1. Introduction
Let
be the polynomial algebra over the prime field of two elements,
, in
variables
, each of degree 1. The mod-2 Steenrod algebra
acts on
by the formula
and subject to the Cartan formula
for
(see Steenrod – Epstein [8]).
Many authors study the hit problem of determination of the minimal set of generators for
as a module over the Steenrod algebra, or equivalently, a basis of
. A polynomial
in
is called hit if it can be written as a finite sum
for some polynomials
. That means
belongs to
whehe
denotes the augmentation ideal in
. This problem has first been studied by Peterson [4], Wood [12], Singer [7], Priddy [5], who show its relationship to several classical problems in homotopy theory. Peterson conjectured in [4] that as a module over the Steenrod algebra
,
is generated by monomials in degrees
that satisfy
, where
denotes the number of ones in dyadic expansion of
, and proved it for
. The conjecture was established in general by Wood [12]. This is a useful tool for determining
-general for
. The tensor product
has explicitly been calculated by Peterson[4] for
, Kameko for
in his thesis [2] and Sum [10] for
. The hit prolem were then investigated by many authors. (See Nam [3], Singer [7], Silverman [6], Hung [1], Walker-Wood [12] and others)
One of important tools in Kameko’s computation of
-general for
is the squaring operation
which is determined for all positive intergers
. Kameko showed in [2] that if
then
is an isomorphism of
-vector spaces, where
. From this result and Wood’s theorem, the hit problem is reduced to the cases of degree
with
.
In this paper, we explicitly determine
for
and degree
. We have
Theorem 1.1.
is an
-vector space of dimension 174 with a basis consisting of all the classes represented by the following admissible monomials:
i1) The 20 monomials are all permutations of the spikes
i2) The 10 monomials of the form 
i3) The 10 monomials of the form 
i4) The 10 monomials of the form 
i5) The 10 monomials of the form 
i6) The 20 monomials of the form 
i7) The 15 monomials of the form 
i8) The 20 monomials of the form 
i9) The 30 monomials are all permutations of the spikes 
i10) 15 monomials are some permutations of the monomial
, namely:
i11) 4 monomials are some permutations of the monomial
, namely:
In Section 2, we recall some results on the admissible monomials and hit monomials in
. Theorem 1.1 will be proved in Section 3.
2. Preliminaries
In this section, we recall some results in Sum[10], Singer [7] on the admissible monomials and the hit monomials in
. Let
denote the
-th coefficient in dyadic expansion of a nonnegativeinteger
. That means
for
or 1 and
. Let
. Following Kameko [2], we define two sequences associated with
by
where
.
Definition 2.1. Let
be the monomials in
. We say that
if and only if one of the following holds
i1) 
i2)
and 
Here, the order on the set of sequences of non-negative intergers is the lexicographical one.
Let
be homogeneous polynomials of the same degree in
. We denote
if and only if
. If
then
is called hit.
Definition 2.2. A monomial
is said to be inadmissible if there exists the monomials
such that
and
.
A monomial
is said to be admissible if it is not inadmissible.
Obviously, the set of all admissible monomials in
is a minimal set of
-generators of
.
Definition 2.3. A monomial
is said to be strictly inadmissible if and only if there exists the monomials
such that
and
, with
and
.
The following propositon is one of our main tools.
Proposition 2.4 (Sum [10]). Let
be an admissible monomial in
. Then, we have:
i1) If there is
such that
then
, for all
.
i2) If there is
such that
then
, for all
.
Now, we recall a result of Singer [7] on the hit monomials in
.
Definition 2.5. A monomial
is called the spike if
for
a non-negative interger and
. If
is the spike with
and
for
then it is called the minimal spike.
The following is a criterion for the hit monomials in
.
Theorem 2.6 (Singer [7]). Suppose
is a monomial of degree
, where
. Let
be the minimal spike of degree
. If
then
is hit.
For later use, we set
It is easy to see that
and
are the
-submodules of
. Furthermore, we have
. For a polynomial
in
, we denote by
the class in
represented by
.
3. Proof of Theorem 1.1
For
, define the homomorphism
of algebras by substituting
Notation
is also the admissible monomials set of degree
in
. It is clear that, for
, we have
.Hence,
.
We recall the following result.
