In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Burnside ring of a
finite group
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or ma ...
is an algebraic construction that encodes the different ways the group can
act on finite sets. The ideas were introduced by
William Burnside
:''This English mathematician is sometimes confused with the Irish mathematician William S. Burnside (1839–1920).''
__NOTOC__
William Burnside (2 July 1852 – 21 August 1927) was an English mathematician. He is known mostly as an early res ...
at the end of the nineteenth century. The algebraic
ring structure is a more recent development, due to Solomon (1967).
Formal definition
Given a
finite group
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or ma ...
''G'', the generators of its Burnside ring Ω(''G'') are the formal sums of isomorphism classes of finite
''G''-sets. For the
ring structure, addition is given by
disjoint union of ''G''-sets and multiplication by their
Cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is
: A\t ...
.
The Burnside ring is a free Z-
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Modul ...
, whose generators are the (isomorphism classes of)
orbit types of ''G''.
If ''G'' acts on a finite set ''X'', then one can write
(disjoint union), where each ''X''
''i'' is a single ''G''-orbit. Choosing any element ''x''
''i'' in ''X''
i creates an isomorphism ''G''/''G''
''i'' → ''X''
''i'', where ''G
i'' is the stabilizer (isotropy) subgroup of ''G'' at ''x''
''i''. A different choice of representative ''y''
''i'' in ''X''
''i'' gives a conjugate subgroup to ''G''
''i'' as stabilizer. This shows that the generators of Ω(''G'') as a Z-module are the orbits ''G''/''H'' as ''H'' ranges over
conjugacy classes of subgroups of ''G''.
In other words, a typical element of Ω(''G'') is
where ''a''
''i'' in Z and ''G''
1, ''G''
2, ..., ''G''
''N'' are representatives of the conjugacy classes of subgroups of ''G''.
Marks
Much as
character theory simplifies working with
group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used ...
s, marks simplify working with
permutation representation
In mathematics, the term permutation representation of a (typically finite) group G can refer to either of two closely related notions: a representation of G as a group of permutations, or as a group of permutation matrices. The term also refers ...
s and the Burnside ring.
If ''G'' acts on ''X'', and ''H'' ≤ ''G'' (''H'' is a
subgroup of ''G''), then the mark of ''H'' on ''X'' is the number of elements of ''X'' that are fixed by every element of ''H'':
, where
:
If ''H'' and ''K'' are conjugate subgroups, then ''m''
''X''(''H'') = ''m''
''X''(''K'') for any finite ''G''-set ''X''; indeed, if ''K'' = ''gHg''
−1 then ''X''
''K'' = ''g'' · ''X''
''H''.
It is also easy to see that for each ''H'' ≤ ''G'', the map ''Ω''(''G'') → Z : ''X'' ↦ ''m''
''X''(''H'') is a homomorphism. This means that to know the marks of ''G'', it is sufficient to evaluate them on the generators of ''Ω''(''G''), ''viz.'' the orbits ''G''/''H''.
For each pair of subgroups ''H'',''K'' ≤ ''G'' define
:
This is ''m''
''X''(''H'') for ''X'' = ''G''/''K''. The condition ''HgK'' = ''gK'' is equivalent to ''g''
−1''Hg'' ≤ ''K'', so if ''H'' is not conjugate to a subgroup of ''K'' then ''m''(''K'', ''H'') = 0.
To record all possible marks, one forms a table, Burnside's Table of Marks, as follows: Let ''G''
1 (= trivial subgroup), ''G''
2, ..., ''G''
''N'' = ''G'' be representatives of the ''N'' conjugacy classes of subgroups of ''G'', ordered in such a way that whenever ''G''
''i'' is conjugate to a subgroup of ''G''
''j'', then ''i'' ≤ ''j''. Now define the ''N'' × ''N'' table (square matrix) whose (''i'', ''j'')th entry is ''m''(''G''
''i'', ''G''
''j''). This matrix is lower triangular, and the elements on the diagonal are non-zero so it is invertible.
It follows that if ''X'' is a ''G''-set, and u its row vector of marks, so ''u''
''i'' = ''m''
''X''(''G''
''i''), then ''X'' decomposes as a
disjoint union of ''a''
''i'' copies of the orbit of type ''G''
''i'', where the vector a satisfies,
:a''M'' = ''u'',
where ''M'' is the matrix of the table of marks. This theorem is due to .
Examples
The table of marks for the cyclic group of order 6:
The table of marks for the symmetric group ''S
3'':
The dots in the two tables are all zeros, merely emphasizing the fact that the tables are lower-triangular.
(Some authors use the transpose of the table, but this is how Burnside defined it originally.)
The fact that the last row is all 1s is because
'G''/''G''is a single point. The diagonal terms are ''m''(''H'', ''H'') = , ''N''
''G''(''H'')/''H'' , . The numbers in the first column show the degree of the representation.
The ring structure of ''Ω''(''G'') can be deduced from these tables: the generators of the ring (as a Z-module) are the rows of the table, and the product of two generators has mark given by the product of the marks (so component-wise multiplication of row vectors), which can then be decomposed as a
linear combination of all the rows. For example, with ''S''
3,
:
as (3, 1, 0, 0).(2, 0, 2, 0) = (6, 0, 0, 0).
Permutation representations
Associated to any finite set ''X'' is a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
''V = V
X'', which is the vector space with the elements of ''X'' as the basis (using any specified field). An action of a finite group ''G'' on ''X'' induces a linear action on ''V'', called a permutation
representation. The set of all finite-dimensional representations of ''G'' has the structure of a ring, the
representation ring, denoted ''R(G)''.
For a given ''G''-set ''X'', the
character
Character or Characters may refer to:
Arts, entertainment, and media Literature
* ''Character'' (novel), a 1936 Dutch novel by Ferdinand Bordewijk
* ''Characters'' (Theophrastus), a classical Greek set of character sketches attributed to The ...
of the associated representation is
:
where
is the cyclic group generated by
.
The resulting map
:
taking a ''G''-set to the corresponding representation is in general neither injective nor surjective.
The simplest example showing that β is not in general injective is for ''G = S
3'' (see table above), and is given by
:
Extensions
The Burnside ring for
compact groups is described in .
The
Segal conjecture relates the Burnside ring to
homotopy
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deform ...
.
See also
*
Burnside category In category theory and homotopy theory the Burnside category of a finite group ''G'' is a category whose objects are finite ''G''-sets and whose morphisms are (equivalence classes of) spans of ''G''-equivariant maps. It is a categorification of t ...
References
*
*
*
*
*{{Citation , last1=Solomon , first1= L. , title=The Burnside algebra of a finite group , journal=J. Comb. Theory , year=1967 , pages=603–615 , volume=1
Group theory
Finite groups
Permutation groups
Representation theory of groups