# The role of steric effects in Hydrogen Atom Transfer reactions

Yi Sun, Jacob Sanders, K.N.Houk<EMAIL_ADDRESS>Department of Chemistry
and Biochemistry, University of California, Los Angeles, California 90095,
United States

###### Abstract

We explored how steric effects infuences the rate of Hydrogen Abstraction
Reactions (HAT) between oxy radicals and alkanes. Quantum chemical
computations of the transition state (TS) show that the barrier heights and
reaction enthalpies are both influenced by bulky substituents on the radical,
and less on the alkane. Additionally, it has been discovered that the
activation barrier is well connected to the reaction enthalpy via the Evans-
Polanyi relationship.

##  I Introduction

Hydrogen atom transfer (HAT) reactions are one of the most fundamental
chemical reactions:

Figure 1: Hydrogen atom transfer reaction.

These reactions are key steps in a wide range of chemical, environmental, and
biological processes. Recently, a number of photoinduced Zhang ; Deng ; Cao
and electrochemical Niko  HAT reactions that have value in synthetic organic
chemistry have been reported. There are also numerous case studies of HAT
reactions that involve metal species Joseph ; Zass ; Cheong ; Laxmi .

There have been many studies aimed at understanding the barrier heights of HAT
reactions. A celebrated example is the Bell-Evans-Polanyi principle, Bell ; EP
which can be expressed mathematically as

$\Delta H^{{\ddagger}}=E_{0}+\alpha\Delta H$ (1)

in which $\Delta H^{{\ddagger}}$ is the activation barrier of the reaction,
i.e., barrier height, $\Delta H$ is the enthalpy change of the reaction, and
$E_{0}$ and $\alpha$ are empirical fitting parameters. In particular, the
parameter $\alpha$ is related to the position of the transition state along
the reaction coordinate.

A refinement of the Bell-Evans-Polanyi principle is the Roberts-Steel
relationship, that adds polar and conjugation effects, RS . Recent work by Liu
_et al._ Liu  has proposed a simplified Roberts-Steel relationship in which
the activation barrier of the reaction,$\Delta H^{{\ddagger}}$, depends on two
quantities: the reaction enthalpy $\Delta H$ and the difference in the
electronegativity $\Delta\chi$ of the radicals. This gives rise to the
equation

$\Delta H^{{\ddagger}}=\alpha\Delta H_{rxn}+\beta\Delta\chi^{2}+\gamma$ (2)

in which $\alpha$, $\beta$ and $\gamma$ are empirical fitting parameters.

Another strategy to understand the barriers of HAT reactions is to use the
Marcus cross relationship, which was shown by Mayer Mayer  to relate the HAT
reaction barrier heights to the self-exchange rate constants and the chemical
equilibrium constant of the HAT reaction via the equation below:

$k_{AH/B}=\sqrt{k_{AH/A}k_{BH/B}K_{eq}f}$ (3)

where $k_{AH/A}$ and $k_{BH/B}$ are the rate constants for the respective
self-exchange HAT reactions, and $K_{eq}$ is the equilibrium constant of the
HAT reaction, which can be calculated from the change of the Gibbs energy,
$\Delta G$ of the reaction; $f$ is a dimensionless proportionality factor.
Thus, Marcus Theory relates HAT reaction barriers to kinetic factors, namely
the intrinsic barriers of self-exchange, and to thermodynamic factors, namely
the free energy of the HAT reaction. By contrast, the modified Roberts-Steel
relationship uses intrinsic properties of reactant and product radicals,
namely their electronegativities, along with thermodynamic factors, namely the
enthalpy of reaction, to understand reaction barriers.

More recently, the Hong group has developed machine learning models to
correlate rates of HAT reactions with many different properties of the
reactants and products.Hong  We have been carrying out parallel efforts to
understand the fundamental factors controlling rates. In this paper, we
explore the role, if any, of steric effects on the rates of HAT reactions.

##  II STERIC EFFECTS IN ORGANIC CHEMISTRY

Although the relationships discussed in the previous section have been used to
explore many types of HAT reactions, these studies have included few
sterically hindered substrates or radicals. Accordingly, we sought to
determine whether steric effects are influential in HAT reactions. On the one
hand, bulky groups might slow HAT by increasing the energies of transition
states via steric repulsion as atoms attempt to approach each other more
closely than the sum of their van der Waals radii. On the other hand, if atoms
do not approach each other beyond the sum of their van der Waals radii in the
transition states, then dispersion interactions among bulky groups may
actually accelerate the HAT reaction. Such dispersion effects have been noted
in a number of hemolysis and radical recombination reactions Peter .

Steric effects are well known throughout organic chemistry. When atoms come
too close, such that the distance between them is shorter than the sum of
their van der Waals radii, the energy rises Bondi ; in this way steric
hindrance can slow down or even prevent a reaction. More recently, the
importance of steric attraction, which arises from London dispersion forces
between large substituents, has been emphasized Peter . There are many
examples where steric effects influence the reactivity of molecules and the
selectivity of substitution reactions Tolman ; Crossley ; Bo ; Zhou . ; they
can even determine whether certain radicals are stable in the first place
Hicks ; Power . Nevertheless, systematic studies of steric effects in HAT
reactions across a wide range of substrates are scarce.

In this paper, we investigate the role of steric effects in HAT reactions
between alkoxy and phenoxy radicals and alkane substrates. We observe the
largest steric effects for alkoxy and phenoxy radicals and only smaller steric
effects for alkane substrates. In particular, steric attraction is often
important in controlling barrier heights. Remarkably, we discover that steric
effects influence not only barrier heights but also reaction enthalpies, even
though reaction enthalpies are calculated as differences between the
enthalpies of separated species. Because steric effects influence reaction
enthalpies, we find that that activation barriers remain well-connected to
reaction enthalpies via the Evans-Polanyi relationship even in the presence of
bulky substituents.

##  III Computational method

Ground state and transition state structures were optimized at the
$\omega$B97xD/6-31G(d) level of theory Francl ; Harihanan ; Chai ; Chai2
using Gaussian09 Gaussian . Optimized structures were confirmed via frequency
calculations, with ground states having all positive frequencies and
transition states having one imaginary frequency corresponding to the reaction
coordinate. These frequencies were also used to compute enthalpies and Gibbs
free energies at 298 K and a standard state of 1 atm. Energies were corrected
via single-point computations performed at the more accurate
$\omega$B97xD/6-311G++(d,p) level of theory.

##  IV The investigated reactions and the results

We explored 96 HAT reactions between 10 radicals and 12 alkanes. The
structures of the alkoxy, phenoxy, and cyclohexoxy radicals are shown in
Figure 2, while the structures of alkane substrates are shown in Figure 3. All
HAT reactions studied are between oxy radicals in Figure 2 and the illustrated
C–H bonds of the alkanes in Figure 3. To minimize polar effects, no polar
substituents or heteroatoms were studied. All barrier heights are shown in
Table I, with the corresponding reaction enthalpies shown in Table II.

Figure 2: The 10 radicals studied in this work. Figure 3: The 12 alkanes studied in this work with the abstracted hydrogen drawn explicitly. Alkane Radical | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10
---|---|---|---|---|---|---|---|---|---|---
A | 6.9 | 7.7 | 6.1 | 5.1 | 19.5 | 21.5 | 20.6 | 26.3 | 5.7 | 8.7
B | 7.1 | 8.2 | 6.6 | 5.4 | 19.9 | 22.0 | 21.4 | 26.1 | 6.0 | 8.8
C | 6.5 | 7.6 | 5.9 | 5.0 | 19.3 | 23.0 | 20.6 | 25.5 | 5.3 | 8.1
D | 6.1 | 7.1 | 5.3 | 4.4 | 19.9 | 22.7 | 21.0 | 25.5 | 5.0 | 8.0
E | 4.1 | 4.6 | 3.0 | 2.7 | 15.7 | 18.4 | 17.1 | 22.0 | 2.7 | 5.4
F | 4.2 | 5.2 | 3.9 | 2.6 | 16.3 | 18.5 | 17.5 | 21.8 | 3.2 | 6.0
G | 3.6 | 5.0 | 3.2 | 2.4 | 16.3 | 19.1 | 17.6 | 23.1 | 1.9 | 5.3
H | 3.2 | 4.1 | 2.1 | 1.3 | 15.1 | 17.5 | 16.6 | 22.2 | 1.6 | 5.5
I | 1.8 | 2.7 | 0.9 | 0.6 | 13.3 | 15.8 | 14.7 | 23.0 | 1.0 | 4.2
J | 2.0 | 3.0 | 1.3 | 0.8 | 13.2 | 15.2 | 14.6 | 23.9 | 1.1 | 4.5
K | 1.2 | 2.1 | 0.2 | -0.4 | 12.2 | 15.2 | 13.5 | 21.7 | -0.2 | 4.0
L | 0.7 | 1.7 | -0.1 | -0.4 | 12.2 | 15.1 | 13.5 | 22.7 | 0.1 | 4.2
Table 1: The barrier heights in kcal/mol. Alkane Radical | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10
---|---|---|---|---|---|---|---|---|---|---
A | -0.8 | -0.5 | -2.1 | -2.9 | 14.9 | 19.2 | 16.5 | 22.2 | -1.8 | -0.4
B | 0.2 | 0.5 | -1.0 | -1.9 | 16.0 | 20.2 | 17.6 | 23.3 | -0.8 | 0.7
C | 0.5 | 1.1 | -0.7 | -1.6 | 16.3 | 20.5 | 17.9 | 23.6 | -0.5 | 1.0
D | 0.8 | 1.1 | -0.5 | -1.3 | 16.5 | 20.8 | 18.1 | 23.8 | -0.2 | 1.2
E | -4.0 | -3.7 | -5.3 | -6.1 | 11.7 | 16.0 | 13.3 | 19.0 | -5.0 | -3.6
F | -3.1 | -2.8 | -4.4 | -5.2 | 12.6 | 16.9 | 14.2 | 19.9 | -4.1 | -2.6
G | -3.4 | -2.8 | -4.7 | -5.5 | 12.3 | 16.6 | 13.9 | 19.6 | -4.4 | -3.0
H | -3.1 | -2.8 | -4.3 | -5.2 | 12.7 | 16.9 | 14.3 | 19.9 | -4.1 | -2.6
I | -6.2 | -5.9 | -7.5 | -8.3 | 9.5 | 13.8 | 11.2 | 16.8 | -7.2 | -5.7
J | -5.8 | -5.5 | -7.1 | -7.9 | 9.9 | 14.2 | 11.5 | 17.2 | -6.8 | -5.3
K | -6.4 | -5.8 | -7.7 | -8.6 | 9.3 | 13.5 | 10.9 | 16.6 | -7.5 | -6.0
L | -6.0 | -5.8 | -7.3 | -8.2 | 9.7 | 14.0 | 11.3 | 17.0 | -7.1 | -5.6
Table 2: The reaction enthalpies in kcal/mol.

