# Formula for Schur multiplier of free nilpotent group

## Contents

## Statement

### Description as a lower central series quotient

Suppose is the free nilpotent group of class on a generating set . If is the free group on , is explicitly the quotient group , where is the member of the lower central series of . Note that the numbering starts with , , and so on.

Then, the Schur multiplier of is the group .

### Explicit formula for dimension

Suppose is a free nilpotent group of class on generators. Then, the Schur multiplier 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 graded component for the free Lie algebra with generators. The explicit formula is:

## Related description of Schur covering group

The *unique* Schur covering group of the free nilpotent group of class on a set is the free nilpotent group of class on the same set . Explicitly, if is the free group on and is the free class group on , then:

- The Schur covering group of is

The corresponding short exact sequence is:

## Related description of exterior square

The exterior square of the free nilpotent group of class on a set is defined as follows. Suppose is the free group on and . is the free class group on , then:

Here, .

This group is *not* in general a free nilpotent group (note that the case is special). In general, the group is nilpotent of class .

## 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 | ||||||
---|---|---|---|---|---|---|

0 | ||||||

0 | ||||||

0 | . | |||||

0 | ||||||

0 |

### Description of exterior square

Positive integer such that we are interested in studying the free nilpotent group of class on generators (sum of the second and third graded component dimensions except in the case) | Hirsch length (equals sum of dimensions of graded components from the second to the ) | Minimum size of generating set for the exterior square of the group | Nilpotency class of the exterior square (equals | Comments |
---|---|---|---|---|

1 | 1 | we are starting with the free abelian group and taking its exterior square as an abelian group to get | ||

2 | 1 | we are starting with a free class two group but we end up with a free abelian group. | ||

3 | 2 | we are starting with the free class three group, but we end up with a non-free class two group. | ||

4 | 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 Schur multiplier of free powered nilpotent group

## Facts used

## Proof

Fact (1) says that if can be written as where is a free group, then we have:

In our case, . We thus get:

Since , . Further, by the definition of lower central series, . Thus, we get: