Lifts of Brauer Characters of Solvable Groupsmath.uakron.edu/~cossey/Akron talk.pdfFor a group G, many representations of G over a eld F are essentially the same. De nition If Xand

Representations of finite groupsBrauer characters and the Fong-Swan theorem

New results and open questions

Lifts of Brauer Characters of Solvable Groups

J.P. CosseyUniversity of Arizona

January 25, 2008

J.P. Cossey University of Arizona Lifts of Brauer Characters of Solvable Groups



The basicsThe important definitionsAn exampleTwo paths diverge...

Throughout, all of our groups will be finite, and all of our fieldswill be algebraically closed.

DefinitionLet G be a group and F a field. A representation X of G over Fof degree n is a homomorphism

X : G → GLn(F).





Throughout, all of our groups will be finite, and all of our fieldswill be algebraically closed.

DefinitionLet G be a group and F a field. A representation X of G over Fof degree n is a homomorphism

X : G → GLn(F).





Why should we care about representations?

I Representations are interesting in their own right (McKayconjecture, Alperin weight conjecture).

I Representations are useful in studying groups (odd ordertheorem, classification of finite simple groups).

I Representations of finite groups arise in number theory,geometry, combinatorics, and elsewhere.

I They have ”real world” applications to cell phone design andcard shuffling problems.









































Before we move on, we’ll need to give an equivalent version of thedefinition of a representation.

Notice that by our definition, each element g ∈ G induces aninvertible linear transformation X (g) on the vector space Fn. If Gis a group and V is a vector space over F, then we say V is aG -module (or abuse notation and say V is a representation) if eachelement of G induces a linear action on V in such a way that theidentity element acts trivially and group multiplication correspondsto composition of transformations.





Before we move on, we’ll need to give an equivalent version of thedefinition of a representation.

Notice that by our definition, each element g ∈ G induces aninvertible linear transformation X (g) on the vector space Fn. If Gis a group and V is a vector space over F, then we say V is aG -module (or abuse notation and say V is a representation) if eachelement of G induces a linear action on V in such a way that theidentity element acts trivially and group multiplication correspondsto composition of transformations.





For a group G , many representations of G over a field F areessentially the same.

DefinitionIf X and Y are representations of the group G over the field F, wesay that X and Y are equivalent if there exists an invertiblematrix M such that, for every g ∈ G , we have

M−1X (g)M = Y(g).

Equivalently, we say two G -modules V and W are equivalent ifthere exists an invertible linear map T : V →W such thatT (vg) = T (v)g for all g ∈ G and v ∈ V .







M−1X (g)M = Y(g).








M−1X (g)M = Y(g).






Most representations break into pieces. We now define the”smallest” representations.

DefinitionA representation V of G is irreducible if there does not exist aproper nontrivial subspace W of V such that W is also arepresentation of G .

Notice that this is equivalent to saying that given a representationX : G → GLn(F), one cannot find a basis for Fn such that X canbe written in block diagonal form with representations on thediagonal, i.e. X is reducible if, for all g ∈ G , X can be written as

X (g) =

(Y1(g) ∗

0 Y2(g)

),

where Y1 and Y2 are representations of G .








X (g) =

(Y1(g) ∗

0 Y2(g)

),









X (g) =

(Y1(g) ∗

0 Y2(g)

),






Combining the above two definitions, we can see that theequivalence classes of irreducible representations of G form the”building blocks” for all of the representations of G .

We thus have the following very important classical result: For agroup G , the number of equivalence classes of irreduciblerepresentations over the complex numbers is equal to the numberof conjugacy classes of G .

As we shall see later, a similar result is true for fields ofcharacteristic p.



















We need one more definition, for now.

DefinitionGiven a representation X of G over a field F, the associatedcharacter

χ : G → F

is defined by χ(g) = trace(X (g)).

Notice that by definition, if X and Y are equivalentrepresentations, then X and Y give the same character. Moreover,the value of the character χ is constant on conjugacy classes. Wedenote the set of characters of G that come from irreduciblerepresentations over C by Irr(G ), the ordinary irreduciblecharacters of G .