##  V Discussion

###  V.1 The Evans-Polanyi relationship for sterically unhindered reactions

To assess the accuracy of the Evans-Polanyi relationship for sterically
unhindered reactions, we compare the reactivities of unhindered primary (A),
secondary (E), and tertiary (I) C–H bonds in alkanes for three unhindered
alkoxy (1), cycloalkoxy (9), and phenoxy (5) radicals. Taking differences of
the Evans-Polanyi relationship for two reactions with the same $E_{0}$ and
$\alpha$ parameters yields:

$\Delta\Delta H^{{\ddagger}}=\alpha\Delta\Delta H$ (4)

If, further, the two HAT reactions being compared involve the same oxy
radical, then $\Delta\Delta H$ depends only on the alkane (it is the
difference in bond dissociation enthalpies of the C–H bonds in the alkanes of
the two reactions), whereas $\Delta\Delta H^{{\ddagger}}$ is different for
different oxy radicals.

Table III shows the changes in reaction enthalpies and barrier heights in
moving from sterically unhindered primary (A) to secondary (E) to tertiary (I)
alkane C–H bonds. While the relative reaction enthalpies are independent of
the specific oxy radical, the relative barrier heights are shown for
sterically unhindered alkoxy (1), cycloalkoxy (9), and phenoxy (5) radicals.

| EtH ($1^{\circ}$) (A) | n-PrH ($2^{\circ}$) (E) | i-BuH ($3^{\circ}$) (I)
---|---|---|---
$\Delta\Delta H$ | 0 | -3.2 | -5.4
Methoxy (1) $\Delta\Delta H^{{\ddagger}}$ | 0 | -2.8 | -5.1
Cyclohexoxy (9) $\Delta\Delta H^{{\ddagger}}$ | 0 | -3.0 | -4.7
Phenoxy (5) $\Delta\Delta H^{{\ddagger}}$ | 0 | -3.8 | -6.2
Table 3: Relative reaction enthalpies and relative barrier heights in moving
from $1^{\circ}$ to $2^{\circ}$ to $3^{\circ}$ alkanes for hydrogen
abstraction in kcal/mol.

As shown in the first row of the table, the relative reaction enthalpies
$\Delta\Delta H$ becomes more negative as the alkane C–H bond becomes more
substituted because the product radical contains more alkyl groups which can
stabilize the radical via hyperconjugation. Surprisingly, with the exception
of the reaction between phenoxy radical 5 and isobutane I, the relative
barrier heights $\Delta\Delta H^{{\ddagger}}$ track the relative reaction
enthalpies $\Delta\Delta H$ quite closely for all three unhindered alkoxy
radicals. This implies that $\alpha$ in the Evans-Polanyi relationship is
approximately 1. The negative deviation in $\Delta\Delta H^{{\ddagger}}$ for
the reaction between the phenoxy radical and isobutane may indicate dispersion
stabilization in the transition state.

###  V.2 Electronic effects of phenoxy radicals in HAT reactions

In preparation for studying the steric effects of phenoxy radicals on the
rates of HAT reactions, we began by isolating the electronic effects and
studying them on their own. Thus, we compared the reactivities of unhindered
phenoxy (5) and p-tert-butylphenoxy (7) radicals for a series of increasingly
substituted alkanes, namely ethane (A), propane (B), isobutane (C), and
neopentane (D). Since, this time, we are making comparisons between different
oxy radicals with individual alkanes, $\Delta\Delta H$ depends only on the oxy
radicals (it is the difference in bond dissociation enthalpies of the O–H
bonds in the phenols of the two reactions), whereas $\Delta\Delta
H^{{\ddagger}}$ is different for different alkane substrates. The results are
shown in Table IV below:

| PhO (5) |  p-t-BuPhO (7)
---|---|---
$\Delta\Delta H$ | 0 | 1.6
ethane (A) $\Delta\Delta H^{{\ddagger}}$ | 0 | 1.1
propane (B) $\Delta\Delta H^{{\ddagger}}$ | 0 | 1.5
isobutane (C) $\Delta\Delta H^{{\ddagger}}$ | 0 | 1.3
neopentane (D) $\Delta\Delta H^{{\ddagger}}$ | 0 | 1.1
Table 4: Relative reaction enthalpies and relative barrier heights of hydrogen
abstraction for phenoxy and p-t-Bu-phenoxy radicals (in kcal/mol).

Because para-substitution in 7 does not create any additional steric
interactions in the HAT transition state or in the newly formed phenol O–H
bond in the product, comparing the HAT reactivity of phenoxy radicals 5 and 7
isolates the electronic effects. In particular, the p-tBu group in 7 simply
stabilizes the phenoxy radical starting material via hyperconjugation, thus
increasing the relative enthalpies of both the transition state and the
product. In particular, the reaction enthalpy ($\Delta H$) increases by 1.6
kcal/mol, while barrier heights increase by about 1.3 kcal/mol on average,
consistent with an Evans-Polanyi fitting parameter $\alpha$ of approximately
0.8.

###  V.3 Steric effects of phenoxy radicals in HAT reactions

Having considered electronic effects in isolation, we are ready to investigate
steric effects of phenoxy radicals on the rates of HAT reactions by
considering ortho-substituted phenoxy radicals. In particular, we compare the
reactivities of increasingly hindered phenoxy (5), 2,6-dimethylphenoxy (6),
and 2,6-di-tert-butylphenoxy (8) with ethane (A), propane (B), isobutane (C),
and neopentane (D). Results are shown in Table V. As in the previous section,
$\Delta\Delta H$ is the same for any alkane substrate while $\Delta\Delta
H^{{\ddagger}}$ is not.

| PhO (5) | di-MePhO (6) | di-t-BuPhO (8)
---|---|---|---
$\Delta\Delta H$ | 0 | 4.3 | 7.3
ethane (A) $\Delta\Delta H^{{\ddagger}}$ | 0 | 2.0 | 6.8
propane (B) $\Delta\Delta H^{{\ddagger}}$ | 0 | 2.1 | 6.2
isobutane (C) $\Delta\Delta H^{{\ddagger}}$ | 0 | 3.7 | 6.2
neopentane (D) $\Delta\Delta H^{{\ddagger}}$ | 0 | 2.8 | 5.6
Table 5: The relative barriers of hydrogen abstraction in kcal/mol.

As expected, for all four alkanes considered, the barrier heights $\Delta
H^{{\ddagger}}$ increase as the ortho-substituents become bulkier from H (5)
to Me (6) to t-Bu (7), owing to increasing steric repulsion in the transition
states between these ortho-substituents and atoms of the alkane. However,
perhaps more surprisingly, we also observe a corresponding increase in the
reaction enthalpies even though $\Delta H$ reflects the difference in enthalpy
between separated products and separated starting materials, meaning that no
interactions between the ortho-substituents and the atoms of the alkane is
possible.

To understand why bulky ortho-substituents lead to higher reaction enthalpies,
we begin by noting that the same electronic stabilization described in the
previous section for para-substituted phenoxy radical 7 also exists for ortho-
substituted 6 and 8. Given that 6 and 8 each have two alkyl substituents, we
can estimate this electronic stabilization as 2 × 1.6 kcal/mol = 3.2 kcal/mol.
However, this still leaves approximately 1.1 kcal/mol in 6 and 5.1 kcal/mol in
8 which cannot be explained by electronic effects.

To explain these remaining contributions to $\Delta\Delta H$, we identify a
substantial steric effect present in the alcohol product simply by virtue of
transferring an H atom to the phenoxy radical. In particular, as the ortho-
substituents beside the alcohol become bulkier, i.e., from H to Me to tBu, the
hydrogen on the phenol experiences more steric repulsion, raising the enthalpy
of the alcohol product. Figure 4 compares the shortest H–H distance between
the phenol hydrogen and the ortho-substituents in the phenol products of 5, 6,
and 8.

Figure 4: Comparing the shortest H–H distance (in Å) between the phenol
hydrogen and the ortho-substituents in phenol (top), 2,6-dimethylphenol
(center), and 2,6-di-tert-butylphenol (bottom). For comparison, twice the van
der Waals radius of hydrogen is 2.4 Å.

As the figure shows, the H–H bond distance decreases from 2.29 Å in phenol to
2.24 Å in 2,6-dimethylphenol to 1.83 Å in 2,6-di-tert-butylphenol. Given that
twice the van der Waals radius of hydrogen is 2.4 Å, this represents
considerable steric repulsion between the newly-formed O–H bond and the ortho-
substituents in the product phenols which grows in magnitude as the ortho-
substituents become bulkier. In this way, bulky ortho-substituents destabilize
product phenols relative to the starting phenoxy radicals and thus increase
the HAT reaction enthalpy $\Delta H$. Thus, perhaps unexpectedly at first
glance, ortho-substituents destabilize both the transition states and the
phenol products relative to the starting materials, resulting in an Evans-
Polanyi relationship between $\Delta H^{{\ddagger}}$ and $\Delta H$ that is
largely maintained even in the presence of steric bulk.

Quantitatively, ortho-methyl groups increase barrier heights by an average of
2.7 kcal/mol and reaction enthalpies by 4.3 kcal/mol, while ortho-tert-butyl
groups increase barrier heights by an average of 6.2 kcal/mol and reaction
enthalpies by 7.3 kcal/mol, leading to an Evans-Polanyi fitting parameter
$\alpha$ in the range of 0.6 to 0.85, not far from the estimated value of 0.8
in the absence of steric bulk.

Moving beyond phenoxy radicals, we observe a similar concomitant increase in
HAT barrier heights and reaction enthalpies when comparing the cyclohexoxy
radical (9) with the sterically-hindered 2,2,6,6-tetramethylcyclohexoy radical
(10), as shown in Table VI below.

| CyO (9) | 2,2,6,6-tetra-MeCyO (10)
---|---|---
$\Delta\Delta H$ | 0 | 1.5
ethane (A) $\Delta\Delta H^{{\ddagger}}$ | 0 | 3.0
propane (B) $\Delta\Delta H^{{\ddagger}}$ | 0 | 2.8
isobutane (C) $\Delta\Delta H^{{\ddagger}}$ | 0 | 2.8
neopentane (D) $\Delta\Delta H^{{\ddagger}}$ | 0 | 3.0
Table 6: Relative reaction enthalpies and relative barrier heights for
hydrogen abstraction for cyclohexoxy and 2,2,6,6-tetramethylcyclohexoxy
radicals (kcal/mol).