Theorem 3.1 (Sum[10])
is an
-vector space of dimension 55 with a basis consisting of all the classes represented by the following admissible monomials:
i1) 
i2) 
i3) 
i4) 
i5)
,
i6) 
i7) 
i8) 
From this Theorem, by a direct computation, we easily obtain
Proposition 3.2.
is an
-vector space of dimension 155 with a basis consisting of all the classes represented by the admissible monomials
.
Now Theorem 1.1 is proved by computing
. From now on, let us write the monomial
by
for abbreviation.
Proposition 3.3.
is generated by 19 elements
.
Proposition 3.4. The elements
, for
are linearly independent in
.
We prepare some lemma for the proof of this propositions.
By a direct computation, we have
Lemma 3.5. We have the relations in 
Lemma 3.6. The following monomials are inadmissible:
Proof. We have
Since
and 
We get
is admissible. The other monomials are proved similar.
Proof of Proposition 3.3. There are 35 monomials of degree eight in
; they are
and their permutations.
There are 5 monomials in
with
-sequence associated (4;0;1); they are
and their permutations. Using Proposition 2.4, we see that the following monomials are inadmissible:
There are 20 monomials in
with
- sequence associated (4;2); they are
and their permutations. Using Lemma 3.6, we see that the following monomials are inadmissible:
There are 10 monomials in
with
-sequence associated (2;3); they are
and their permutations. Using Lemma 3.6, we see that the following monomials are inadmissible:
So,
is generated by 19 elements
.
Proof of Proposition 3.4. Suppose there is a linear relation
 | (1) |
with 
Consider the homomorphisms
,
induced by the ring homomorphisms
, which are determined by the following tables:
By the direct computation, we see that the homomorphism
send the relation (1), we obtain
Then, we get
Similarly, the homomorphisms
send the relation (1), we obtain
Hence, we get
for
The proposition is proved.
Acknowledgements
It is a pleasure for me to express my hearty gratitude to Professor Nguyen Sum for his valuable suggestions and constant encouragement. My thanks also go to all colleagues at the Foundation Sciences Faculty, University of Technical Education of Ho Chi Minh city.
References
[1] | N. H. V. Hung, The cohomology of the Steenrod algebra and representations of the generalinear groups, Trans. Amer. Math. Soc. 357(2005), 4065-4089. |
| In article | CrossRef |
|
[2] | M. Kameko, Products of projective spaces as Steenrod modules, Thesis Johns Hopkins University, 1990. |
| In article | |
|
[3] | T. N. Nam, A-générateurs génériquess pour l’algèbre polynomiale, Adv. Math. 186(2004), 334-362. |
| In article | CrossRef |
|
[4] | F. P. Peterson, Generators of H*(RP∞ × RP∞) as a module over the Steenrod algebra, Abstracts Amer. Math. Soc. No.833 April 1987. |
| In article | |
|
[5] | S. Priddy, On characterizing summands in the classifying space of a group, I,Amer. Jour. Math. 112(1990), 737-748. |
| In article | CrossRef |
|
[6] | J. H. Silverman, Hit polynomials and the canonical antimonomorphism of the Steenrod algebra, Proc.Amer.Math.Soc. 123(1995), 627-637. The transfer in homological algebra, Math.Zeit, 202 (1989), 493-523. |
| In article | CrossRef |
|
[7] | W. M. Singer, On the action of the Steenrod squares on polynomial algebras, Proc. Amer. Math. Soc. 111(1991), 577-583. |
| In article | CrossRef |
|
[8] | N. E. Steenrod, Cohomology operations, Lectures by N. E. Steenrod written and revised by D. B. A. Epstein, Annals of Mathematics, No.50, Princeton University Press, Princeton N.J(1962). |
| In article | |
|
[9] | N. Sum, The negative answer to Kameko’s conjecture on the hit problem, Advances in Mathematics,(2010) 225:5, 2365-2390. |
| In article | CrossRef |
|
[10] | N. Sum, On the Peterson hit problem, preprint, 2013, 59pages (submitted). |
| In article | |
|
[11] | N. Sum, On the hit problem for the polynomial algebra, C. R. Math. Acad. Sci. Pari, Ser. I, 351 ( 2013), 565-568. |
| In article | |
|
[12] | G. Walker and R. M. W. Wood, Young tableaux and the Steenrod algebra, Proceedings of the International School and Conferenca in Algebraic Topology, Ha Noi 2004, Geom.Topol. Monogr., Geom.Topol. Publ.,Coventry 11(2007), 379-397. |
| In article | |
|