We need one more definition, for now.

DefinitionGiven a representation X of G over a field F, the associatedcharacter

χ : G → F

is defined by χ(g) = trace(X (g)).

Notice that by definition, if X and Y are equivalentrepresentations, then X and Y give the same character. Moreover,the value of the character χ is constant on conjugacy classes. Wedenote the set of characters of G that come from irreduciblerepresentations over C by Irr(G ), the ordinary irreduciblecharacters of G .





We now look at an example: Let G = Sn, and let {e1, e2, . . . , en}be the standard basis for Cn.

Let Sn act on Cn in the natural way: if

v = α1e1 + · · ·+ αnen,

then for π ∈ Sn,

v · π = α1eπ(1) + · · ·+ αneπ(n).

For instance, if n = 4, then under this representation,

(143) −→

0 0 0 10 1 0 01 0 0 00 0 1 0





We now look at an example: Let G = Sn, and let {e1, e2, . . . , en}be the standard basis for Cn.Let Sn act on Cn in the natural way: if

v = α1e1 + · · ·+ αnen,

then for π ∈ Sn,

v · π = α1eπ(1) + · · ·+ αneπ(n).


(143) −→

0 0 0 10 1 0 01 0 0 00 0 1 0





We now look at an example: Let G = Sn, and let {e1, e2, . . . , en}be the standard basis for Cn.Let Sn act on Cn in the natural way: if

v = α1e1 + · · ·+ αnen,

then for π ∈ Sn,

v · π = α1eπ(1) + · · ·+ αneπ(n).


(143) −→

0 0 0 10 1 0 01 0 0 00 0 1 0





This representation is not irreducible. For instance, the subspace

W = {α1e1 + · · ·+ αnen|α1 + · · ·+ αn = 0}

is a proper subspace of V that is also a representation.

In fact, it turns out that V = W ⊕ X , where W is as above and

X = {α1e1 + · · ·+ αnen|α1 = α2 = . . . = αn},

is a decomposition of V into irreducible representations.





This representation is not irreducible. For instance, the subspace

W = {α1e1 + · · ·+ αnen|α1 + · · ·+ αn = 0}

is a proper subspace of V that is also a representation.

In fact, it turns out that V = W ⊕ X , where W is as above and

X = {α1e1 + · · ·+ αnen|α1 = α2 = . . . = αn},

is a decomposition of V into irreducible representations.





At this point, we have to start talking about specific kinds of finitegroups. Most of my research is on solvable groups, so I will focuson those.

However, I should mention that many of the topics discussed belowapply equally well to the symmetric group, which I also work on.The methods used in studying these issues in the symmetric groupare different and very combinatorial in nature (and are the subjectof another talk for another day).





At this point, we have to start talking about specific kinds of finitegroups. Most of my research is on solvable groups, so I will focuson those.

However, I should mention that many of the topics discussed belowapply equally well to the symmetric group, which I also work on.The methods used in studying these issues in the symmetric groupare different and very combinatorial in nature (and are the subjectof another talk for another day).




Brauer charactersThe Fong-Swan theoremConstruction of lifts

Given a group G , we will primarily be interested in the interplaybetween the representations of G over C and the representations ofG over Fp, an algebraically closed field of characteristic p.

Our main tool for doing this will be Brauer characters and theirlifts. In order to understand these, we first need to make couple ofobservations.





Given a group G , we will primarily be interested in the interplaybetween the representations of G over C and the representations ofG over Fp, an algebraically closed field of characteristic p.

Our main tool for doing this will be Brauer characters and theirlifts. In order to understand these, we first need to make couple ofobservations.





We first point out some useful facts. Throughout, p is a fixedprime, and Fp is an algebraically closed field of characteristic p.

I If X is a a representation of G over Fp, and g ∈ G has ordernot divisible by p, it can be shown that X (g) is similar to adiagonal matrix, and thus the trace of X (g) is equal to thesum of the eigenvalues of X (g), and the eigenvalues must benon-zero.