###  V.4 Steric attraction in HAT reactions of alkoxy radicals and alkanes

Finally, we consider the role of steric attraction, or dispersion
interactions, in the barrier heights of HAT reactions between alkoxy radicals
and alkanes. Among the alkoxy radicals in our study, the bulkiest is tert-
butoxy radical 4 while the least bulky is methoxy radical 1. With these two
alkoxy radicals, we can compare the most hindered and least hindered reactions
in our study for primary, secondary, and tertiary alkane C–H bonds. For
primary alkane C–H bonds, the least hindered HAT reaction is between methoxy
radical 4 and ethane A, while the most hindered reaction is between tert-
butoxy radical 4 with neopentane D. Similarly, for secondary alkane C–H bonds,
the least hindered reaction is between 1 and E and the most hindered reaction
is between 4 and H; for tertiary alkane C–H bonds, the least hindered reaction
is between 1 and I and the most hindered reaction is between 4 and L. Table
VII compares the relative barrier heights and relative reaction enthalpies
between these least hindered and most hindered HAT reactions for primary,
secondary, and tertiary C–H bonds.

|  Primary C–H (A,1) to (D,4) |  Secondary C–H (E,1) to (H,4) |  Tertiary C–H (I,1) to (L,4)
---|---|---|---
$\Delta\Delta H$ | -0.5 | -1.2 | -2.0
$\Delta\Delta H^{{\ddagger}}$ | -2.5 | -2.8 | -2.2
Table 7: Comparing reaction enthalpies and barrier heights for least hindered
and most hindered HAT reactions between alkoxy radicals and primary,
secondary, and tertiary alkane C–H bonds in kcal/mol.

As the table shows, for primary, secondary, and tertiary C–H bonds, both the
reaction enthalpy and the barrier heights decrease when the reactants go from
unhindered to bulky. Presumably, the decrease in reaction enthalpy $\Delta H$
is caused by increased hyperconjugative stabilization of the product alkyl
radical when the alkane is bulkier. However, the even greater decrease in the
barrier heights $\Delta H^{{\ddagger}}$ suggests the presence of stabilizing
dispersion interactions between the two bulky reactants in the transition
state.

To summarize, as the reactants in HAT reactions become more bulky, the
decrease in reaction enthalpy is primarily an electronic effect caused by
larger hyperconjugative stabilization of the product alkyl radical, while the
decrease in barrier heights is primarily a steric attraction effect caused by
stronger dispersion interactions that stabilize the transition state. Given
that the diverging origins of these two effects, it is not surprising that a
direct Evans-Polanyi relationship no longer holds when steric attraction in
the transition state becomes important.

##  VI Conclusion

In this study, the role of steric effects in the HAT process is explored by
analyzing 96 HAT reactions. The barrier height and reaction enthalpy is
controlled by three different effects, steric repulsion, electronic
stabilization and steric attraction or dispersion effects.

## Acknowledgements

We thank Huiling Shao for helpful discussions. K. N. H. acknowledges the
support of the National Science Foundation under award number CHE-2153972.
Computational resources were provided by the University of Cambridge.

## References

  * (1) Zhang, C., Chen, S., Ye, C. X., Harms, K., Zhang, L., Houk, K. N., Meggers, E. (2019). Asymmetric Photocatalysis by Intramolecular Hydrogen-Atom Transfer in Photoexcited Catalyst–Substrate Complex. Angewandte Chemie, 131(41), 14604-14608.
  * (2) Deng, Y., Nguyen, M. D., Zou, Y., Houk, K. N., Smith III, A. B. (2019). Generation of dithianyl and dioxolanyl radicals using photoredox catalysis: application in the total synthesis of the danshenspiroketallactones via radical relay chemistry. Organic letters, 21(6), 1708-1712.
  * (3) Cao, H., Tang, X., Tang, H., Yuan, Y., Wu, J. (2021). Photoinduced intermolecular hydrogen atom transfer reactions in organic synthesis. Chem Catalysis, 1(3), 523-598.
  * (4) Nikolaienko, P., Jentsch, M., Kale, A. P., Cai, Y., Rueping, M. (2019). Electrochemical and Scalable Dehydrogenative C(sp3)-H Amination via Remote Hydrogen Atom Transfer in Batch and Continuous Flow. Chemistry-A European Journal, 25(29), 7177-7184.
  * (5) Samec, J. S., Bäckvall, J. E., Andersson, P. G., Brandt, P. (2006). Mechanistic aspects of transition metal-catalyzed hydrogen transfer reactions. Chemical Society Reviews, 35(3), 237-248.
  * (6) Zassinovich, G., Mestroni, G., Gladiali, S. (1992). Asymmetric hydrogen transfer reactions promoted by homogeneous transition metal catalysts. Chemical Reviews, 92(5), 1051-1069.
  * (7) Cheong, P. H. Y., Morganelli, P., Luzung, M. R., Houk, K. N., Toste, F. D. (2008). Gold-catalyzed cycloisomerization of 1, 5-allenynes via dual activation of an ene reaction. Journal of the American Chemical Society, 130(13), 4517-4526.
  * (8) Laxmi, Y. S., Bäckvall, J. E. (2000). Mechanistic studies on ruthenium-catalyzed hydrogen transfer reactions. Chemical Communications, (7), 611-612.
  * (9) Bell, R. P. (1936). The theory of reactions involving proton transfers. Proceedings of the Royal Society of London. Series A-Mathematical and Physical Sciences, 154(882), 414-429.
  * (10) Evans, M. G., Polanyi, M. (1937). On the introduction of thermodynamic variables into reaction kinetics. Transactions of the Faraday Society, 33, 448-452.
  * (11) Roberts, B. P., Steel, A. J. (1994). An extended form of the Evans–Polanyi equation: a simple empirical relationship for the prediction of activation energies for hydrogen-atom transfer reactions. Journal of the Chemical Society, Perkin Transactions 2, (10), 2155-2162.
  * (12) Liu, F., Ma, S., Lu, Z., Nangia, A., Duan, M., Yu, Y., Xu, G., Mei, Y., Bietti, M., Houk, K.N. (2022). Hydrogen Abstraction by Alkoxyl Radicals: Computational Studies of Thermodynamic and Polarity Effects on Reactivities and Selectivities. Journal of the American Chemical Society, 144(15), 6802-6812.
  * (13) Mayer, J. M. (2011). Understanding hydrogen atom transfer: from bond strengths to Marcus theory. Accounts of Chemical Research, 44(1), 36-46.
  * (14) Yang, L. C., Li, X., Zhang, S. Q., Hong, X. (2021). Machine learning prediction of hydrogen atom transfer reactivity in photoredox-mediated C–H functionalization. Organic Chemistry Frontiers, 8(22), 6187-6195.
  * (15) Wagner, J. P., Schreiner, P. R. (2015). London dispersion in molecular chemistry—reconsidering steric effects. Angewandte Chemie International Edition, 54(42), 12274-12296.
  * (16) Bondi, A. V. (1964). van der Waals volumes and radii. The Journal of physical chemistry, 68(3), 441-451.
  * (17) Tolman, C. A. (1977). Steric effects of phosphorus ligands in organometallic chemistry and homogeneous catalysis. Chemical reviews, 77(3), 313-348.
  * (18) Crossley, M. J., Field, L. D., Forster, A. J., Harding, M. M., Sternhell, S. (1987). Steric effects on atropisomerism in tetraarylporphyrins. Journal of the American Chemical Society, 109(2), 341-348.
  * (19) Li, B., Xu, H., Dang, Y., Houk, K. N. (2022). Dispersion and steric effects on enantio-/diastereoselectivities in synergistic dual transition-metal catalysis. Journal of the American Chemical Society, 144(4), 1971-1985.
  * (20) Zhou, B., Haj, M. K., Jacobsen, E. N., Houk, K. N., Xue, X. S. (2018). Mechanism and origins of chemo-and stereoselectivities of aryl iodide-catalyzed asymmetric difluorinations of $\beta$-substituted styrenes. Journal of the American Chemical Society, 140(45), 15206-15218.
  * (21) Hicks, R. (Ed.). (2011). Stable radicals: fundamentals and applied aspects of odd-electron compounds. John Wiley & Sons.
  * (22) Power, P. P. (2003). Persistent and stable radicals of the heavier main group elements and related species. Chemical reviews, 103(3), 789-810.
  * (23) Francl, M. M., Pietro, W. J., Hehre, W. J., Binkley, J. S., Gordon, M. S., DeFrees, D. J., Pople, J. A. (1982). Self-consistent molecular orbital methods. XXIII. A polarization-type basis set for second-row elements. The Journal of Chemical Physics, 77(7), 3654-3665.
  * (24) Hariharan, P. C., Pople, J. A. (1973). The influence of polarization functions on molecular orbital hydrogenation energies. Theoretica chimica acta, 28(3), 213-222.
  * (25) Chai, J. D., Head-Gordon, M. (2008). Long-range corrected hybrid density functionals with damped atom–atom dispersion corrections. Physical Chemistry Chemical Physics, 10(44), 6615-6620.
  * (26) Chai, J. D., Head-Gordon, M. (2008). Systematic optimization of long-range corrected hybrid density functionals. The Journal of chemical physics, 128(8), 084106.
  * (27) Gaussian 09, Revision A.02, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, G. A. Petersson, H. Nakatsuji, X. Li, M. Caricato, A. Marenich, J. Bloino, B. G. Janesko, R. Gomperts, B. Mennucci, H. P. Hratchian, J. V. Ortiz, A. F. Izmaylov, J. L. Sonnenberg, D. Williams-Young, F. Ding, F. Lipparini, F. Egidi, J. Goings, B. Peng, A. Petrone, T. Henderson, D. Ranasinghe, V. G. Zakrzewski, J. Gao, N. Rega, G. Zheng, W. Liang, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, K. Throssell, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. Bearpark, J. J. Heyd, E. Brothers, K. N. Kudin, V. N. Staroverov, T. Keith, R. Kobayashi, J. Normand, K. Raghavachari, A. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, J. M. Millam, M. Klene, C. Adamo, R. Cammi, J. W. Ochterski, R. L. Martin, K. Morokuma, O. Farkas, J. B. Foresman, and D. J. Fox, Gaussian, Inc., Wallingford CT, 2016.


# TransformerFAM: Feedback attention is working memory

Dongseong Hwang  Weiran Wang  Zhuoyuan Huo  Khe Chai Sim  Pedro Moreno
Mengibar


# Interactive Latent Interpolation on MNIST Dataset

Mazeyar Moeini Feizabadi  Ali Mohammed Shujjat  Sarah Shahid  Zainab Hasnain

###### Abstract

This paper will discuss the potential of dimensionality reduction with a web-
based use of GANs. Throughout a variety of experiments, we show synthesizing
visually-appealing samples, interpolating meaningfully between samples, and
performing linear arithmetic with latent vectors. GANs have proved to be a
remarkable technique to produce computer generated images, very similar to an
original image. This is primarily useful when coupled with dimensionality
reduction as a effective application of our algorithm. We proposed a new
architecture for GANs, which ended up not working for mathematical reasons
later explained. We then proposed a new web-based GAN that still takes
advantage of dimensionality reduction to speed generation in the browser to .2
milliseconds. Lastly, we made a modern UI with linear interpolation to present
the work. With the speedy generation we can generate so fast that we can
create a animation type effect that we have never seen before that works on
both web and mobile.

