On the transfer between the Dickson algebras as modules over the Steenrod algebra

On the transfer between the Dickson algebras as modules over the Steenrod algebra tài liệu, giáo án, bài giảng , luận vă...

DOI 10.1007/s40062-014-0097-0

On the transfer between the Dickson algebras

as modules over the Steenrod algebra

Võ T N Quỳnh · Lưu X Trường

Received: 25 June 2014 / Accepted: 8 November 2014

© Tbilisi Centre for Mathematical Sciences 2014

Abstract Let D  := D(W) be the Dickson–Mùi algebra of W for an elementary abelian p-group W of rank , which consists of all invariants in the mod p cohomol-

ogy ofW under the general linear group GL(W) Hưng (Math Ann 353:827–866,

2012) determined explicitly all the homomorphisms between the Dickson–Mùi bras (regarded as modules over the Steenrod algebra,A) He showed that the compo-

alge-sitions of the restrictions r es ,m : D  → D m and the transfers tr m ,n : D m → D nfor

m ≤ min{, n} form a basis of Hom A (D  , Dn) The restriction res ,mhas explicitly

been known (from Lemmas 3.3 and 3.4 for p = 2 and Lemma 9.1 for p > 2 of the cited article), while the transfer tr m ,n has only been computed for m = 1 in some

degrees (see Lemma 9.2 of the article) In this paper, we study tr m ,n for p = 2 in

general We determine completely tr1 ,n for any n, and compute the image of tr m ,n

for arbitrary m , n on some powers of multilinear and alternating invariants Then, we

recognize some families of invariants in D m on which the transfer tr m ,n vanishes.Keywords Steenrod algebra· Modular representations · Invariant theory ·

Dickson algebra

Communicated by Lionel Schwartz.

Dedicated to Professor Nguy˜ên H V Hưng on the occasion of his 60th birthday.

The work was supported in part by a grant of the NAFOSTED.

V T N Quỳnh (B) · L X Trường

Department of Mathematics, Vietnam National University, Hanoi,

334 Nguyễn Trãi Street, Hanoi, Vietnam

e-mail: quynhvtn@vnu.edu.vn

L X Trường

e-mail: lxtruong.lt@gmail.com

Mathematics Subject Classification Primary 55S10· 55S05 · 20G10 · 20G05

1 Introduction

LetV be an elementary abelian 2-group rank n Then V can also be regarded as an n-dimensional vector space over F2, the field of two elements Let H∗(V) denote

the cohomology of the groupV As it is well-known, H∗(V) ∼ = S(V∗) where S(V∗)

denote the symmetric algebra over the spaceV∗ Throughout the paper, the

coeffi-cient ring for homology and cohomology is alwaysF2 Let x1, , xnbe a basis of

V∗, we have

H∗(V) ∼= F2[x1, , xn ].

The general linear group G L (V) ∼ = GL(n, F2) acts regularly on V and therefore

on H∗(V) The Dickson algebra is the algebra of all invariants of H∗(V) under the

action of G L (V) It is explicitly determined by Dickson in [2] as follows:

D (V) := H∗(V) G L (V)∼= F2[x1, , xn]G L (n,F2 )= F2[Qn ,0 , Qn ,1 , , Qn ,n−1 ], where Q n ,i denotes the Dickson invariant of degree 2n− 2i (A precise definitionfor the Dickson invariants will be given in Sect.2.)

Being the cohomology of the classifying space BV, the group H∗(V) is equipped

with a structure of module over the mod 2 Steenrod algebra,A Each γ ∈ GL(V)

induces anA-isomorphism γ∗on H∗(V) The map γ → γ∗gives rise to the regular

action of G L (V) on H∗(V) So, the actions of GL(V) and A on H∗(V) commute

with each other Hence, the Dickson algebra D (V) inherits a structure of module

over the Steenrod algebraA from H∗(V).

LetU be a F2-vector space of dimension m with m≤ n The subject of the present paper is the transfer tr U,V : D(U) → D(V), which is also denoted by tr m ,n : D m →

D n defined in [3] For fixed n and , the set {trm ,n | m < min{, n}} is known

as a key component forming allA-homomorphisms from D  to D n in the sense asfollows Let W be a vector space of dimension  The restriction res ,m : D  →

D m is the homomorphism induced from an inclusionU → W (in [3, Lemma 3.1],Hưng showed that any inclusion U → W induces the same homomorphism on

H∗(W) G K (W)) Let ¯D (V) or ¯Dn be the augmentation ideal of all positive degree

elements in the Dickson algebra D n

Theorem 1.1 [3, Theorem 1.1] The A-module homomorphisms {trm ,n r es ,m: 1 ≤

m ≤ min{, n}} form a basis of the vector space Hom A ( ¯D  , ¯Dn) of A-module morphisms from ¯ D  to ¯ D n In particular, dim Hom A ( ¯D  , ¯Dn) = min{, n}.

