Medical, Pharma, Engineering, Science, Technology and Business

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 2*m* + 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 *b _{i}* ≤

**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 [4]

(1)

Though we will not need them, we mention that the individual Betti numbers are given in
[1]; 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,

(2)

This relation is easily deduced from (1). Using induction, we assume that *S _{m}* is log-concave.
Since (1+

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 *x*_{1}, . . . , * x*_{7} 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.

- CairnsG,JamborS (2008) The cohomology of the Heisenberg Lie algebras over elds of nite characteristic. ProcAmer Math Soc136: 3803-3807.
- PouseeleH (2005)On the cohomology of extensions by a Heisenberg Lie algebra. Bull Austral Math Soc71: 459-470.
- SantharoubaneLJ (1983)Cohomologyof Heisenberg Lie algebras.ProcAmer Math Socpp: 23-28.
- SkoldbergE (2005) The homology of Heisenberg Lie algebras over elds of characteristic two. MathProc R IrAcad105A: 47-49.
- StanleyRP (1989) Log-concave and unimodal sequences in algebra, combinatorics, and geometry. In\Graph Theory and Its Applications: East and West"(Jinan, 1986). M. F. Capobianco, M. G.Guan, D. F. Hsu, and F. Tian, Eds. Annals of the New York Academy of Sciences 576, New YorkAcademy of Sciences, New Yorkpp: 500-535.

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