##  1 Introduction

The interactive display of latent interpolation on MNIST dataset is done using
GANs. Generative Adversarial Networks, as its name suggests, work like on the
basis of two adversarial frameworks. The first is the generative model which
learns from the original datasets and tries to copy it. It learns from the
input images in the process of producing ’fake’ images.

On the other had, the discriminative model tries to distinguish between the
original image and the ’fake’ image produced by the generative model. Both the
models work simultaneously until the optimal efficiency is achieved.

The required level of accuracy is an amalgamation of the generative and
discriminative model both producing high level of accuracy. The better the
generative model works, the more the ’fake’ image resembles the original
image. Consequently, the error rate is high in the result of the
discriminative model.

Initially, the discriminator gives higher values to the original images which
is used by the generative model to learn better and produce better quality
’fake’ images. This in turn reduces the accuracy of the discriminative model
as it fails to discriminate between the original and the fake image. This is
when we know that both out neural networks are successfully achieving their
purpose.
This is briefly explained in the methodology part later and is shown in the
diagram. We finally used latent space interpolation to get the desired
results.

##  2 Initial idea

The initial proposal presented was to train a generative adversarial network
(GAN) on the MNIST dataset with its dimensionality reduced by using principle
component analysis(PCA).

For certain models features or dimensions can decrease it’s accuracy due to
the large amount of data that needs to be generalized, famously known as the
curse of dimensionality. The reduction of dimensionality is a way to reduce
the complexity of a model and to avoid overfitting, a condition when the model
learns the training data too well and fails to provide useful output when
exposed to new unseen data. This is where PCA comes in.

"Principal component analysis is a mathematical algorithm that reduces the
dimensionality of the data while retaining most of the variation in the data
set."(Jolliffe & Cadima, 2016)

GANs is used to generate completely new datapoints, to present photorealistic
images. However it trains two networks, the generator and the discriminator,
simultaneously on the images provided which proves to be very computationally
expensive and in some cases overfits the data.

The objective of the project was to apply principle component analysis on our
selected dataset and provide the reduced dimensionality dataset to train the
GAN. This would essentially reduced the intensive operation, making it more
functional. Unfortunately, we soon ran into some issues. We managed to
successfully implement PCA to the data and extract the most important
principal components, reducing the dimensionality in the process.

###  2.1 Failure

However, the new dataset was no longer representative of the original dataset,
using this as an input to the GAN did not yield an expected result as it could
not learn the features needed to generate a new image resembling the original
dataset, therefore we ended up with a distorted image that we could not use
and eventually decided to pivot.

This is partially due to the nature of PCA. Most of the energy is stored in
the first few variables, however at least 200 variables are needed for
denoising. This means often the top 10% variables have weights x10,000 times
bigger than the bottom 90% percentile. Given that our dataset needs to be
normalized for training the data started to become ambiguous.

As seen the PCA only holds most of its information inside of the first few
layers. Another problem of variance is created.

This can also be seen with the PCA representation of the 8’s in the top left
the figure 8’s are distinctive however later on they become more ambiguous and
more useful for noise reduction.

##  3 New idea

Our new idea continues to work with the MNIST data set. However, we now work
towards the idea of showcasing linear interpolation on latent space using a
generative adversarial network.

In its simplest form linear interpolation involves the generation of new
values or points based on the existing set of values. Linear interpolation is
primarily achieved by geometrically rendering a straight line between two
adjacent points on a graph or plane. All points on the line connecting the two
points other than the original two can be considered interpolated values.

Using this idea we can interpolate data in the latent space and generate new
data samples using our GANs model. The latent space is a representation of
compressed data in which similar data points are closer together in space.
This is particularly useful for learning data features, such as writing
patterns in our dataset and for finding simpler representations of data for
analysis.

Applying this to the context of our project would mean that through latent
space interpolation along with GANs we would be able to see new handwritten
numbers be generated in real-time as digits shift from one to another.

Finally, we plan on making this an interactive experience for the user by
making it available through a website. Our model is trained in browser to
facilitate mobility and accessibility for users and can be accessed on
desktops and mobile phones alike.

##  4 Tools

###  4.1 PyTorch

PyTorch is an open source machine learning library developed by Facebook’s AI
Research lab. Its applications vary from computer vision and natural language
processing to deep convolutional neural networks. It is a Python-based
scientific computing package targeted to users as a deep learning research
platform that provides maximum flexibility and speed.

###  4.2 ONNX

ONNX, Open Neural Network Exchange, is an open format built to represent
machine learning models. ONNX defines a common set of operators used in
machine and deep learning models along with a common file format to enable AI
developers to use models with a variety of frameworks, tools, runtimes, and
compilers. This enables us to design a deep learning model using the PyTorch
framework and deploy it in a browser.

###  4.3 Generative Adversarial Networks - GANs

Generative Adversarial Networks (GANs) are a class of neural networks that are
used for unsupervised learning. It was developed and introduced by Ian J.
Goodfellow in 2014. GANs consist of a system of two competing neural network
models that work together to capture and analyze properties in a dataset and
produce a new unseen dataset. GANs can also be used to created new datasets
for supervised learning for certain classification problems.

Generate realistic photographs, face aging, photo blending, super resolution
and even deep fakes are just some of the impressive applications of GANs.

###  4.4 MNIST

MNIST classification is a classic problem in machine learning. The problem is
to look at greyscale 28x28 pixel images of handwritten digits and determine
which digit the image represents, for all the digits from zero to nine.

The dataset is available in different testing and training sizes which are
accessible here.

##  5 Literature review

Since Ian J. Goodfellow proposed and developed the idea of Generative
Adversarial Nets i.e. GANs in 2014, there have been variety of experiments and
researches involving this topic. GANs were described as a framework for
training two models (i.e. the Generator G and the Discriminator) through an
adversarial process. Both the models were had their own functionality with
respect to the dataset. The Generator was required to’ capture’ the data
distribution whereas the Discriminator was supposed to correctly discriminate
between the generated points and the ones from the real dataset. Both the
models display a minimax game functionality, one model maximizes the
classification error (i.e. the Generator tries to fool the Discriminator and
create images that are difficult to classify) and the Discriminator tries to
minimize the classification error. This results in a saddle-like graph of the
functionality of both models.

The work talks about how both the models are trained through unsupervised
learning and follow back propagation and compete with each other in an
adversarial modeling framework. The theoretical results are more probability
based than deterministic and further algorithms are identified to apply
practicality to the theory based approach.

Since our research was not just GANs based but GANs in Latent Space, the
second literature piece was relevant to a great extent. Our research involved
linear interpolation in latent space i.e. a space where all newly generated
samples are in the form of compressed data where similar data points are
closer to each other. The paper explores how success of GANs is specific to
their performance in saddle point optimization and parameterizing the
Generator and Discriminator as Deep Convolutional Neural Networks. The
research went on to introduce Generative Latent Optimization (GLOs) which are
similar in functioning to GANs but perform without the basic adversarial
optimization scheme of GANs.

##  6 Novelty

Even though there has not been much work done on the latent space of GANs, we
managed to successfully model it using linear interpolation and with the other
tools at hand. We are proud to say that this is the first browser based
interactive latent extrapolation. Users can generate new hand-written digits
and see digits morph into others all with the press of a button. This is just
the first roll-out, we hope to continue working and release updates to improve
user experience.

##  7 Methodology

###  7.1 Training GANs

The GANs model is based on two neural networks working with each other. The
neural network corresponding to the generative part of the model follows the
unsupervised learning approach on the data that we provide. This is then
responsible for creating new data instances. It then captures the distribution
of data and maximises the final classification error.

The neural network corresponding to the discriminator part is responsible for
minimizing the final classification error and discriminate between the real
images and the ones generated i.e. opposing the generator neural network. The
discriminator does this by using binary classification with the help of an
activation function that gives a certain probability for the authenticity of
the image.

In our context of the MNIST dataset we could say that the goal of the
generator is to generate passable hand-written digits that are similar to the
original dataset. Whereas the goal of the discriminator is to identify images
coming from the generator as fake.

The GANs are formulated as a minimax game with value function V(D,G), We train
the discriminator, D, to maximize the probability of assigning the correct
label to both training examples and samples from the generator, G. We
simultaneously train G to produced instances that minimize the chance of being
identified by D. It can be mathematically described by the formula below:

$min_{G}max_{D}V(D,G)=E_{x_{\textasciitilde}p_{data}(x)}[logD(x)]+E{z_{\textasciitilde}p_{z}(z)}[log(1-D(G(z)))].$
(1)

Our final generator is just a dense neural networks as seen below. This
network has 1,486,352 parameters, we will try to reduce this later on once
ONNX supports more advanced operators.

###  7.2 Latent Space Interpolation

Latent space Interpolation can be thought of as a continuous walk between to
points. With vectors, this can be thought of as a walk all the points to the
other. If the known points are given by the vectors $S$ and $T$, the linear
interpolation is the straight line between these points. This is amazing
because it allows us to see in between the numbers, and now the GAN interprets
similar numbers.

$X(t)_{i}=(1-t)S_{i}+tT_{i},0\leq t\leq 1$ (2)

##  8 Final Product

After much revision we present the final version of our project. It is now
live and can be accessed with the link to it here. Shown below is a screenshot
of the user interface encountered once you arrive on the web-page.

Pressing the ’Generate’ button will generate a single new image for a hand-
written digit. Pressing the ’Explore’ button begins the interpolation action
and you get to see digits shift from one number to another. It must be noted
that the first time the button is pressed it may take a few seconds before the
result appears on screen as the model is being trained in the background and
requires some time to perform all the computations needed to accomplish the
task.

##  9 Limitations / Challenges

Since we were using ONNX, we were limited to only the operators it provided.
Despite the many functionalities ONNX provides, it does not have an operator
for conv2dtranspose therefore we had to settle for convtranspose which limits
the upsampling operation in contrast to its 2d version.

Furthermore, even though we managed to successfully implement the model and
deploy it on the web the images are not as clear as we wanted. A way to
sharpen the images would be to use a deep convolutional GAN (DCGAN). These are
one of the most popular adaptations of GANs. It is composed of convolutional
neural networks in place of multi-layer perceptrons. Although ONNX does
provide the capabilities of implementing such GANs, there are many work
arounds required and much understanding of the documentation is needed. We
were limited by time and could not meet due to the pandemic, therefore we
settled for the success we achieved with the traditional GANs.

##  10 Conclusion

In conclusion, GANs are very useful for many application, one of which we have
used on hand-written digits. We also provided a unique method for making this
more interactive through a web-page and making it more accessible through any
browser on any device.. However, there is much more that could and we look
forward to in the future.

