The relative Burnside kernel - the elementary abelian case
We give a conjectural description for the kernel of the map assigning to each finite -free -set its rational permutation module where is a finite -group. We prove that this conjecture is true when is an elementary abelian -group or a cyclic -group.
Given a prime and a finite -group , there is a natural map between the Burnside ring and the rational representation ring taking finite -sets to their permutation modules . It was shown by J. Ritter and G. Segal [5, 6] that when is a -group this map is a surjection and we can study the connection between and by looking at generators for the kernel denoted of the map. These generators were shown by Tornehave in  to always be induced by “small” subquotients of .
From a topological viewpoint we have the Segal conjecture proved by G. Carlsson in  which states that the ring is isomorphic up to completion to the stable cohomotopy group of . In addition it was shown by M. Atiyah in  that the complex representation ring is isomorphic up to completion to the complex -theory of the classifying space . Thus we gain a connection between stable cohomotopy theory and complex -theory of which can be studied by purely algebraically means.
In a generalization of these ideas, we can consider for a finite -group , the natural map between the relative Burnside ring and the relative rational representation ring which takes finite, -free, -sets to the -free, permutation -modules . We give their definition in section 2.2. It was show by M. Anton in  that this map is surjective.
The purpose of this paper is to give a conjecture describing the kernel denoted of the map in the relative case and to prove the conjecture for some specific -groups . We begin in section 2 by giving a brief overview of the classical case, including the construction of induction maps. In section 3 we consider the case when is an elementary abelian -group. In this case we decompose both the classical and relative Burnside rings into graded modules and compute their ranks. The main theorem is given in section 3.3 and describes the generators of the relative kernel . In section 4 we offer a conjecture for describing the kernel of the relative map for any -group and offer proofs in the cases of a cyclic -group and an elementary abelian -group.
Acknowledgments I am grateful to the department at the University of Kentucky for funding over the summer to work on this problem and to Dr. M. Anton for his enlightening discussions.
2.1. Burnside and representation rings
For a finite group the isomorphism classes of finite -sets form a semiring with respect to disjoint union and direct product. The Burnside ring is defined to be the Grothendieck construction of the semiring . In fact is a free -module with a basis given by the set of left coset spaces where runs thru conjugacy class representatives of subgroups . For each such subgroup we define an induction map by sending an -set to the -set where for all in . This definition extends to induction maps via the pullback map where is a subquotient of . The induction maps are -linear but do not preserve the product.
Likewise let be the semiring of isomorphism classes of finitely generated -modules with respect to direct sum and tensor product. The rational representation ring is defined to be the Grothendieck construction of . For , the induction map sends a -module to the -module and extends to subquotients as in the Burnside ring case.
The Burnside and representation rings are related by a natural ring homomorphism sending a -set to the permutation -module . It is immediate that commutes with the induction maps.
The Burnside kernel is the kernel of the map .
2.2. Relative Burnside and representation modules
If is a direct product of two finite groups then a -set is thought of with acting on the left and on the right. Let be the monoid of isomorphism classes of finite -free -sets with respect to disjoint union. The relative Burnside module is the Grothendieck construction of the monoid . Then is a free -submodule with a basis given by twisted products where runs thru conjugacy class representatives of homomorphisms with and for all in .
Similarly, a -module is thought of with acting on the left and on the right. Let be the monoid of isomorphism classes of finitely generated -free -modules with respect to direct sum. The relative rational representation module is the Grothendieck construction of the monoid . Then the natural ring homomorphism will restrict to a module homomorphism .
The relative Burnside kernel is the kernel of .
2.3. Relative induction
The relative induction maps are defined by the usual induction restricted to the submodule made of those elements of that land in where is a subquotient of . The same observation applies to the relative induction map . It is immediate that the natural map from the relative Burnside module to the relative representation module commutes with the relative induction maps. We conclude by proving from scratch the following
All induction maps are injective.