homo-The restrictions r es ,m are explicitly determined on each Dickson monomial in[3, Lemma 3.4]; so, in order to compute HomA ( ¯D  , ¯Dn) we need to compute the

transfers tr m ,n

First, the transfer tr1 ,n is concretely computed in [3, Lemma 4.2] Then, it isfurther studied in the undergraduate thesis of Phạm H -Dăng under the guidance ofNguyễn H V Hưng (see [1])

In the case m = 1, we have D m = F2[x] in which deg x = 1 The followingtheorem is one of main results of the paper

say that f (x1, , xm ) is multilinear on V∗ if f is m-multilinear The invariant

f (x1, , xm ) is called alternating on V∗if the map f is alternating.

The set of invariants in D mwhich are multilinear and alternating forms a subspace

of D m Furthermore, this subspace is subset of the ideal of D m generated by Q m ,0

(see Corollary 4.7) It is infinity dimensional and can be explicitly determined as

follows For each set of m distinct non-negative numbers β = {β1, , βm }, let  m

be the symmetric group on{1, , m} and let

Proposition 1.3 The set

{ω β (x1, , xm ) | β = (β1, , βm) ∈ N m , βi = β j for all i = j}

forms a basis for the subspace of Dm generated by multilinear and alternating ants.

invari-The following theorem shows the image of the transfer tr m ,m+1 on powers of

multilinear and alternating invariants in D m For each Q ∈ D m and y1 , , ym ∈

F[x1 , , xm+1], we denote by Q(y1 , , ym ) the polynomial in F[x1, , xm+1]

obtained from Q by the substitution y j for x j ( j = 1, , m).

Theorem 1.4 Suppose that Q is a multilinear and alternating invariant in D m Let

y j = Q(x1 , , x j−1, x j+1, , xm+1) for j = 1, , m + 1 Then, for r > 0 we have

Since Q m ,0 , Q m ,0 Q m ,i are multilinear and alternating invariants (see Example4.8),

we get the two following explicit formulas for image of tr m ,m+1 on some powers ofthese invariants

The following proposition gives some families of invariants on which the transfer

is zero

Proposition 1.7 tr m ,n (Q) is zero for each of the following cases.

(i) Q ∈ D m is a multilinear and alternating invariant;

2,0 Q22s1 ,1 for any nonnegative integers s0, s1.

The statement in Proposition1.7(iii) is no longer true for the case n= 3 In fact,

tr2 ,3 (Q2s0

2,0 Q22s1 ,1 ) = 0 for s0− s1 > 1 or s0− s1 < 0 (see Remark7.3)

The paper is divided into seven sections The introduction in Sect.1 is followed

by the preliminary in Sect.2, where we recall definition of the transfer between theDickson algebras Section3is a proof of Theorem1.2 The concepts of multilinearand alternating invariants are introduced in Sect.4, where we prove Proposition1.3

In Sect.5, we study the transfer on powers of multilinear and alternating invariants,then prove Theorem 1.4 and Proposition 1.7(i) Section 6 deals with a method offinding invariant monomials in the image of the transfer by finding “leading ele-ments” of it By using this method, we prove Lemma5.5, the most technical lemma

of the paper that is used in the proofs of Corollaries1.5and1.6 Finally, in Sect.7,

we prove Proposition1.7(ii) and (iii)

The paper was in progress while the first named author was visiting to the VietnamInstitute for Advanced Study in Mathematics (VIASM) She would like to thankVIASM for the financial support and the warm hospitality

2 Preliminary

In this section, we exploit Hưng’s definition of the transfer tr m ,n

Let U and V be a F2-vector spaces of dimensions m and n with m ≤ n Let

K be a subspace of V and πK: V → U be an epimorphism with ker πK = K

Suppose that π : V → U is another epimorphism whose kernel is also K Then

there is an isomorphism α: U → U such that πK = απ It follows that π∗

K: H∗(U) G L (U) → H∗(V) does not depend on the choice of the epimorphism πK

It only depends on the kernel ofπK,K On the other hand, the group GL(V) permute

the subspacesK of dimension n − m in V So the sum of π∗

Kmaps H∗(U) G L (U)to

the G L (V)-invariants.

Definition 2.1 [3, Definition 4.1] tr U,V =Kπ∗

K: H∗(U) G L (U) → H∗(V) G L (V),

where the sum runs over all the subspacesK of dimension (dim V − dim U) in V

As it is well known that H1(V) ∼= V∗and H∗(V) ∼ = S(V∗) where S(V∗) denote

the symmetric algebra over the spaceV∗ Let x1 , , xnbe a basis ofV∗, we have

H∗(V) ∼= F2[x1, , xn ].