##  11 Future Work

For the future we hope to build up on the work we have done here and continue
to make such fascinating applications of machine learning more accessible to
the greater public.

We plan on implementing this again using a DCGAN for a clearer image so that
we can observe the latent space interpolation in action more distinctly.
However, we do not want to compromise user experience and we will deploy the
model using ONNX so that it may be accessed using a browser.

## References

  * [1] Bojanowski, P., Joulin, A., Lopez-Paz, D., & Szlam, A. (2017). Optimizing the latent space of generative networks. arXiv preprint arXiv:1707.05776.
  * [2] Goodfellow, I., Pouget-Abadie, J., Mirza, M., Xu, B., Warde-Farley, D., Ozair, S., … & Bengio, Y. (2014). Generative adversarial nets. In Advances in neural information processing systems.
  * [3] Wang, Z. (2020, February 16). Interpolation with Generative Models. Retrieved June 17, 2020, from https://towardsdatascience.com/interpolation-with-generative-models-eb7d288294c
  * [4] Zhang, A. (2019). Dong shou xue shen du xue xi = Dive into deep learning. Beijing Shi: Ren min you dian chu ban she.
  * [5] Strang, G. (2019). Linear algebra and learning from data. Wellesley, MA: Wellesley-Cambridge Press.
  * [6] Zeiler, M. D., & Fergus, R. (2014, September). Visualizing and understanding convolutional networks. In European conference on computer vision (pp. 818-833). Springer, Cham.


# Steenrod operations on the modular
invariants

Nguyễn Sum

###### Abstract.

In this paper, we compute the action of the mod $p$ Steenrod operations on the
modular invariants of the linear groups with $p$ an odd prime number.

###### Key words and phrases:

Polynomial algebra, cohomology operations, modular invariants

###### 2010 Mathematics Subject Classification:

Primary 55S10; Secondary 55S05

## Introduction

Fix an odd prime $p$. Let $A_{p^{n}}$ be the alternating group on $p^{n}$
letters. Denote by $\Sigma_{p^{n},p}$ a Sylow $p$-subgroup of $A_{p^{n}}$ and
$E^{n}$ an elementary abelian $p$-group of rank $n$. Then we have the
restriction homomorphisms

$\displaystyle\mbox{Res}(E^{n},\Sigma_{p^{n},p}):H^{*}(B\Sigma_{p^{n},p})\longrightarrow
H^{*}(BE^{n}),$
$\displaystyle\mbox{Res}(E^{n},A_{p^{n}}):H^{*}(BA_{p^{n}})\longrightarrow
H^{*}(BE^{n}),$

induced by the regular permutation representation
$E^{n}\subset\Sigma_{p^{n},p}\subset A_{p^{n}}$ of $E^{n}$ (see Mùi [4]). Here
and throughout the paper, we assume that the coefficients are taken in the
prime field $\mathbb{Z}/p$. Using modular invariant theory of linear groups,
Mùi proved in [3, 4] that

$\displaystyle\mbox{ImRes}(E^{n},\Sigma_{p^{n},p})=E(U_{1},\ldots,U_{n})\otimes
P(V_{1},\ldots,V_{n}),$
$\displaystyle\mbox{ImRes}(E^{n},A_{p^{n}})=E(\tilde{M}_{n,0},\ldots,\tilde{M}_{n,n-1})\otimes
P(\tilde{L}_{n},Q_{n,1},\ldots,Q_{n,n-1}),$

Here and in what follows, $E(.,\ldots,.)$ and $P(.,\ldots,.)$ are the exterior
and polynomial algebras over $\mathbb{Z}/p$ generated by the variables
indicated. $\tilde{L}_{n},\,Q_{,s}$ are the Dickson invariants of dimensions
$p^{n},\,2(p^{n}-p^{s})$, and $\tilde{M}_{n,s},,U_{k},\,V_{k}$ are the Mùi
invariants of dimensions $p^{n}-2p^{s},\,p^{k-1},\,2p^{k-1}$ respectively (see
Section 1).

Let $\mathcal{A}$ be the mod $p$ Steenrod algebra and let $\tau_{s},\,\xi_{i}$
be the Milnor elements of dimensions $2p^{s}-1,\,2p^{i}-2$ respectively in the
dual algebra $\mathcal{A}_{*}$ of $\mathcal{A}$. In [7], Milnor showed that,
as an algebra

$\mathcal{A}_{*}=E(\tau_{0},\tau_{1},\ldots)\otimes
P(\xi_{1},\xi_{2},\ldots).$

Then $\mathcal{A}_{*}$ has a basis consisting of all monomials
$\tau_{S}\xi^{R}=\tau_{s_{0}}\ldots\tau_{s_{k}}\xi^{r_{1}}\ldots\xi^{r_{m}}$,
with $S=(s_{1},\ldots,s_{k}),\,0\leqslant s_{1}<\ldots<s_{k}$,
$R=(r_{1},\ldots,r_{m}),\,r_{i}\geqslant 0$. Let $St^{S,R}\in\mathcal{A}$
denote the dual of $\tau_{S}\xi^{R}$ with respect to that basis. Then
$\mathcal{A}$ has a basis consisting all operations $St^{S,R}$. For
$S=\emptyset,\,R=(r),\,St^{\emptyset,(r)}$ is nothing but the Steenrod
operation $P^{r}$.

Since $H^{*}(BG),\ G=E^{n},\,\Sigma_{p^{n},p}$ or $A_{p^{n}}$, is an
$\mathcal{A}$-module (see [13, Chap. VI]) and the restriction homomorphisms
are $\mathcal{A}$-linear, their images are $\mathcal{A}$-submodules of
$H^{*}(BE^{n})$.

The purpose of the paper is to study the module structures of
$\mbox{ImRes}(E^{n},\Sigma_{p^{n},p})$ and $\mbox{ImRes}(E^{n},A_{p^{n}})$
over the Steenrod algebra $\mathcal{A}$. More precisely, we prove a duality
relation between $St^{S,R}(\tilde{M}_{n,s}^{\delta}Q_{n,s}^{1-\delta})$ and
$St^{S^{\prime},R^{\prime}}(U_{k+1}^{\delta}V_{k+1}^{1-\delta})$ for
$\delta=0,1,\,\ell(R)=k$ and $\ell(R^{\prime})=n$. Here by the length of a
sequence $T=(t_{1},\ldots,t_{q})$ we mean the number $\ell(T)=q$. Using this
relation we explicitly compute the action of the Steenrod operations $P^{r}$
on $U_{k+1},\,V_{k+1},\,M_{n,s}$ and $Q_{n,s}$.

The analogous results for $p=2$ have been announced in [11].

The action of $P^{r}$ on $V_{k+1}$ and $Q_{n,s}$ has partially studied by
Campbell [1], Madsen [5], Madsen-Milgram [6], Smith-Switzer [12], Wilkerson
[14]. Eventually, this action was completly determined by Hung-Minh [10] and
by Hai-Hung [8], Hung [9] for the case of the coefficient ring $\mathbb{Z}/2$.

The paper contains 3 sections. After recalling some needed information on the
invariant theory, the Steenrod homomorphism $d_{n}^{*}P_{n}$ and the
operations $St^{S,R}$ in Section 1, we prove the duality theorem and its
corollaries in Section 2. Finally, Section 3 is an application of the duality
theorem to determine the action of the Steenrod operations on the Dickson and
Mùi invariants.

## Acknowledgement

The author expresses his warmest thanks to Professor Huỳnh Mùi for generous
help and inspiring guidance. He also thanks Professor Nguyễn H.V. Hưng for
helpful suggestions which lead him to this paper.

##  1\. Preliminaries

As is well-known $H^{*}(BE^{n})=E(x_{1},\ldots,x_{n})\otimes
P(y_{1},\ldots,y_{n})$ where $\dim x_{i}=1,\,y_{i}=\beta x_{i}$ with $\beta$
the Bockstein homomorphism. Following Dickson [2] and Mùi [3], we define

$\displaystyle[e_{1},\ldots,e_{k}]=\det(y_{i}^{p^{e_{j}}}),$
$\displaystyle[1;e_{2},\ldots,e_{k}]=\begin{vmatrix}x_{1}&\cdots&x_{k}\\\
y_{1}^{p^{e_{2}}}&\cdots&y_{k}^{p^{e_{2}}}\\\ \vdots&\cdots&\vdots\\\
y_{1}^{p^{e_{k}}}&\cdots&y_{k}^{p^{e_{k}}}\end{vmatrix}.$

for every sequence of non-negative integers $(e_{1},\ldots,e_{k}),\,1\leqslant
k\leqslant n$. We set

$\displaystyle
L_{k,s}=[0,\ldots,\hat{s},\ldots,k],\,L_{k}=L_{k,k}=[0,\ldots,k-1],\,L_{0}=1,$
$\displaystyle M_{k,s}=[1;0,\ldots,\hat{s},\ldots,k-1],\,0\leqslant
s<k\leqslant n.$

Then $\tilde{L}_{n},\,Q_{n,s},\,\tilde{M}_{n,s},\,U_{k},\,V_{k}$ are defined
by

$\displaystyle\tilde{L}_{n}=L_{n}^{h},\,h=(p-1)/2,\,Q_{n,s}=L_{n,s}/L_{n},\,0\leqslant
s\leqslant n,$
$\displaystyle\tilde{M}_{k,s}=M_{n,s}L_{n}^{h-1},\,U_{k}=M_{k,k-1}L_{k-1}^{h-1},\,V_{k}=L_{k}/L_{k-1},\,1\leqslant
k\leqslant n.$

Note that $Q_{n,0}=\tilde{L}_{n}^{2}$, $Q_{n,n}=1$ for any $n>0$.

Let X be a topological space. Then we have the Steenrod power map

$P_{n}:H^{q}(X)\,\longrightarrow
H^{p^{n}q}(EA_{p^{n}}\underset{A_{p^{n}}}{\times}X^{p^{n}}),$

which sends $u$ to $1\otimes u^{p^{n}}$ at the cochain level (see [13, Chap.
VII]). We also have the diagonal homomorphism

$d_{n}^{*}:H^{*}(EA_{p^{n}}\underset{A_{p^{n}}}{\times}X^{p^{n}})\longrightarrow\,H^{*}(BE^{n})\otimes
H^{*}(X)$

induced by the diagonal map of $X$, the inclusion $E^{n}\subset A_{p^{n}}$ and
the Künneth formula.

$d_{n}^{*}P_{n}$ has the following fundamental properties.