Let be any of the monoids , , , defining the Burnside and representation modules and their relative versions. The induction map is defined by a homomorphism of monoids extended to the Grothendieck constructions where is a subquotient of . The Grothendieck construction of consists of fomal differences of elements in such that if and only if
for some in . In particular, if then
for some in . By restricting the -structure to an -structure we have a restriction map such that . In particular,
Since each element of has a unique decomposition into a sum of irreducible elements, we conclude that proving the injectivity of the induction map. ∎
2.4. The -group case
For a finite -group it was shown by Tornehave  that is generated by the induced kernels where runs thru all subquotients of that are isomorphic to the elementary abelian group , the dihedral group, or the nonabelian group of order and exponent . Combining this with the Ritter-Segal [5, 6] proof for the surjectivity of we get a well understood short exact sequence:
In the abelian case for instance, it is shown in  that is the free cyclic group generated by
where runs thru all proper cyclic subgroups of . In the relative case with a finite -group and it is known by  only that we have a short exact sequence
and the purpose of this paper is to study .
2.5. A useful trick
Consider the chain complex of finitely generated free -modules
with injective and surjective. If the cokernel of is a free module and the rank of the image of equals the rank of the kernel of , then the sequence is exact.
Since and is free, we have the free -submodule . But the rank of the image of equals the rank of the kernel of so that is torsion. Therefore . ∎
3. The Relative Burnside Kernel for Elementary Abelian Groups
In this section let and so that . Also, we denote
and define to be generated by all with of dimension . Thus,
and a similar decomposition holds for with .
3.2. Rank calculations
Let denote the number of -dimensional subspaces of the vector space . Then by [7, p. 28] we have
The ranks and of and are given by the formulas
The basis elements for are in one-to-one correspondence with the -dimensional subspaces . Hence, we get the first formula.
The basis elements for are in one-to-one correspondence with pairs with a -dimensional subspace and a homomorphism. Given , is uniquely determined by its kernel and an automorphism of its image. If is -dimensional, the kernel of is either or any -dimensional subspace of . In the later case the image admits automorphisms. Hence, for a given -dimensional there are different ’s. For a given dimension the number of pairs is thus given by the formula:
Let denote a primitive -root of unity and be the associated cyclotomic field. For each let be the -module obtained by letting the canonical generator of act on via the automorphism sending to where is the coordinate of .
The ranks and of and are given by the formulas
With the above notations, two isomorphism classes are equal if and only if or for some unit in . In the later case we say that and represent the same point in the projective -space over . With this observation indexed by and the trivial module form a basis for . Thus we get the first formula.
For the second formula we claim that a basis for is given by the elements
indexed by and . Let denote the set of these elements and the -module generated by . Since it follows that by forgetting the -action, the elements:
are all represented by the -free modules or . Thus and it is immediate that is a linearly independent set. Now by inspection is the free module generated by and implies . Thus the rank of equals the rank of . In particular Lemma 2.4 applies to the sequence:
implying that . ∎
The ranks and of and are given by the formulas
3.3. The main theorem
It is convenient to identify each basis element of where with the projective subspace generated by . Also, let denote the distinguished vector . Then we have the following characterization for the basis elements of :
The submodule is the free abelian group on the set of projective subspaces with of dimension not containing .
It is easy to see that the basis elements of associated with a pair is of the form where , is a homomorphism, and is a linear subspace of not containing .
Conversely, let with of dimension not containing and define to be the image of the canonical projection . If is an element in which maps to 0 under the projection, then . This would imply so . Thus the projection induces an isomorphism . Let be the inverse and define by composing with the canonical projection . We can then check that . ∎
Given a subspace of codimension at least 2 in we define to be a distinguished subspace such that the following two conditions are both satisfied:
contains and has rank 2
If does not contain and has codimension at least 3 then does not contain
Now we observe that always exists subject to the two conditions. In particular, if has codimension exactly 2 then is the only choice without violating condition 2.
For each such define
where the sum is over all proper subspaces .
In particular, define to be the set of all with an -dimensional subpace where . By Lemma 3.4, this set is also a basis for
Let be the submodule of generated by all those differences of elements in that are subject to the relation
The rank of is and we have the following commutative diagram of short exact sequences:
where the vertical arrows are all inclusions and , are the restrictions of , .
