In the mathematical area of
group theory
In abstract algebra, group theory studies the algebraic structures known as groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as ...
, the covering groups of the alternating and symmetric groups are groups that are used to understand the
projective representation In the field of representation theory in mathematics, a projective representation of a group ''G'' on a vector space ''V'' over a field ''F'' is a group homomorphism from ''G'' to the projective linear group
\mathrm(V) = \mathrm(V) / F^*,
where ...
s of the
alternating and
symmetric groups. The covering groups were classified in : for , the covering groups are 2-fold covers except for the alternating groups of degree 6 and 7 where the covers are 6-fold.
For example the
binary icosahedral group In mathematics, the binary icosahedral group 2''I'' or Coxeter&Moser: Generators and Relations for discrete groups: : Rl = Sm = Tn = RST is a certain nonabelian group of order 120.
It is an extension of the icosahedral group ''I'' or (2,3,5) of o ...
covers the
icosahedral group, an alternating group of degree 5, and the
binary tetrahedral group
In mathematics, the binary tetrahedral group, denoted 2T or , Coxeter&Moser: Generators and Relations for discrete groups: : Rl = Sm = Tn = RST is a certain nonabelian group of order 24. It is an extension of the tetrahedral group T or (2,3,3) ...
covers the
tetrahedral group
150px, A regular tetrahedron, an example of a solid with full tetrahedral symmetry
A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection a ...
, an alternating group of degree 4.
Definition and classification
A group homomorphism from ''D'' to ''G'' is said to be a
Schur cover of the finite group ''G'' if:
# the kernel is contained both in the
center
Center or centre may refer to:
Mathematics
*Center (geometry), the middle of an object
* Center (algebra), used in various contexts
** Center (group theory)
** Center (ring theory)
* Graph center, the set of all vertices of minimum eccentricity ...
and the
commutator subgroup
In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.
The commutator subgroup is important because it is the smallest normal ...
of ''D'', and
# amongst all such homomorphisms, this ''D'' has maximal size.
The
Schur multiplier of ''G'' is the kernel of any Schur cover and has many interpretations. When the homomorphism is understood, the group ''D'' is often called the Schur cover or Darstellungsgruppe.
The Schur covers of the symmetric and alternating groups were classified in . The symmetric group of degree ''n'' ≥ 4 has
Schur covers of order 2⋅''n''! There are two isomorphism classes if ''n'' ≠ 6 and one isomorphism class if ''n'' = 6.
The alternating group of degree ''n'' has one isomorphism class of Schur cover, which has order ''n''! except when ''n'' is 6 or 7, in which case the Schur cover has order 3⋅''n''!.
Finite presentations
Schur covers can be described using
finite presentations. The symmetric group ''S''
''n'' has a presentation on ''n''−1 generators ''t''
''i'' for ''i'' = 1, 2, ..., n−1 and relations
:''t''
''i''''t''
''i'' = 1, for 1 ≤ ''i'' ≤ ''n''−1
:''t''
''i''+1''t''
''i''''t''
''i''+1 = ''t''
''i''''t''
''i''+1''t''
''i'', for 1 ≤ ''i'' ≤ ''n''−2
:''t''
''j''''t''
''i'' = ''t''
''i''''t''
''j'', for 1 ≤ ''i'' < ''i''+2 ≤ ''j'' ≤ ''n''−1.
These relations can be used to describe two non-isomorphic covers of the symmetric group. One covering group
has generators ''z'', ''t''
1, ..., ''t''
''n''−1 and relations:
:''zz'' = 1
:''t''
''i''''t''
''i'' = ''z'', for 1 ≤ ''i'' ≤ ''n''−1
:''t''
''i''+1''t''
''i''''t''
''i''+1 = ''t''
''i''''t''
''i''+1''t''
''i'', for 1 ≤ ''i'' ≤ ''n''−2
:''t''
''j''''t''
''i'' = ''t''
''i''''t''
''j''''z'', for 1 ≤ ''i'' < ''i''+2 ≤ ''j'' ≤ ''n''−1.