######  Proposition 1.1 (Mùi [3, 4]).

(i) $d_{n}^{*}P_{n}$ is natural monomorphism preserving cup product up to a
sign, more precisely

$d^{*}_{n}P_{n}(uv)=(-1)^{nhqr}d^{*}_{n}P_{n}ud^{*}_{n}P_{n}v\ ,$

where $q=\dim u,\,r=\dim v,\,h=(p-1)/2.$

(ii) $d^{*}_{n}P_{n}=d^{*}_{n-s}P_{n-s}d^{*}_{s}P_{s}\ ,\ 0\leqslant
s\leqslant n.$

(iii) For $H^{*}(E^{1})=E(x)\otimes P(y)$, we have

$\displaystyle
d_{n}^{*}P_{n}x=(-h!)^{n}U_{n+1}=(h!)^{n}\Big{(}\tilde{L}_{n}x+\sum_{s=0}^{n-1}(-1)^{s+1}\tilde{M}_{n,s}y^{p^{s}}\Big{)},$
$\displaystyle
d_{n}^{*}P_{n}y=V_{n+1}=(-1)^{n}\sum_{s=0}^{n}(-1)^{s}Q_{n,s}y^{p^{s}}.$

where $U_{n+1}=U_{n+1}(x_{1},\ldots,x_{n},x,y_{1},\ldots,y_{n},y)$,
$V_{n+1}=V_{n+1}(y_{1},\ldots,y_{n},y)$.

The following is a description of $d_{n}^{*}P_{n}$ in terms of modular
invariants and cohomology operations.

######  Theorem 1.2 (Mùi [4, 1.3]).

Let $z\in H^{q}(X)$, $\mu(q)=(h!)^{q}(-1)^{hq(q-1)/2}$. We then have

$d_{n}^{*}P_{n}z\,=\,\mu(q)^{n}\sum_{S,R}(-1)^{r(S,R)}\tilde{M}_{n,s_{1}}\ldots\tilde{M}_{n,s_{k}}\tilde{L}_{n}^{r_{0}}Q_{n,1}^{r_{1}}\ldots
Q_{n,n-1}^{r_{n-1}}\ \otimes\ St^{S,R}z\ .$

Here the sum rwns over all $(S,R)$ with $S=(s_{1},\ldots,s_{k})$, $0\leqslant
s_{1}<\ldots<s_{k}$, $R=(r_{1},\ldots,r_{n}),\,r_{i}\geqslant 0$,
$r_{0}=q-k-2(r_{1}+\ldots\ +r_{n})\geqslant 0,\ r(S,R)=k+s_{1}+\ldots\
+s_{k}+r_{1}+2r_{2}+\ \ldots\ +nr_{n}\ .$

##  2\. The duality theorem

Let $\tilde{m}_{m,s},\,\tilde{\ell}_{m},\,q_{m,s},\,m=n$ or $k$, (resp.
$u_{k+1},\,v_{k+1}$) be the dual of
$\tilde{M}_{m,s},\,\tilde{L}_{m},\,Q_{m,s}$ (resp. $U_{k+1},\,V_{k+1}$) in

$E(\tilde{M}_{m,0},\ldots,\tilde{M}_{m,m-1})\otimes
P(\tilde{L}_{m},Q_{m,1}\ldots,Q_{m,m-1})$

(resp. $E(U_{k+1})\otimes P(V_{k+1})$) with respect to the basis consisting of
all monomials

$\displaystyle\tilde{M}_{S}\tilde{Q}^{H}$
$\displaystyle=\tilde{M}_{m,s_{1}}\ldots\tilde{M}_{m,s_{k}}\tilde{L}_{m}^{h_{0}}Q_{m,1}^{h_{1}}\ldots
Q_{m,m-1}^{h_{m-1}},$

with $S=(s_{1},\ldots,s_{k}),\,0\leqslant
s_{1}<\ldots<s_{k},\,H=(h_{0},\ldots,h_{m-1}),\,h_{i}\geqslant 0,$ (resp.
$U_{k+1}^{e}V_{k+1}^{j};\,e=0,1,\ j\geqslant 0$). Let
$\Gamma(\tilde{\ell}_{m},q_{m,1},\ldots,q_{m,m-1})$ (resp. $\Gamma(v_{k+1})$)
be the divided polynomial algebra with divided power $\gamma_{i},\,i\geqslant
0$ generated by $\tilde{\ell}_{m},\ q_{m,1},\ldots$, $q_{m,m-1}$ (resp.
$v_{k+1}$). We set

$\tilde{m}_{S}\tilde{q}_{H}=\tilde{m}_{m,s_{1}}\ldots\tilde{m}_{m,s_{k}}\gamma_{h_{0}}(\tilde{\ell}_{m})\gamma_{h_{1}}(q_{m,1})\ldots\gamma_{h_{m-1}}(q_{m,m-1}).$

For $q\geqslant 0$ and $R=(r_{1},\ldots,r_{m})$, set

$R_{q}^{*}=(q-2(r_{1}+\ldots+r_{m}),r_{1},\ldots,r_{m-1}).$

Let $V$ be a vector space over $\mathbb{Z}/p$ and $V^{*}$ be its dual. Denote
by

$\langle.,.\rangle:V\otimes V^{*}\longrightarrow\mathbb{Z}/p$

the dual pairing.

The main result of the section is

######  Theorem 2.1.

Suppose given $e,\delta=0,1,\,j\geqslant 0,\,(S,R)$, and
$(S^{\prime},R^{\prime})$ with
$\ell(R)=k,\,\ell(R^{\prime})=n,\,\ell(S)=t\leqslant
k,\,\ell(S^{\prime})=t^{\prime}\leqslant n$. Set
$\sigma=r(S,R)+r(S^{\prime},R^{\prime})+s+\delta+(t+[-2p^{s}])t^{\prime}+nhk\delta,$
with $-\delta\leqslant s\leqslant n-\delta$. Then we have

$\langle\tilde{m}_{S}\tilde{q}_{R^{*}_{(2-\delta)p^{n}-e-2j-t}}\otimes
u_{k+1}^{e}\gamma_{j}(v_{k+1}),St^{S^{\prime},R^{\prime}}\big{(}U_{k+1}^{\delta}V_{k+1}^{1-\delta}\big{)}\rangle\\\
=\begin{cases}(-1)^{\sigma}\langle\tilde{m}_{S^{\prime}}\tilde{q}_{{R^{\prime}}^{*}_{(2-\delta)p^{k}-t^{\prime}}},St^{S,R}\big{(}\tilde{M}_{n,s}^{\delta}Q_{n.s}^{1-\delta}\big{)}\rangle,&e+2j=-[-2p^{s}],\\\
0,&\text{otherwise.}\end{cases}$

Here, by convention, $\tilde{M}_{n,-1}=\tilde{L}_{n}.$

###### Proof.

We prove the theorem for $\delta=1$. For $\delta=0$, it is similarly proved.
We set

$\displaystyle U$
$\displaystyle=U_{n+k+1}(x_{1},\ldots,x_{k},x^{\prime}_{1},\ldots,x^{\prime}_{n},x,y_{1},\ldots,y_{k},y^{\prime}_{1},\ldots,y^{\prime}_{n},y),$
$\displaystyle U^{\prime}$
$\displaystyle=U_{n+k+1}(x^{\prime}_{1},\ldots,x^{\prime}_{n},x_{1},\ldots,x_{k},x,y^{\prime}_{1},\ldots,y^{\prime}_{n},y_{1},\ldots,y_{k},y).$

It is easy to verify that

(a) $\displaystyle U\ =\ (-1)^{nkh}U^{\prime}\ .$

Computing directly from Proposition 1.1 gives

(b) $\displaystyle U$
$\displaystyle=(-h!)^{-k}d^{*}_{k}P_{k}U_{n+1}(x^{\prime}_{1},\ldots,x^{\prime}_{n},x,y^{\prime}_{1},\ldots,y^{\prime}_{n},y)$
$\displaystyle=(-h!)^{-k}(-1)^{n}d_{k}^{*}P_{k}\big{(}\sum_{s=-1}^{n-1}(-1)^{s+1}\tilde{M}_{n,s}y^{p^{s}}\big{)}$
$\displaystyle=(-1)^{n}\sum_{s=-1}^{n-1}(-h!)^{-(s+1)k/(|s|+1)}(-1)^{s+1}(d_{k}^{*}P_{k}\tilde{M}_{n,s})V_{k+1}^{p^{s}}\
.$

Here by convention $y^{1/p}=x$, and $V_{k+1}^{1/p}=U_{k+1}$.

We observe that $\dim\tilde{M}_{n,s}=p^{n}+[-2p^{s}]$. According to Theorem
1.2 we have

(c) $\displaystyle
d^{*}_{k}P_{k}\tilde{M}_{n,s}=\mu(p^{n}+[-2p^{s}])^{k}\sum_{S,R}(-1)^{r(S,R)}\tilde{M}_{S}\tilde{Q}^{R^{*}_{p^{n}+[-2p^{s}]-t}}St^{S,R}\tilde{M}_{n,s}.$

A simple computation shows that

(d) $\displaystyle(-h!)^{-(s+1)/(|s|+1)}\mu(p^{n}+\ [-2p^{s}])=(-1)^{nh}.$

Combining (b),(c) and (d) we get

$U=\sum_{s=-1}^{n-1}\Big{(}\sum_{S,R}(-1)^{n(kh+1)+r(S,R)+s+1}\tilde{M}_{S}\tilde{Q}^{R^{*}_{p^{n}+[-2p^{s}]-t}}St^{S,R}\tilde{M}_{n,s}\Big{)}V_{k+1}^{p^{s}}.$

From this, we see that it implies

(e)
$\displaystyle(-1)^{r(S,R)+n(hk+1)+s+1}\langle\tilde{m}_{S}\tilde{q}_{R^{*}_{p^{n}-e-2j-t}}\otimes\tilde{m}_{S^{\prime}}\tilde{q}_{{R^{\prime}}^{*}_{p^{k}-t^{\prime}}}\otimes
u_{k+1}^{e}\gamma_{j}(v_{k+1}),U\rangle$
$\displaystyle\qquad=\begin{cases}(-1)^{tt^{\prime}}\langle\tilde{m}_{S^{\prime}}\tilde{q}_{{R^{\prime}}^{*}_{p^{k}-t^{\prime}}},St^{S,R}(\tilde{M}_{n,s})\rangle,&e+2j=-[-2p^{s}],\\\
0\ ,&\text{otherwise.}\end{cases}$

On the other hand, from Proposition 1.1 and Theorem 1.2 we have

$\displaystyle U^{\prime}$
$\displaystyle=(-h!)^{-n}d_{n}^{*}P_{n}U_{k+1}(x_{1},\ldots,x_{k},x,y_{1},\ldots,y_{k},y)$
$\displaystyle=(-h!)^{-n}\mu(p^{k})^{n}\sum_{S^{\prime},R^{\prime}}(-1)^{r(S^{\prime},R^{\prime})}\tilde{M}_{S^{\prime}}\tilde{Q}^{{R^{\prime}}^{*}_{p^{k}-t^{\prime}}}St^{S^{\prime},R^{\prime}}U_{k+1}.$