Given in ,
where and the sum is taken over all not containing . By Definition 3.6, if is a generator of then
From subsection 2.4 we know that the kernel of is generated by the induced kernels where . In particular, by applying to equation (2) of Section 2.4 with we deduce that is generated by elements of the form
where is any subquotient of isomorphic to and the sum runs over all proper subgroups . By Definition 3.5 the above elements with replaced by generate the image of so that the composition and are both zero.
We are left to prove the injectivity of the map and the inclusions of the kernels of and inside the images of and respectively. Under the map , each basis element of is mapped to an element inside whose first component is again . Therefore the matrix representation of is upper triangular with cokernel which is free. Hence , and therefore , are injective.
Regarding the exactness at observe that by Proposition 3.1 and the injectivity of it follows that the rank of is the sum for . The same sum by Corollary 3.3 is the rank of the kernel of . Since the cokernel of is a free module we conclude by Lemma 2.4 that the bottom sequence is exact at .
To determine exactness at we must first determine the rank of . As calculated in Proposition 3.1, the rank of which is also the order of is equal to . We observe that breaks into equivalence classes relative to the equivalence relation if and only if
By Lemma 3.4 and Proposition 3.1 with replaced by , each -subspace containing contains subspaces of dimension not containing . This is the number of elements in any of the equivalence classes. Hence, since each equivalence class produces basis elements for and there are equivalence classes, we conclude that the rank of is given by the formula
Combining this with the fact that is injective it follows that the image of has rank equal to the kernel of . Moreover when considering the generators of , if we allow any given basis element to play the role of an in the difference at most once, then we see that the matrix of will be upper triangular as maps a difference to an element inside with first component . Therefore the cokernel of is a free module and by Lemma 2.4, the top row is exact.
3.4. An illustration for and
Order such that and order lexicographically. Then for we gain a labeling of the basis of ) such that . With this labeling of the basis of , the subgroup lattice of can be represented by the graph below and offers a visual description of the relationship between basis elements and .
Theorem 3.7 implies that we have the commutative diagram of short exact sequences:
We see using our basis that
Hence is well defined on while we define:
Define the subgraph to be the full subgraph of where the vertices are the terms occurring in . Then the image is associated with the subgraph whose vertices are those in and . For example, if and these subgraphs are:
Conversely, given a subgraph the image is uniquely determined by taking a weighted sum of the vertices of . Moreover, given a subgraph , the image is also uniquely determined by the vertices of . It follows that the kernel of is generated by all of the subgraphs for and the kernel of is generated by all the non-singular subgraphs .
4. Final Remarks
We would like to develop a description for the kernel with similar to that given by Tornehave in  for with an arbitrary finite -groups . Define for a subquotient of to be the intersection of with the submodule of that lands inside under the induction of Section 2.3.
Let be a prime, any finite -group, and . Then
where the sum is taken over subquotients of isomorphic to where is the elementary abelian group , the dihedral group, or the nonabelian group of order and exponent .
For elementary abelian or cyclic this conjecture can readily be checked using Theorem 3.7 and rank arguments.
Let be any prime, be an elementary abelian -group, and . Then
with the sum taken over all subquotients .
From Theorem 3.7 we know that the image of generates the kernel . If with , then there exists subgroups such that where is the distinguished element used to define in Definition 3.5. In addition, regardless of our choice of ,
If , let
We see immediately that and also that is an element of . Hence we conclude that
The converse is immediate. ∎
Let be any prime, the cyclic -group with order , and . Then is an isomorphism between and .
Let and be the primitive -root of unity. Since is cyclic, easily the rank of is equal to as has subgroups and for a nontrivial subgroup , there are homomorphisms . Let be the -module with the generators of and acting by multiplication by and respectively where and . Then the irreducible -modules as seen from the decomposition of the group ring are:
This implies the rank of is .
For , define if we have . Using this relation we immediately gain the following equivalences from :
The equivalences imply that the rank of is less than or equal to 1. In addition, since is surjective, the rank of is , and the rank of is , we see that the rank of is at least 1. Thus the rank of is exactly 1 which implies the rank of . As is a free module, the rank of is equal to the rank of , and is a surjection, we see that is an isomorphism.
As a corollary, Conjecture 4.1 is true for a cyclic -group.
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