I If Fxp is the set of nonzero elements of Fp, and if U is the set

of roots of unity in C of order not divisible by p, then there isa multiplicative isomorphism ∗ from Fx

p to U.

Let G o denote the set of elements of G of order not divisible by p.









p to U.










p to U.










p to U.






We are now ready to define a Brauer character. Intuitively, aBrauer character ϕ is a complex-valued function that arises from acharacteristic p representation.

DefinitionLet G be a finite group, and X a representation of G over Fp ofdegree n. For g ∈ G o , let ξ1, ξ2, . . . , ξn be the eigenvalues ofX (g). Then the Brauer character ϕ : G → C corresponding to Xis defined by

ϕ(g) = ξ∗1 + · · ·+ ξ∗n.





We are now ready to define a Brauer character. Intuitively, aBrauer character ϕ is a complex-valued function that arises from acharacteristic p representation.

DefinitionLet G be a finite group, and X a representation of G over Fp ofdegree n. For g ∈ G o , let ξ1, ξ2, . . . , ξn be the eigenvalues ofX (g). Then the Brauer character ϕ : G → C corresponding to Xis defined by

ϕ(g) = ξ∗1 + · · ·+ ξ∗n.





Notice that a Brauer character behaves much like a regularcharacter. For instance, it is constant on conjugacy classes of G .Moreover, if ϕ is a Brauer character coming from an irreduciblerepresentation, then we say that ϕ is an irreducible Brauercharacter, and we denote the set of irreducible Brauer charactersby IBrp(G ).

In fact, the connection between Brauer characters and ordinarycharacters are much deeper.





Notice that a Brauer character behaves much like a regularcharacter. For instance, it is constant on conjugacy classes of G .Moreover, if ϕ is a Brauer character coming from an irreduciblerepresentation, then we say that ϕ is an irreducible Brauercharacter, and we denote the set of irreducible Brauer charactersby IBrp(G ).

In fact, the connection between Brauer characters and ordinarycharacters are much deeper.





Recall that there are as many ordinary irreducible characters asthere are conjugacy classes of G . In fact, the set Irr(G ) forms abasis for the vector space of complex-valued class functions on G .

Similarly, there are as many irreducible Brauer characters as thereare conjugacy classes in G o . Again, the set IBrp(G ) forms a basisfor the vector space of complex-valued class functions on G o .





Recall that there are as many ordinary irreducible characters asthere are conjugacy classes of G . In fact, the set Irr(G ) forms abasis for the vector space of complex-valued class functions on G .

Similarly, there are as many irreducible Brauer characters as thereare conjugacy classes in G o . Again, the set IBrp(G ) forms a basisfor the vector space of complex-valued class functions on G o .





Still more is true. For an ordinary character χ of G , we will denoteby χo the restriction of χ to G o . Since IBrp(G ) forms a basis forthe vector space of class functions on G o , then naturally

χo =∑

ϕ∈IBrp(G)

cχϕϕ.

However, the cχϕ turn out to be nonnegative integers.

The cχϕ are called the ”decomposition numbers” of G , and onewould like to understand them better.

In fact, our motivating question from now on is the following:When is it the case that χo = ϕ?





Still more is true. For an ordinary character χ of G , we will denoteby χo the restriction of χ to G o . Since IBrp(G ) forms a basis forthe vector space of class functions on G o , then naturally

χo =∑

ϕ∈IBrp(G)

cχϕϕ.

However, the cχϕ turn out to be nonnegative integers.

The cχϕ are called the ”decomposition numbers” of G , and onewould like to understand them better.

In fact, our motivating question from now on is the following:When is it the case that χo = ϕ?





DefinitionLet ϕ be an irreducible Brauer character. If there exists an ordinaryirreducible character χ ∈ Irr(G ) such that χo = ϕ, then we saythat χ is a lift of ϕ. We denote the set of lifts of ϕ by Lϕ, i.e.

Lϕ = {χ ∈ Irr(G )|χo = ϕ}.