From this and the fact that $(-h!)^{-1}\mu(p^{k})=(-1)^{hk}$, we get

(f)
$\displaystyle(-1)^{r(S^{\prime},R^{\prime})+n(hk+1)}\langle\tilde{m}_{S}\tilde{q}_{R^{*}_{p^{n}-e-2j-t}}\otimes\tilde{m}_{S^{\prime}}\tilde{q}_{{R^{\prime}}^{*}_{p^{k}-t^{\prime}}}\otimes
u_{k+1}^{e}\gamma_{j}(v_{k+1}),U^{\prime}\rangle$
$\displaystyle\quad=(-1)^{t^{\prime}e}\langle\tilde{m}_{S}\tilde{q}_{R^{*}_{p^{n}-e-2j-t}}\otimes
u_{k+1}^{e}\gamma_{j}(v_{k+1}),St^{S^{\prime},R^{\prime}}U_{k+1}\rangle.$

Comparing (e) with (f) and using (a), we obtain the theorem for $\delta=1$. ∎

Since the basis $\\{\tilde{M}_{S^{\prime}}\tilde{Q}^{H^{\prime}}\\}$ of
$E(\tilde{M}_{n,0},\ldots,\tilde{M}_{n,n-1})\otimes
P(\tilde{L}_{n},Q_{n,1},\ldots,Q_{n,n-1})$ is dual to the basis
$\\{\tilde{m}_{S^{\prime}}\tilde{q}^{H^{\prime}}\\}$ of
$E(\tilde{m}_{n,0},\ldots,\tilde{m}_{n,n-1})\otimes\Gamma(\tilde{\ell}_{n},q_{n,1},\ldots,q_{n,n-1})$.
Hence, we easily obtain from Theorem 2.1

######  Corollary 2.2.

Set

$C_{S^{\prime},R^{\prime}}=\langle\tilde{m}_{S}\tilde{q}_{R^{*}_{(2-\delta)p^{n}+[-2p^{s}]-t}}\otimes\gamma_{p^{s}}(v_{k+1}),St^{S^{\prime},R^{\prime}}\big{(}U_{k+1}^{\delta}V_{k+1}^{1-\delta}\big{)}\rangle.$

We have

$St^{S,R}\big{(}\tilde{M}_{n,s}^{\delta}Q_{n,s}^{1-\delta}\big{)}=\sum_{S^{\prime},R^{\prime}}(-1)^{\sigma}C_{S^{\prime},R^{\prime}}\tilde{M}_{S^{\prime}}\tilde{Q}^{{R^{\prime}}^{*}_{(2-\delta)p^{k}-t^{\prime}}}.$

Here, by convention, $\gamma_{1/p}(v_{k+1})=u_{k+1}.$

By an analogous argument we obtain

######  Corollary 2.3.

Set
$C_{s,S,R}=\langle\tilde{m}_{S^{\prime}}\tilde{q}_{{R^{\prime}}^{*}_{(2-\delta)p^{k}-t^{\prime}}},St^{S,R}\big{(}\tilde{M}_{n,s}^{\delta}Q_{n,s}^{1-\delta}\big{)}\rangle.$
We have

$St^{S^{\prime},R^{\prime}}\big{(}\tilde{U}_{k+1}^{\delta}V_{k+1}^{1-\delta}\big{)}=\sum_{s=-\delta}^{n-\delta}\Big{(}\sum_{S,R}(-1)^{\sigma}C_{s,S,R}\tilde{M}_{S}\tilde{Q}^{R^{*}_{(2-\delta)p^{n}+[-2p^{s}]-t}}\Big{)}V_{k+1}^{p^{s}}\
,$

Here, by convention, $V_{k+1}^{1/p}=U_{k+1}$ .

##  3\. Applications

Fix a non-negative integer $r$. Let $\alpha_{i}=\alpha_{i}(r)$ denote the
$i$-th coefficient in $p$-adic expansion of $r$. That means

$r=\alpha_{0}p^{0}+\alpha_{1}p^{1}+\ldots$

with $0\leqslant\alpha_{i}<p,\,i\geqslant 0$. Set $\alpha_{i}=0$ for $i<0$.

The aim of the section is to prove the following four theorems:

######  Theorem 3.1.

Set $c=\frac{(h-1)!}{(h-\alpha_{k-1})!\prod_{0\leqslant
i<k}(\alpha_{i}-\alpha_{i-1})!},\ t_{i}=\alpha_{i}-\alpha_{i-1},\ 0\leqslant
i<k$. We have

$\displaystyle P^{r}U_{k+1}$
$\displaystyle=\begin{cases}\displaystyle{c\Big{(}hU_{k+1}+\sum_{u=0}^{k-1}t_{u}V_{k+1}\tilde{M}_{k,u}Q_{k,u}^{-1}\Big{)}\prod_{i=0}^{k-1}Q_{k,i}^{t_{i}}},\
&2r<p^{k},\,t_{i}\geqslant 0,\ i<k,\\\ 0\ ,&\text{otherwise.}\end{cases}$

######  Theorem 3.2.

Set
$c=\frac{(h-\alpha_{s})(h-1)!}{(h-\alpha_{n-1})!(\alpha_{s}+1-\alpha_{s-1})!\prod_{s\not=i<n}(\alpha_{i}-\alpha_{i-1})!},t_{i}=\alpha_{i}-\alpha_{i-1},\
-1\leqslant i\neq s,\ t_{s}=\alpha_{s}+1-\alpha_{s-1}$, with $-1\leqslant
s\leqslant n-1$. We have

$\displaystyle P^{r}\tilde{M}_{n,s}$
$\displaystyle=\begin{cases}\displaystyle{c\sum_{u=-1}^{s}t_{u}\tilde{M}_{n,u}Q_{n,u}^{t_{u}-1}\prod_{u\not=i<n}Q_{n,i}^{t_{i}}},&2r\leqslant
p^{n}+[-2p^{s}],\alpha_{i}\geqslant\alpha_{i-1},\\\ &s\not=i<n,\
\alpha_{s}+1\geqslant\alpha_{s-1}\ ,\\\ 0\ ,&\text{otherwise.}\end{cases}$

The following two theorems were first proved in [10] by another method.

######  Theorem 3.3 (Hưng-Minh [10]).

$\displaystyle P^{r}V_{k+1}=\begin{cases}V_{k+1}^{p},&r=p^{k},\\\
\frac{(-1)^{\alpha_{k-1}}\alpha_{k-1}!}{\prod_{\scriptstyle 0\leqslant
i<k}(\alpha_{i}-\alpha_{i-1})!}\displaystyle{V_{k+1}\prod_{i=0}^{k-1}Q_{k,i}^{\alpha_{i}-\alpha_{i-1}}},&r<p^{k},\
\alpha_{i}\geqslant\alpha_{i-1},\ i<k,\\\ 0\ ,&\text{otherwise.}\end{cases}$

######  Theorem 3.4 (Hưng-Minh [10]).

Set
$c=\frac{(-1)^{\alpha_{n-1}}\alpha_{n-1}!(\alpha_{s}+1)}{(\alpha_{s}+1-\alpha_{s-1})!\prod_{s\neq
i<n}(\alpha_{i}-\alpha_{i-1})!}.$ Then

$\displaystyle P^{r}Q_{n,s}=\begin{cases}Q_{n,s}^{p},&r=p^{n}-p^{s},\\\
\displaystyle{cQ_{n,s}\prod_{0\leqslant
i<n}Q_{n,i}^{\alpha_{i}-\alpha_{i-1}}},&r<p^{n}-p^{s},\alpha_{i}\geqslant\alpha_{i-1},\\\
&s\neq i<n,\ \alpha_{s}+1\geqslant\alpha_{s-1},\\\ 0\
,&\text{otherwise.}\end{cases}$

To prove these theorems we need

######  Notation 3.5.

Let $R=(r_{1},\ldots,r_{n})$ be a sequence of arbitrary integers and
$b\geqslant 0$. Denote by $|R|=\sum_{i=1}^{n}(p^{i}-1)r_{i}$ and
$\binom{b}{R}$ the coefficient of $y_{1}^{r_{1}}\ldots y_{n}^{r_{n}}$ in
$(1+y_{1}+\ldots y_{n})^{b}$. That means,

$\binom{b}{R}=\begin{cases}\dfrac{b!}{(b-r_{1}-\ldots-r_{n})!r_{1}!\ldots
r_{n}!},&r_{1}+\ldots+r_{n}\leqslant b,\\\
0,&r_{1}+\ldots+r_{n}>b.\end{cases}$

The proofs of Theorems 3.1 and 3.3 are based on the duality theorem and the
following

######  Lemma 3.6.

Let $b$ be a non-negative integer and $\varepsilon=0,\,1$. We then have

$\displaystyle
St^{S,R}(x^{\varepsilon}y^{b})=\begin{cases}\displaystyle{\binom{b}{R}x^{\varepsilon}y^{b+|R|},}&S=\emptyset,\\\
\displaystyle{\varepsilon\binom{b}{R}y^{b+|R|+p^{u}}},&S=(u),\ u\geqslant
0,\\\ 0\ ,&\text{otherwise.}\end{cases}$

Here $x$ and $y$ are the generators of $H^{*}(B\mathbb{Z}/p)=E(x)\otimes
P(y)$.

