. Since (1+t)2 is log-concave, (1+t)2Sm(t) is thus also log-concave (see ). So in view of (2), to establish the log-concavity of Sm+1, it remains to verify it for the middle terms; that is, for we require that . But by Poincare duality, bm+1 = bm+2, and so we only require bm+1 ≥ bm, and this is given by the unimodality of the Betti numbers, which was shown in . This completes the proof.
The following example shows that, despite the above result, log-concavity is not a route for establishing unimodality in the general setting of nilpotent Lie algebras in characteristic two.
Example 2. Let denote the 7-dimensional Lie algebra with basis x1, . . . , x7 and dening relations:
Clearly is nilpotent (and actually graded and liform). Direct calculations using Mathematica show that in characteristic two, the Betti numbers are
As , the Betti numbers are not log-concave.
The author thanks Peter Cameron for asking whether the Betti numbers of the Heisenberg Lie algebras are log-concave.