Recall that the algebra of the G L (V)-invariants H∗(V) G L (V)was computed by

Dick-son [2] He showed in [2] that H∗(V) G L (V)is also a polynomial algebra and denoted

is the Mùi invariant under the Sylow 2-subgroup T n of G L n consisting of all upper

triangular n × n-matrices with 1 on the main diagonal (see Mùi [5])

Since the transfer tr U,Vonly depends on the dimensions ofU and V, it is denoted

by tr m ,n : D m → D n , where m = dim U and n = dim V For each epimorphism

π: V → U, the induced homomorphism in cohomological degree 1, π∗: H1(U) →

H1(V) is a monomorphism So W := π∗H1(U) is a subspace of dimension m in

H1(V) Let x1, , xm be a basis of H1(U) Then π∗(x1), , π∗(xm) a basis for

π∗H1(U) And for each Q ∈ Dmwe haveπ∗(Q) = Q(π∗(x1), , π∗(xm )) Since

Q is invariant under any invertible transformation on U, Q(π∗(x1), , π∗(xm )) =

Q (y1, , ym ) for any basis y1, , ym of W Hence Q (π∗(x1), , π∗(xm )) only

depends on W ; so, we denote it by Q (W) for short Therefore, by Definition2.1wehave

tr m ,n (Q) =

W

Q (W)

where the sum runs over all m-dimensional subspaces W in H1(V) For example,

with m = 1, n = 3 let x1 , x2, x3be a basis forV∗we have

3 The transfer from D1to D n

The aim of this section is to prove the following theorem, which is also numbered asTheorem1.2in the introduction

For n = 1, as tr1 ,1 is the identity map, we have tr1 ,1 (x +1 ) = x +1 =

Q1 ,0 tr1 ,1 (x  ) So the statement holds for n = 1 Suppose inductively that it is true

for n Let

U = {c1x1 + · · · + c n x n |c i ∈ F2 for i = 1, , n},

W = {c1x1 + · · · + c n x n + x n+1|c i ∈ F2 for i = 1, , n}.

(a) n−1

i=0 Q2n ,i tr1 ,n+1 (x +2 i+1 −1) = R U

+2 n+1 −1+n−1

i=0 Q2n ,i R +2 W i+1 −1, (b) Vn+1n

n ,i−1 + Q n ,i V n+1for i ≥ 0 and Q n ,i = 0 for i < 0)

So, the lemma is proved

Now, we prove the two formulas (a) and (b) above

On the other hand, V n+1(x1, , xn, xn+1) = Vn+1(x1, , xn, x) for all x ∈ W; so,

= k As a consequence, the right hand side of (1) is equal to zero On the other hand,

by Lemma3.2, tr1 ,n (x k ) = 0 as k < 2 n− 1 Therefore, we get (1) for this case

Case k ≥ 2n− 1 Suppose inductively that the theorem holds at degree less than

So, (1) holds at degree k+ 1 Therefore, the theorem is proved

4 Multilinear and alternating invariants

Recall that U and V are vector spaces of dimensions m and n respectively with

m < n, tr U,V=Kπ∗

K: H∗(U) G L (U) → H∗(V) G L (V), where the sum runs over all

the subspacesK of dimension (dim V−dim U) in V Let x1 , , xmbe a basis ofU∗.

Then H∗(U) = S(U∗) = F2[x1, , xm ] For each f (x1 , , xm ) ∈ F2[x1, , xm],

we define by f the map

Since f (x1, , xm) is is multilinear and alternating, we get

This sum is invariant underm

Lemma 4.4 (i) ω β (x1, , xm ) is multilinear and alternating.

i j = d If d is the degree of ω β (x1, , xm), let α(d) be

the number 1’s occurring in the dyadic expansion of d, then d= 2β1+ · · · + 2β m is

the dyadic expansion of d and α(d) = m Therefore, in this case, ω β (x1, , xm) is

actually the elementωm (d) in [4, Definition 2.1] However ifα(d) < m then ωm(d)

maybe a G L (m, F2)-invariant but not multilinear and alternating For example, let

ω β (x1, , xm ) = ωm (d) with α(d) = k On the other hand, the part (ii) is also

a corollary of the part (i) and Lemma4.2 So, we only need to show the part (i)

We have ω β (x1, , xm) is symmetric, so it is enough to show that the map

t1 → ω β (t1, , tm) is linear and ω β (t1, t2 , tm) = 0 when t1= t2.

Thus, if t1 = t2then(t2a1

1 t22a2 + t2a2

1 t22a1 ) = 2t2a1

1 t22a2 = 0 So ω β (t1, t1 , tm) = 0.

The lemma is proved

The following proposition is also numbered as Proposition1.3in the introduction

Proposition 4.5 The set

{ω β (x1, , xm ) | β = (β1, , βm) ∈ N m , βi = β j for all i = j}

forms a basis for the subspace ofF2[x1, , xm ] generated by multilinear and

alter-nating invariants.