A first natural question, then, is: Is the set Lϕ even nonempty?

In general, the answer is ”no”. For instance, in the symmetricgroup, ”most” Brauer characters do not have a lift.






Lϕ = {χ ∈ Irr(G )|χo = ϕ}.








Lϕ = {χ ∈ Irr(G )|χo = ϕ}.







However, in the case that G is solvable, (or, actually, p-solvable),the answer is ”yes”.

Theorem(Fong, Swan) Let G be solvable (or p-solvable), and letϕ ∈ IBrp(G ). Then there necessarily exists a lift of ϕ.





Let’s back up a little and recall what solvable means. Recall that achief series of G is a sequence of normal subgroups

1 = N0 ⊆ N1 ⊆ N2 ⊆ . . . ⊆ Nk = G

such that if M C G and Ni ⊆ M ⊆ Ni+1, then either M = Ni orM = Ni+1. The groups Ni+1/Ni are called the chief factors of G .

Of course, every group has a chief series. A group G is solvable ifevery chief factor of G is an elementary abelian group. A group isp-solvable if every chief factor is either a p-group or a p′-group.

Intuitively, a group is solvable if there are ”many” normalsubgroups, but not so many as to make things too easy (likenilpotent or abelian groups).






1 = N0 ⊆ N1 ⊆ N2 ⊆ . . . ⊆ Nk = G









1 = N0 ⊆ N1 ⊆ N2 ⊆ . . . ⊆ Nk = G








The first proof of the Fong-Swan theorem was not constructive.Through a series of papers in the 1980’s, my adviser I.M. Isaacsdeveloped a proof that was both constructive, and generalized theresult by replacing the prime p with a set of primes π.

We now briefly discuss Isaacs’ construction of a ”canonical” set oflifts. To do this, we’ll need a few definitions, and we’ll have towaive our hands at a few things along the way.





The first proof of the Fong-Swan theorem was not constructive.Through a series of papers in the 1980’s, my adviser I.M. Isaacsdeveloped a proof that was both constructive, and generalized theresult by replacing the prime p with a set of primes π.

We now briefly discuss Isaacs’ construction of a ”canonical” set oflifts. To do this, we’ll need a few definitions, and we’ll have towaive our hands at a few things along the way.





First, we’ll need the notion of p-special characters and p′-specialcharacters. Intuitively, a p-special character is a character that”looks like” a character of a p-group, and a p′-special character isa character that ”looks like” a character of a group of order notdivisible by p.

DefinitionLet G be a solvable group. A character α ∈ Irr(G ) is p′-special ifboth (i) α(1) is not divisible by p and (ii) for every subnormalsubgroup S C CG and every constituent γ of αS , thedeterminantal order of γ is not divisible by p.

A similar definition holds for p-special, replacing ”not divisible byp” with ”a power of p”.



















In general, when one multiplies two irreducible characters, onedoes not necessarily get an irreducible character. However,Gajendragadkar showed the following:

Theorem(Gajendragadkar, 1979) Suppose G is solvable, and let α be ap′-special character and let β be a p-special character. If χ = αβ,then χ is irreducible, and the factorization is unique: If χ = α1β1

with α1 p′-special and β1 p-special, then α = α1 and β = β1.

One more notion we’ll waive our hands at is inducing characters. IfH is a subgroup of G , and γ is a character of H, then one can”induce” γ to a character γG of G in a certain useful way. Even ifγ is an irreducible character of H, it is usually not the case that γG

is irreducible, though in certain cases it is.





In general, when one multiplies two irreducible characters, onedoes not necessarily get an irreducible character. However,Gajendragadkar showed the following:

Theorem(Gajendragadkar, 1979) Suppose G is solvable, and let α be ap′-special character and let β be a p-special character. If χ = αβ,then χ is irreducible, and the factorization is unique: If χ = α1β1

with α1 p′-special and β1 p-special, then α = α1 and β = β1.

One more notion we’ll waive our hands at is inducing characters. IfH is a subgroup of G , and γ is a character of H, then one can”induce” γ to a character γG of G in a certain useful way. Even ifγ is an irreducible character of H, it is usually not the case that γG

is irreducible, though in certain cases it is.





