Formula for Schur multiplier of free nilpotent group

Description as a lower central series quotient
Suppose $$G$$ is the free nilpotent group of class $$c \ge 1$$ on a generating set $$S$$. If $$F$$ is the free 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 $$\gamma_{c+1}(F)/\gamma_{c+2}(F)$$.

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 abelian group 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 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 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 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 $$\lceil c/2 \rceil$$.

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:

Related facts

 * Formula for nilpotent multiplier of free nilpotent group
 * Formula for Schur multiplier of free powered nilpotent group

Facts used

 * 1) uses::Hopf's formula for Schur multiplier

Proof
Fact (1) says that if $$G$$ can be written as $$F/R$$ where $$F$$ is a free 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:

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