Proof For each β = (β1, , βm) ∈ N m with βi = β j for all i = j, the sum

2β1 + · · · + 2β m is the dyadic expansion of degω β (x1, , xm ) So, in each degree

d, there exists at most an element of the form ω β (x1, , xm) with d = m

j=12sj.

Therefore the set of elementsω β (x1, , xm ) is linearly independent.

Suppose that Q (x1, , xm) is a multilinear and alternating invariant of degree d

in D m We prove that Q (x1, , xm ) = ω β (x1, , xm) for some β = (β1, , βm )

such thatβi = β j for all i = j and d = 2 β1 + · · · + 2β m Let x i1

= 0 for some 1 ≤ i ≤ i1 Therefore, there exists

a(m + 1)-tuple ( j, j1, j2, , jm ) with j1> 0, 1 ≤ j ≤ j1− 1 andj1

j



= 0 suchthat (i, i2, , im , i1− i) = ( j, j2 , , jm , j1− j) It follows that (i1 , , im ) = ( j1, , jm) This is a contradiction Hence i1is a power of 2 Similarly, we also get

i2 , , im are powers of 2 We write i1= 2β1 , i2= 2β2 , , im = 2β m for some negative integersβ1, β2, , βm Now, we show that i s = i t for all s = t Without lost of generality, suppose to the contrary that i1 = i2 As Q (x1, x2, , xm) ∈ Dm,

non-it is invariant under the transposnon-ition of x1 and x2 that keeps the other variables

fixed It follows that Q (x1, x2, , xm) has the form

Hence, there is a tuple ( j1, j2, , jm) such that j1 = j2 , j1+ j2 = 2i1 , jk =

i k for k ≥ 3 It implies that (i1 , i2, i3, , im ) = ( j1, j2, j3, , jm) This is a

contradiction Therefore i1 = i2 Similarly, we get i s = i t for all s = t.

It has shown that if x i1

1x i2

2 x i m

m is a monomial of Q (x1, x2, , xm ) then

i j = 2β j for 1 ≤ j ≤ m and β i = β j for i = j As Q(x1 , x2, , xm ) is

invari-ant under the symmetric group m , it contains all permutations of x i1

1x i2

2 x i m m

as terms Since β1, , βm are distinct, the sum of powers of x1 , , xm in the

monomial x i1

1x i2

2 x im

m, 2β1 + 2β2 + · · · + 2β m, is the the dyadic expansion of

deg Q (x1, x2, , xm) Therefore Q(x1, x2, , xm ) only contains x i1

So, the proposition is proved

As we had known thatω β (x1, , xm ) is ωm(d) with α(d) = m in [4, Definition

2.1], where d is the degree of ω β (x1, , xm ), the following is a corollary of [4,Corollary 2.7]

Corollary 4.6 Suppose that β = {β1, , βm } with β1 < β2 < · · · < βm Then (Qm ,0 )2β1 | ω β (x1, , xm ).

From Proposition4.5and Corollary4.6, we get the following result

Corollary 4.7 If Q ∈ D m is multilinear and alternating, then Qm ,0 divides Q Example 4.8 (i) Qm ,0 = ω β (x1, , xm ) with β = (0, 1, , m − 1).

(ii) For each 0 ≤ i ≤ m − 1, Q m ,0 Q m ,i = ω β (x1, , xm ) with β = (0, 2,

i − 1, i + 1, , m).

1 x22m−2 x2

m−1x m +(symmetried) Then, we obtain (i)

withβ = (0, , 2 i−1, 2 i+1, , 2 m ) Thus, (ii) is proved.

5 Transfer on powers of multilinear and alternating invariants

In this section, we first study the transfer tr m ,n in which n = m + 1 Recall that

tr U,V : H∗(U) G L (U) → H∗(V) G L (V) , or tr m ,m+1 : D m → D m+1, is given by

sub-vector spaces W of V∗ and the set of (m + 1)-tuple of scalars (c1, , cm+1) in which not all c i are zero such that Q (W) = c1Q (X1) + · · · + cm+1(Xm+1) for all multilinear and alternating invariants Q of D m

Proof Suppose that W is an m-dimensional vector space ofV∗and y1 , , ym is a

basis of W We have (y1 ym) = (x1 xm+1)C where C is a (m + 1) × m-matrix

and rankC = m So y j =m+1

i=1 c i j x i Setting c j =i = j∀,i k =i c i1,1 cim+1,m

for 1≤ j ≤ m + 1, for arbitrary multilinear and alternating Q of D m, we get

Q (W) = Q(y1, , ym ) = c1Q(X1) + · · · + cm+1Q (Xm+1).

Conversely, we show by induction on m that for each (m + 1)-tuple of scalars (c1, , cm+1) in which not all ci are zero, there exists exactly one m-dimensional