Rather than discussing Isaacs’ canonical lift construction, it willactually be slightly easier to discuss a similar construction ofNavarro. We begin by constructing the ”normal nucleus” of acharacter χ ∈ Irr(G ).

Our goal will be to attach to χ a subgroup W and a factoredcharacter αβ ∈ Irr(W ) such that χ = (αβ)G in a canonical way.





Rather than discussing Isaacs’ canonical lift construction, it willactually be slightly easier to discuss a similar construction ofNavarro. We begin by constructing the ”normal nucleus” of acharacter χ ∈ Irr(G ).

Our goal will be to attach to χ a subgroup W and a factoredcharacter αβ ∈ Irr(W ) such that χ = (αβ)G in a canonical way.





G χ

N θ0 = α0β0 maximal factorable normal ”pair” under χ

•T0 = Gθ ψ0 Clifford correspondent for χ

= ψG0

•N1 θ1 = α1β1 maximal (in T ) ”pair” under ψ0

•T1 = Tθ1 ψ1 Clifford correspondent for ψ0

= ψT01

= ψG1

•

•

•

•

•

•

•Tk = W ψk = αβ terminates with factorable character

= . . . = (αβ)G





G χ



= ψG0



= ψT01

= ψG1

•

•

•

•

•

•


= . . . = (αβ)G





G χ



= ψG0



= ψT01

= ψG1

•

•

•

•

•

•


= . . . = (αβ)G





G χ



= ψG0



= ψT01

= ψG1

•

•

•

•

•

•


= . . . = (αβ)G





G χ



= ψG0



= ψT01

= ψG1

•

•

•

•

•

•


= . . . = (αβ)G





G χ



= ψG0



= ψT01

= ψG1

•

•

•

•

•

•


= . . . = (αβ)G





G χ



= ψG0



= ψT01

= ψG1

•

•

•

•

•

•


= . . . = (αβ)G





G χ



= ψG0



= ψT01

= ψG1

•

•

•

•

•

•


= . . . = (αβ)G





G χ



= ψG0



= ψT01

= ψG1

•

•

•

•

•

•


= . . . = (αβ)G





G χ



= ψG0



= ψT01

= ψG1

•

•

•

•

•

•


= . . . = (αβ)G





G χ



= ψG0



= ψT01

= ψG1

•

•

•

•

•

•


= . . . = (αβ)G





G χ



= ψG0



= ψT01

= ψG1

•

•

•

•

•

•


= . . . = (αβ)G





G χ



= ψG0



= ψT01

= ψG1

•

•

•

•

•

•


= . . . = (αβ)G





G χ



= ψG0



= ψT01

= ψG1

•

•

•

•

•

•


= . . . = (αβ)G





G χ



= ψG0



= ψT01

= ψG1

•

•

•

•

•

•


= . . . = (αβ)G





G χ



= ψG0



= ψT01

= ψG1

•

•

•

•

•

•


= . . . = (αβ)G





To recap: For each ordinary irreducible character χ ∈ Irr(G ), wehave now associated a subgroup W and a factorable characterαβ ∈ Irr(W ) such that χ = (αβ)G .

Along the way, choices weremade (which constituent of the restriction to use?). It can beshown that, up to conjugacy, (W , αβ) is unique.

DefinitionThe pair (W , αβ) constructed above is the normal nucleus of χ.If the normal nucleus character αβ for χ is actually p′-special (i.e.β is trivial) then we say that χ ∈ Np′(G ).





To recap: For each ordinary irreducible character χ ∈ Irr(G ), wehave now associated a subgroup W and a factorable characterαβ ∈ Irr(W ) such that χ = (αβ)G . Along the way, choices weremade (which constituent of the restriction to use?). It can beshown that, up to conjugacy, (W , αβ) is unique.







DefinitionThe pair (W , αβ) constructed above is the normal nucleus of χ.

