Formula for Schur multiplier of free powered nilpotent group

Statement

Description as a lower central series quotient

Suppose $\pi$ is a set of primes.

Suppose $G$ is the free $\pi$-powered nilpotent group of class $c \ge 1$ on a generating set $S$. If $F$ is the free $\pi$-group on $S$, $G$ is explicitly the quotient group $F/\gamma_{c+1}(F)$, where $\gamma_{c+1}(F)$ is the $(c + 1)^{th}$ member of the lower central series of $F$. Note that the numbering starts with $\gamma_1(F) = F$, $\gamma_2(F) = [F,F]$, and so on.

Then, the Schur multiplier $M(G) = H_2(G;\mathbb{Z})$ of $G$ is the group $\gamma_{c+1}(F)/\gamma_{c+2}(F)$. Note that this is an abelian $\pi$-powered group.

Explicit formula for dimension

Suppose $G$ is a free nilpotent group of class $c \ge 1$ on $n$ generators. Then, the Schur multiplier $H_2(G;\mathbb{Z})$ is a free $\mathbb{Z}[\pi^{-1}]$-module whose rank is given by the formula for dimension of graded component of free Lie algebra where we are talking of the $(c+1)^{th}$ graded component for the free Lie algebra with $n$ generators. The explicit formula is:

$\frac{1}{c + 1} \sum_{d | c + 1} \mu(d)n^{(c+1)/d}$

Related description of Schur covering group

The unique Schur covering group of the free $\pi$-powered nilpotent group of class $c$ on a set $S$ is the free nilpotent group of class $c + 1$ on the same set $S$. Explicitly, if $F$ is the free $\pi$-powered group on $S$ and $G = F/\gamma_{c+1}(F)$ is the free class $c$ group on $S$, then:

• $M(G) = \gamma_{c+1}(F)/\gamma_{c+2}(F)$
• The Schur covering group of $G$ is $F/\gamma_{c+2}(F)$

The corresponding short exact sequence $0 \to M(G) \to \mbox{Schur covering group} \to G \to 1$ is:

$0 \to \gamma_{c+1}(F)/\gamma_{c+2}(F) \to F/\gamma_{c+2}(F) \to F/\gamma_{c+1}(F) \to 1$

Related description of exterior square

The exterior square of the free $\pi$-powered nilpotent group of class $c$ on a set $S$ is defined as follows. Suppose $F$ is the free group on $S$ and $G = F/\gamma_{c+1}(F)$. is the free class $c$ group on $S$, then:

$G \wedge G = \gamma_2(F)/\gamma_{c+2}(F)$

Here, $\gamma_2(F) = [F,F]$.

This group is not in general a free nilpotent group (note that the case $c = 1$ is special). In general, the group is nilpotent of class $\lceil c/2 \rceil$.

Particular cases

Description of Schur multiplier

The rows here correspond to the nilpotency class used and the columns correspond to the number of generators. The entry in the cell describes the Schur multiplier for the free nilpotent group of class given by the row and number of generators given by the column:

	generic $n$	$n = 1$	$n = 2$	$n = 3$	$n = 4$	$n = 5$
$c = 1$	$\mathbb{Z}[\pi^{-1}]^{\frac{n^2 - n}{2}}$	0	$\mathbb{Z}[\pi^{-1}]$	$\mathbb{Z}[\pi^{-1}]^3$	$\mathbb{Z}[\pi^{-1}]^6$	$\mathbb{Z}[\pi^{-1}]^{10}$
$c = 2$	$\mathbb{Z}^{\frac{n^3 - n}{3}}$	0	$\mathbb{Z}[\pi^{-1}]^2$	$\mathbb{Z}[\pi^{-1}]^8$	$\mathbb{Z}[\pi^{-1}]^{20}$	$\mathbb{Z}[\pi^{-1}]^{40}$
$c = 3$	$\mathbb{Z}[\pi^{-1}]^{\frac{n^4 - n^2}{4}}$	0	$\mathbb{Z}[\pi^{-1}]^3$	$\mathbb{Z}[\pi^{-1}]^{18}$	$\mathbb{Z}[\pi^{-1}]^{60}$	$\mathbb{Z}[\pi^{-1}]^{150}$.
$c = 4$	$\mathbb{Z}[\pi^{-1}]^{\frac{n^5 - n}{5}}$	0	$\mathbb{Z}[\pi^{-1}]^6$	$\mathbb{Z}[\pi^{-1}]^{48}$	$\mathbb{Z}[\pi^{-1}]^{204}$	$\mathbb{Z}[\pi^{-1}]^{624}$
$c = 5$	$\mathbb{Z}[\pi^{-1}]^{\frac{n^6 - n^3 - n^2 + n}{6}}$	0	$\mathbb{Z}[\pi^{-1}]^9$	$\mathbb{Z}[\pi^{-1}]^{116}$	$\mathbb{Z}[\pi^{-1}]^{670}$	$\mathbb{Z}[\pi^{-1}]^{2580}$

Description of exterior square

Positive integer $c$ such that we are interested in studying the free nilpotent group of class $c$ on $n$ generators (sum of the second and third graded component dimensions except in the $c = 1$ case)	Hirsch length (equals sum of dimensions of graded components from the second to the $(c+1)^{th}$)	Minimum size of generating set for the exterior square of the group	Nilpotency class of the exterior square (equals $\lceil c/2 \rceil$)	Comments
1	$\frac{n^2 - n}{2}$	$\frac{n^2 - n}{2}$	1	we are starting with the free abelian group $\mathbb{Z}^n$ and taking its exterior square as an abelian group to get $\mathbb{Z}^{n(n-1)/2}$
2	$\frac{n^2 - n}{2} + \frac{n^3 - n}{3}$	$\frac{n^2 - n}{2} + \frac{n^3 - n}{3}$	1	we are starting with a free class two group but we end up with a free abelian group.
3	$\frac{n^2 - n}{2} + \frac{n^3 - n}{3}$	$\frac{n^2 - n}{2} + \frac{n^3 - n}{3} + \frac{n^4 - n^2}{4}$	2	we are starting with the free class three group, but we end up with a non-free class two group.
4	$\frac{n^2 - n}{2} + \frac{n^3 - n}{3}$	$\frac{n^2 - n}{2} + \frac{n^3 - n}{3} + \frac{n^4 - n^2}{4} + \frac{n^5 - n}{5}$	2	we are starting with the free class four group, but we end up with a non-free class two group.

Related facts

• Formula for nilpotent multiplier of free nilpotent group
• Formula for nilpotent multiplier of free powered nilpotent group

Facts used

1. Hopf's formula for Schur multiplier

Proof

Fact (1) says that if $G$ can be written as $F/R$ where $F$ is a free $\pi$-powered group, then we have:

$H_2(G;\mathbb{Z}) \cong (R \cap [F,F])/[F,R]$

In our case, $R = \gamma_{c+1}(F)$. We thus get:

$H_2(G;\mathbb{Z}) \cong (\gamma_{c+1}(F) \cap [F,F])/[F,\gamma_{c+1}(F)]$

Since $c \ge 1$, $\gamma_{c+1}(F) \le [F,F]$. Further, by the definition of lower central series, $[F,\gamma_{c+1}(F)] = \gamma_{c+2}(F)$. Thus, we get:

$H_2(G;\mathbb{Z}) \cong \gamma_{c+1}(F)/\gamma_{c+2}(F)$