On the homology of free 2-step nilpotent Lie algebras.
The paper under review offers an explicit formula for the total homology, i.e., the sum of the ranks of the homology groups, of a 2-step nilpotent Lie algebra of a finite dimensional vector space \(V\) over the complex numbers \(\mathbb{C}\). For \(\dim_{\mathbb{C}}(V) = r\), the 2-step nilpotent Lie algebra of \(V\) is \[ N_r = V \oplus \Lambda^2 V , \] where, for \(v\), \(w \in V\), \([v , w] = v \wedge w\), and the bracket is zero otherwise. Using work of \textit{S. Sigg} [J. Algebra 185, No. 1, 144--161 (1996; Zbl 0868.17015)], the Lie algebra homology \(H_*( N_r )\) can be computed in terms of irreducible components of \(GL(V)\) representations. The total homology (as a function of \(r\)) is then \[ T(r) = \sum_{k \geq 0} \dim H_k ( N_r ). \] A key result of the paper is that for \(n \geq 0\), \[ \begin{aligned} T(2n + 1) & = 2^{n+1} \beta (n)^2, \\ T(2n + 2) & = 2^{n+1} \beta (n) \beta (n+1), \\ \beta (n) &= \prod_{1 \leq k \leq n} \frac{(4k)! (k!)^2}{(2k)!^3} . \end{aligned} \] This result is derived by realizing the Poincaré polynomial \[ P(N_r ; t) = \sum_{k \geq 0} \dim H_k (N_r) t^k \] as the determinant of the \(r \times r\) matrix \(G_3\), \[ P(N_r ; t) = {\text{det}} \, G_3(r ; t) , \] where \[ G_3 (r ; t)_{ij} = \binom{r}{j-i} - (-t)^i \binom{r}{j+i-1} \, , \] and relies on the Giambelli formula for the dimension of the irreducible tensor representations corresponding to self-conjugate partitions of Young diagrams. See \textit{W. Fulton} and \textit{J. Harris} [Representation Theory. A First Course. New York: Springer (1991; Zbl 0744.22001)]. Furthermore, the determinant of \(G_3\) is related to \(SO(V)\) characters. By using Stirling's asymptotic series for \(n!\), the authors prove that \[ T(r) \sim 2^{({r^2}/{2})} r^{1/8} k \, , \] where \(k \simeq 1.3814\), which improves the previous lower bound for \(T(r)\) given by \(2^{({r^2}/{2})}\).
scientific article; zbMATH DE number 1867976