If the normal nucleus character αβ for χ is actually p′-special (i.e.β is trivial) then we say that χ ∈ Np′(G ).











We’ve done things a little chronologically out of order here.Originally, Isaacs gave a similar definition for the subnormalnucleus of the character χ, which followed the same general steps,though the proofs of the relevant details were more difficult.

A character χ with a subnormal nucleus (W , αβ) is in Bp′(G ) ifthe p-special character β is the trivial character.





We’ve done things a little chronologically out of order here.Originally, Isaacs gave a similar definition for the subnormalnucleus of the character χ, which followed the same general steps,though the proofs of the relevant details were more difficult.

A character χ with a subnormal nucleus (W , αβ) is in Bp′(G ) ifthe p-special character β is the trivial character.





We can now state two generalizations of the Fong-Swan Theorem:

Theorem(Isaacs, 1984) If G is a solvable (or p-solvable) group, then themap

χ→ χo

is a bijection from Bp′(G ) to IBrp(G ).

Theorem(Navarro, 2002) If G is a solvable (or p-solvable) group, then themap

χ→ χo

is a bijection from Np′(G ) to IBrp(G ).







χ→ χo



χ→ χo








χ→ χo



χ→ χo





Sets of liftsLower boundsUpper boundsThe present and future

We now have a number of different proofs of the Fong-Swantheorem, which guarantees the existence of a lift of a given Brauercharacter ϕ. However, we still know very little about the set of liftsLϕ.

For instance, Mark Lewis recently constructed a set of lifts of theBrauer characters of G that depend on a choice of a chain ofnormal subgroups of G .

Another question: Are the sets Bp′(G ) and Np′(G ) defined aboveequal? And if not, are there conditions that guarantee equality?



















Theorem(C, 2006) If G has odd order, or if p is odd, then Bp′(G ) = Np′(G ).

However, I also showed that if p = 2, then there is acounterexample. The same counterexample shows that theconstituents of a character Np′(G ) restricted to a normal subgroupN C G need not be in Np′(N).

We still know relatively little about the set Lϕ. For instance, howbig can Lϕ be?



















Building on work of Navarro, we can determine a lower bound onthe number of lifts of a Brauer character ϕ.

We know, from before, that given a Brauer character ϕ ∈ IBrp(G ),there is a unique character χ ∈ Np′(G ) such that χ0 = ϕ. If W isthe normal nucleus subgroup for χ, then the normal nucleuscharacter must be a p′-special character α.

Theorem(C, 2007) Let ϕ, χ, W , and α be as above. Then the map

γ → (αγ)G

is an injection from the set of linear p-special characters of W toLϕ. Therefore |W : W ′|p ≤ |Lϕ|.It is easy to find examples of equality and strict inequality in theabove situation.








γ → (αγ)G









γ → (αγ)G

is an injection from the set of linear p-special characters of W toLϕ. Therefore |W : W ′|p ≤ |Lϕ|.

It is easy to find examples of equality and strict inequality in theabove situation.








γ → (αγ)G






In the case that the order of G is odd, we can also find upperbounds for the number of lifts of a Brauer character. We first needa definition.

DefinitionSuppose ϕ ∈ IBrp(G ) and let χ ∈ Np′(G ) be such that χo = ϕ. IfW is the normal nucleus subgroup for ϕ, and if Q is a Sylowp-subgroup of W , then Q is called a vertex of ϕ.

This is not the ”classical” definition of a vertex. However, it canbe shown to be equivalent to the classical definition.



















There are a number of reasons we need the odd order toassumption to state and prove our upper bound. The most basic isthe following lemma:

LemmaSuppose G has odd order, and let χ ∈ Irr(G ) be such thatχ0 = ϕ ∈ IBrp(G ). If (W , αβ) is a normal nucleus for χ, then β isa linear character.

In the above situation, notice that if Q is a Sylow p-subgroup ofW , then β necessarily restricts to a linear character of Q. It canbe shown (though it takes a lot of work) that Q is in fact a vertexof ϕ.



