The same group
can be given the following presentation using the generators ''z'' and ''s''
''i'' given by ''t''
''i'' or ''t''
''i''''z'' according as ''i'' is odd or even:
:''zz'' = 1
:''s''
''i''''s''
''i'' = ''z'', for 1 ≤ ''i'' ≤ ''n''−1
:''s''
''i''+1''s''
''i''''s''
''i''+1 = ''s''
''i''''s''
''i''+1''s''
''i''''z'', for 1 ≤ ''i'' ≤ ''n''−2
:''s''
''j''''s''
''i'' = ''s''
''i''''s''
''j''''z'', for 1 ≤ ''i'' < ''i''+2 ≤ ''j'' ≤ ''n''−1.
The other covering group
has generators ''z'', ''t''
1, ..., ''t''
''n''−1 and relations:
:''zz'' = 1, ''zt''
''i'' = ''t''
''i''''z'', for 1 ≤ ''i'' ≤ ''n''−1
:''t''
''i''''t''
''i'' = 1, for 1 ≤ ''i'' ≤ ''n''−1
:''t''
''i''+1''t''
''i''''t''
''i''+1 = ''t''
''i''''t''
''i''+1''t''
''i''''z'', for 1 ≤ ''i'' ≤ ''n''−2
:''t''
''j''''t''
''i'' = ''t''
''i''''t''
''j''''z'', for 1 ≤ ''i'' < ''i''+2 ≤ ''j'' ≤ ''n''−1.
The same group
can be given the following presentation using the generators ''z'' and ''s''
''i'' given by ''t''
''i'' or ''t''
''i''''z'' according as ''i'' is odd or even:
:''zz'' = 1, ''zs''
''i'' = ''s''
''i''''z'', for 1 ≤ ''i'' ≤ ''n''−1
:''s''
''i''''s''
''i'' = 1, for 1 ≤ ''i'' ≤ ''n''−1
:''s''
''i''+1''s''
''i''''s''
''i''+1 = ''s''
''i''''s''
''i''+1''s''
''i'', for 1 ≤ ''i'' ≤ ''n''−2
:''s''
''j''''s''
''i'' = ''s''
''i''''s''
''j''''z'', for 1 ≤ ''i'' < ''i''+2 ≤ ''j'' ≤ ''n''−1.
Sometimes all of the relations of the symmetric group are expressed as (''t''
''i''''t''
''j'')
''m''''ij'' = 1, where ''m''
''ij'' are non-negative integers, namely ''m''
''ii'' = 1, ''m''
''i'',''i''+1 = 3, and ''m''
''ij'' = 2, for 1 ≤ ''i'' < ''i''+2 ≤ ''j'' ≤ ''n''−1. The presentation of
becomes particularly simple in this form: (''t''
''i''''t''
''j'')
''m''''ij'' = ''z'', and ''zz'' = 1. The group
has the nice property that its generators all have order 2.
Projective representations
Covering group
In mathematics, a covering group of a topological group ''H'' is a covering space ''G'' of ''H'' such that ''G'' is a topological group and the covering map is a continuous group homomorphism. The map ''p'' is called the covering homomorphism. ...
s were introduced by
Issai Schur
Issai Schur (10 January 1875 – 10 January 1941) was a Russian mathematician who worked in Germany for most of his life. He studied at the University of Berlin. He obtained his doctorate in 1901, became lecturer in 1903 and, after a stay at ...
to classify
projective representation In the field of representation theory in mathematics, a projective representation of a group ''G'' on a vector space ''V'' over a field ''F'' is a group homomorphism from ''G'' to the projective linear group
\mathrm(V) = \mathrm(V) / F^*,
where ...
s of groups. A (complex)
''linear'' representation of a group ''G'' is a
group homomorphism ''G'' → GL(''n'',''C'') from the group ''G'' to a
general linear group, while a ''projective'' representation is a homomorphism ''G'' → PGL(''n'',''C'') from ''G'' to a
projective linear group. Projective representations of ''G'' correspond naturally to linear representations of the covering group of ''G''.
The projective representations of alternating and symmetric groups are the subject of the book .
Integral homology
Covering groups correspond to the second
group homology group, H
2(''G'',Z), also known as the
Schur multiplier. The Schur multipliers of the alternating groups ''A''
''n'' (in the case where ''n'' is at least 4) are the cyclic groups of order 2, except in the case where ''n'' is either 6 or 7, in which case there is also a triple cover. In these cases, then, the Schur multiplier is the cyclic group of order 6, and the covering group is a 6-fold cover.