We now see that there is a connection between the lifts of aparticular Brauer character ϕ and the linear characters of its vertexsubgroup. There is no particular reason, at this point, to believethat this connection should be particularly well-behaved or useful.

Theorem(C, 2007) Let G be a group of odd order and let ϕ ∈ IBrp(G ). LetQ be a vertex subgroup for ϕ. Then there is a well-definedinjection from the set Lϕ of lifts of ϕ to the linear characters of Q.Therefore |Lϕ| ≤ |Q : Q ′|.





We now see that there is a connection between the lifts of aparticular Brauer character ϕ and the linear characters of its vertexsubgroup. There is no particular reason, at this point, to believethat this connection should be particularly well-behaved or useful.

Theorem(C, 2007) Let G be a group of odd order and let ϕ ∈ IBrp(G ). LetQ be a vertex subgroup for ϕ. Then there is a well-definedinjection from the set Lϕ of lifts of ϕ to the linear characters of Q.Therefore |Lϕ| ≤ |Q : Q ′|.





Big picture summary:

I The Fong-Swan theorem showed (non-constructively) thateach irreducible Brauer character ϕ of a solvable group G hada lift.

I Isaacs, and later Navarro, (and later Lewis) explicitlyconstructed families of ordinary irreducible characters of Gthat were ”canonical”’ lifts of the Brauer characters of G .

I Now that we know that lifts abound, we have bounded them,and would like to know an exact count for a given Brauercharacter ϕ.





















Any approach to solving this problem will likely take two thingsinto account:

I Groups of odd order seem to be the ”proving ground” forproblems like this - there is a lot of structure there and a lotof nice results.

I The vertex subgroup seems to play an important role, and theexact count will likely be given terms of information about thevertex subgroup of ϕ, and how it is embedded in the group.





Any approach to solving this problem will likely take two thingsinto account:

I Groups of odd order seem to be the ”proving ground” forproblems like this - there is a lot of structure there and a lotof nice results.

I The vertex subgroup seems to play an important role, and theexact count will likely be given terms of information about thevertex subgroup of ϕ, and how it is embedded in the group.





For instance, it would be nice to have a necessary and sufficientcondition for when a character of the vertex subgroup can comefrom a lift.

Theorem(C, 2008) Suppose G has odd order, and ϕ ∈ IBrp(G ) has vertexsubgroup Q, and let λ be a linear character of Q. Then there is alift of ϕ ”coming from” λ if and only if there exists a subgroup Wand a Brauer character ψ of W such that ψG = ϕ and λ extendsto W .

There is almost certainly a better result lurking here somewhere.This result seems unsatisfying because it requires one to know allof the subgroups that ϕ is induced from.





For instance, it would be nice to have a necessary and sufficientcondition for when a character of the vertex subgroup can comefrom a lift.

Theorem(C, 2008) Suppose G has odd order, and ϕ ∈ IBrp(G ) has vertexsubgroup Q, and let λ be a linear character of Q. Then there is alift of ϕ ”coming from” λ if and only if there exists a subgroup Wand a Brauer character ψ of W such that ψG = ϕ and λ extendsto W .

There is almost certainly a better result lurking here somewhere.This result seems unsatisfying because it requires one to know allof the subgroups that ϕ is induced from.





In addition to the study of the set of lifts, there are otherinteresting related questions regarding ”Fong characters” (whichare used in Isaacs’ construction and are interesting in their ownright) that remain tantalizingly unsolved.

Also, little is known about how the vertex and lift structure relatesto the block structure and defect groups (Slattery has done someinteresting work on this).





In addition to the study of the set of lifts, there are otherinteresting related questions regarding ”Fong characters” (whichare used in Isaacs’ construction and are interesting in their ownright) that remain tantalizingly unsolved.

Also, little is known about how the vertex and lift structure relatesto the block structure and defect groups (Slattery has done someinteresting work on this).


Documents

Lifts of Brauer Characters of Solvable Groupsmath.uakron.edu/~cossey/Akron talk.pdfFor a group G, many representations of G over a eld F are essentially the same. De nition If Xand