:H
2(''A''
''n'',Z) = 0 for ''n'' ≤ 3
:H
2(''A''
''n'',Z) = Z/2Z for ''n'' = 4, 5
:H
2(''A''
''n'',Z) = Z/6Z for ''n'' = 6, 7
:H
2(''A''
''n'',Z) = Z/2Z for ''n'' ≥ 8
For the symmetric group, the Schur multiplier vanishes for n ≤ 3, and is the cyclic group of order 2 for n ≥ 4:
:H
2(''S''
''n'',Z) = 0 for ''n'' ≤ 3
:H
2(''S''
''n'',Z) = Z/2Z for ''n'' ≥ 4
Construction of double covers
The double covers can be constructed as spin (respectively, pin) covers of faithful, irreducible, linear representations of ''A''
''n'' and ''S''
''n''. These spin representations exist for all ''n,'' but are the covering groups only for n≥4 (n≠6,7 for ''A''
''n''). For ''n''≤3, ''S''
''n'' and ''A''
''n'' are their own Schur covers.
Explicitly, ''S''
''n'' acts on the ''n''-dimensional space R
''n'' by permuting coordinates (in matrices, as
permutation matrices). This has a 1-dimensional
trivial subrepresentation corresponding to vectors with all coordinates equal, and the complementary (''n''−1)-dimensional subrepresentation (of vectors whose coordinates sum to 0) is irreducible for n≥4. Geometrically, this is the symmetries of the (''n''−1)-
simplex, and algebraically, it yields maps
and
expressing these as discrete subgroups (
point groups). The special orthogonal group has a 2-fold cover by the
spin group
In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when )
:1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1.
As a ...
and restricting this cover to
and taking the preimage yields a 2-fold cover
A similar construction with a
pin group
The PIN Group was a German courier and postal services company. It belonged to PIN Group S.A., a Luxembourg-based corporate affiliation made up of several German postal companies.
History and shareholding
The PIN Group originally traded under ...
yields the 2-fold cover of the symmetric group:
As there are two pin groups, there are two distinct 2-fold covers of the symmetric group, 2⋅''S''
''n''±, also called
and
.
Construction of triple cover for ''n'' = 6, 7
The triple covering of
denoted
and the corresponding triple cover of
denoted
can be constructed as symmetries of a certain set of vectors in a complex 6-space. While the exceptional triple covers of ''A''
6 and ''A''
7 extend to
extensions
Extension, extend or extended may refer to:
Mathematics
Logic or set theory
* Axiom of extensionality
* Extensible cardinal
* Extension (model theory)
* Extension (predicate logic), the set of tuples of values that satisfy the predicate
* Ex ...
of ''S''
6 and ''S''
7, these extensions are not
central
Central is an adjective usually referring to being in the center of some place or (mathematical) object.
Central may also refer to:
Directions and generalised locations
* Central Africa, a region in the centre of Africa continent, also known a ...
and so do not form Schur covers.
This construction is important in the study of the
sporadic groups, and in much of the exceptional behavior of small classical and exceptional groups, including: construction of the Mathieu group M
24, the exceptional covers of the
projective unitary group and the
projective special linear group
In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space ''V'' on the associa ...
and the exceptional double cover of the
group of Lie type .
Exceptional isomorphisms
For low dimensions there are
exceptional isomorphisms with the map from a
special linear group over a
finite field to the
projective special linear group
In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space ''V'' on the associa ...
.
For ''n'' = 3, the symmetric group is SL(2,2) ≅ PSL(2,2) and is its own Schur cover.
For ''n'' = 4, the Schur cover of the alternating group is given by SL(2,3) → PSL(2,3) ≅ ''A''
4, which can also be thought of as the
binary tetrahedral group
In mathematics, the binary tetrahedral group, denoted 2T or , Coxeter&Moser: Generators and Relations for discrete groups: : Rl = Sm = Tn = RST is a certain nonabelian group of order 24. It is an extension of the tetrahedral group T or (2,3,3) ...
covering the
tetrahedral group
150px, A regular tetrahedron, an example of a solid with full tetrahedral symmetry
A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection a ...
. Similarly, GL(2,3) → PGL(2,3) ≅ ''S''
4 is a Schur cover, but there is a second non-isomorphic Schur cover of ''S''
4 contained in GL(2,9) – note that 9=3
2 so this is
extension of scalars of GL(2,3). In terms of the above presentations, GL(2,3) ≅ ''Ŝ''
4