###### Proof.

A direct computation using Proposition 1.1 shows that

$\displaystyle d_{m}^{*}P_{m}(x^{\varepsilon}y^{b})$
$\displaystyle=(-1)^{mb}(h!)^{m\varepsilon}\Big{(}\tilde{L}_{m}^{\varepsilon}x^{\varepsilon}+\varepsilon\sum_{u=0}^{m-1}(-1)^{u+1}\tilde{M}_{m,u}y^{p^{u}}\Big{)}$
$\displaystyle\hskip
85.35826pt\times\Big{(}\sum_{R=(r_{1},\ldots,r_{m})}(-1)^{r(\emptyset,R)}\binom{b}{R}\tilde{Q}^{R^{*}_{2b}}y^{b+|R|}\Big{)}$
$\displaystyle=\mu(2b+\varepsilon)^{m}\Big{(}\sum_{R=(r_{1},\ldots,r_{m})}(-1)^{r(\emptyset,R)}\binom{b}{R}\tilde{Q}^{R^{*}_{2b+\varepsilon}}x^{\varepsilon}y^{b+|R|}$
$\displaystyle\hskip
28.45274pt+\varepsilon\sum_{u=0}^{m-1}\sum_{R=(r_{1},\ldots,r_{m})}(-1)^{r((u),R)}\binom{b}{R}\tilde{M}_{m,u}\tilde{Q}^{R^{*}_{2b}}y^{b+|R|+p^{u}}\Big{)}.$

The lemma now follows from Theorem 1.2. ∎

###### Proof of Theorem 3.1.

Since $\dim U_{k+1}=p^{k}$, it is clear that$P^{r}U_{k+1}=0$ for $2r>p^{k}$.
Suppose $r\leqslant(p^{k}-1)/2$. Applying Corollary 2.3 with $\delta=n=1$ and
using Lemma 3.6 we obtain

$\displaystyle P^{r}U_{k+1}$
$\displaystyle=\sum_{R=(r_{1},\ldots,r_{k})}(-1)^{r(\emptyset,R)+r+hk}\langle\tilde{q}_{(r)^{*}_{p^{k}}},St^{\emptyset,R}\tilde{L}_{1}\rangle
U_{k+1}\tilde{Q}^{R^{*}_{p-1}}$
$\displaystyle\quad+\sum_{u=0}^{k-1}\sum_{R}(-1)^{r((u),R)+r+kh+1}\langle\tilde{q}_{(r)^{*}_{p^{k}}},St^{(u),R}\tilde{M}_{1,0}\rangle
V_{k+1}\tilde{M}_{k,u}\tilde{Q}^{R^{*}_{p-3}}\ .$

Set $\bar{r}_{i}=\alpha_{i}-\alpha_{i-1},\ i<k,\ \bar{r}_{k}=h-\alpha_{k},\
\bar{R}_{0}=(\bar{r}_{1},\ldots,\bar{r}_{k}),\bar{R}_{u}=(\bar{r}_{1},\ldots,\bar{r}_{u}-1,\ldots,\bar{r}_{k}),\
1\leqslant u\leqslant k.$ Computing directly from Lemma 3.6 with
$\varepsilon=0,\ b=h$ or $\varepsilon=1,\ b=h-1$ gives

$\displaystyle\langle\tilde{q}_{(r)^{*}_{p^{k}}},St^{\emptyset,R}\tilde{L}_{1}\rangle$
$\displaystyle=\begin{cases}\dfrac{h!}{\bar{r}_{0}\ldots\bar{r}_{k}},\quad&R=\bar{R}_{0}\\\
0\ ,&\text{otherwise.}\end{cases}$
$\displaystyle\langle\tilde{q}_{(r)^{*}_{p^{k}}},St^{(u),R}\tilde{M}_{1,0}\rangle$
$\displaystyle=\begin{cases}\dfrac{(h-1)!\bar{r}_{u}}{\bar{r}_{0}\ldots\bar{r}_{k}},\quad&R=\bar{R}_{u}\\\
0\ ,&\text{otherwise.}\end{cases}$

A simple computation shows that

$r(\emptyset,\bar{R}_{0})+r=r((u),\bar{R}_{u})+r+1=hk\ (\text{mod}\ 2).$

Hence, the theorem is proved. ∎

###### Proof of Theorem 3.3.

Since $\dim V_{k+1}=2p^{k}$, we have only to prove the theorem for $r<p^{k}$.
Note that $Q_{1,1}=1$. Hence

$St^{S,R}Q_{1,1}=\begin{cases}1\ ,\quad&S=\emptyset,\,R=(0,\ldots,0),\\\ 0,\
&\mbox{otherwise}.\end{cases}$

So, $\langle\tilde{q}_{(r)^{*}_{2p^{k}}},St^{S,R}Q_{1,1}\rangle=0$ for any
$S,\,R$. Remember that $Q_{1,0}=y^{p-1}$. So, applying Corollary 2.3 with
$\delta=0,\,n=1$ and using Lemma 3.6 with $\varepsilon=0,\,b=p-1$, we get

(a) $\displaystyle
P^{r}V_{k+1}=\sum_{R}(-1)^{r(\emptyset,R)+r}\langle\tilde{q}_{(r)^{*}_{2p^{k}}},St^{\emptyset,R}Q_{1,0}\rangle\tilde{Q}^{R^{*}_{2(p-1)}}V_{k+1}.$

From Lemma 3.6, we see that it implies

(b)
$\displaystyle\langle\tilde{q}_{(r)^{*}_{2p^{k}}},St^{\emptyset,R}Q_{1,0}\rangle$
$\displaystyle\qquad=\begin{cases}\displaystyle{\binom{p-1}{R}},&R=(\alpha_{1}-\alpha_{0},\ldots,\alpha_{k-1}-\alpha_{k-2},p-1-\alpha_{k-1}),\\\
0\ ,&\text{otherwise.}\end{cases}$

Suppose that
$R=(\alpha_{1}-\alpha_{0},\ldots,\alpha_{k-1}-\alpha_{k-2},p-1-\alpha_{k-1}).$
Then we can easily observe that

(c) $\displaystyle r(\emptyset,R)+r=0\ (\text{mod}\ 2),$
$\displaystyle\binom{p-1}{R}=\frac{(-1)^{\alpha_{k-1}}\alpha_{k-1}!}{\prod_{0\leqslant
i<k}(\alpha_{i}-\alpha_{i-1})},$ $\displaystyle
R^{*}_{2(p-1)}=(2\alpha_{0},\alpha_{1}-\alpha_{0},\ldots,\alpha_{k-1}-\alpha_{k-2}).$

Theorem 3.3 now follows from (a),(b) and (c). ∎

Following Corollary 2.2,to determine $P^{r}M_{n,s}$ and $P^{r}Q_{n,s}$ we need
to compute the action of $St^{S,R}$ on $U_{2}$ and $V_{2}$.

######  Proposition 3.7.

Suppose given $R=(r_{1},\ldots,r_{n})$, and $0\leqslant u<n$. Set
$w_{s}=r_{s+1}+\ldots+r_{n}$, for $s\geqslant 0$. Then we have

$St^{S,R}U_{2}=\begin{cases}\displaystyle{\binom{h}{R}\big{(}\tilde{L}_{1}^{|R|/h}U_{2}+\overset{n-1}{\underset{s=0}{\sum}}h^{-1}w_{0}\tilde{M}_{1,0}\tilde{L}_{1}^{(|R|-p^{s+1}+1)/h}V_{2}^{p^{s}}\big{)},\quad
S=\emptyset},\\\
\displaystyle{\binom{h}{R}\overset{n-1}{\underset{s=u}{\sum}}h^{-1}w_{s}\tilde{L}_{1}^{(|R|-p^{s+1}+p^{u}+h)/h}V_{2}^{p^{s}},\hskip
73.97733ptS=(u)},\\\ 0\ ,\hskip 170.71652pt\text{otherwise.}\end{cases}$

Here, $|R|$ and $\binom{h}{R}$ are defined in Notation 3.5.

The proposition will be proved by using Theorem 1.2 and the following

######  Lemma 3.8.

Let $u,v$ be non-negative integers with $u\leqslant v$. We have

i) $[u,v]=\sum_{s=u}^{v-1}V_{1}^{p^{v}}-p^{s+1}+p^{u}V_{2}^{p^{s}}$,

ii)
$[1;v]=V_{1}^{p^{v}-h}U_{2}+M_{1,0}\sum_{s=0}^{v-1}V_{1}^{p^{v}-p^{s+1}}V_{2}^{p^{s}}.$

Here $[u,v]$ and $[1;v]$ are defined in Section 1.

The proof is straightforward.

###### Proof of Proposition 3.7.

Recall that $M_{2,1}=x_{1}y_{2}-x_{2}y_{1}$. From Proposition 1.1 we directly
obtain

$d_{n}^{*}P_{n}M_{2,1}=(-h!)^{n}\sum_{v=0}^{n}(-1)^{v}\tilde{L}_{n}Q_{n,v}[1;v]+\sum_{\scriptstyle
0\leqslant u<n\atop\scriptstyle 0\leqslant v\leqslant
n}(-1)^{u+v+1}\tilde{M}_{n,u}Q_{n,v}[u,v].$

Since $L_{1}=y_{1}$ and $2(h-1)=p-3$, using Proposition 1.1(iii) with
$y=y_{1}$ and Notation 3.5 we get

$d_{n}^{*}P_{n}L_{1}^{h-1}=(-1)^{n(h-1)}\sum_{R^{\prime}}(-1)^{r(\emptyset,R^{\prime})}\binom{h-1}{R^{\prime}}\tilde{Q}^{{R^{\prime}}^{*}_{p-3}}y_{1}^{|R^{\prime}|+h-1}.$

We have $U_{2}=M_{2,1}L_{1}^{h-1},\ \dim U_{2}=p$ and $\mu(p)=(-1)^{h}h!$. So,
it implies from the above equalities and Proposition 1.1 that

$\displaystyle d_{n}^{*}P_{n}U_{2}$
$\displaystyle=\mu(p)^{n}\Big{(}\sum_{R}(-1)^{r(\emptyset,R)}\tilde{Q}^{R^{*}_{p}}\binom{h}{R}\sum_{v=0}^{n}h^{-1}r_{v}y_{1}^{|R|+h-p^{v}}[1;v]$
$\displaystyle+\sum_{u=0}^{n-1}\sum_{R}(-1)^{r((u),R)}\tilde{M}_{n,u}\tilde{Q}^{R^{*}_{p-1}}\binom{h}{R}\sum_{v=u}^{n}h^{-1}r_{v}y_{1}^{|R|+h-p^{v}}[u;v]\Big{)}.$

Then by Theorem 1.2 we have

$St^{S,R}U_{2}=\begin{cases}\displaystyle{h^{-1}\binom{h}{R}\sum_{v=0}^{n}r_{v}y_{1}^{|R|+h-p^{v}}[1;v]},\
&S=\emptyset,\\\
\displaystyle{h^{-1}\binom{h}{R}\sum_{v=u}^{n}r_{v}y_{1}^{|R|+h-p^{v}}[u;v]},\
&S=(u),\ u<n,\\\ 0\ ,&\text{otherwise.}\end{cases}$

Now the proposition follows from Lemma 3.8. ∎

###### Proof of Theorem 3.2.

For simplicity, we assume that $0\leqslant s<n$. Applying Corollary 2.2 with
$\delta=k=1$ and using Proposition 3.7 we get

$\displaystyle P^{r}\tilde{M}_{n,s}=\sum_{\scriptstyle 0\leqslant u\leqslant
s\atop\scriptstyle
R=(r_{1},\ldots,r_{n})}(-1)^{r((u),R)+r+s+1+nh}C_{(u),R}\tilde{M}_{n,u}\tilde{Q}^{R^{*}_{p-1}}.$

Here
$C_{(u),R}=\langle\tilde{q}_{(r)^{*}_{p^{n}-2p^{s}}}\otimes\gamma_{p^{s}}(\bar{v}_{2}),St^{(u),R}U_{2}\rangle.$

If $2r>p^{n}-2p^{s}-1$, then $P^{r}\tilde{M}_{n,s}=0$ since
d$\dim\tilde{M}_{n,s}=p^{n}-2p^{s}$. Suppose $2r\leqslant p^{n}-2p^{s}-1$. Set
$t_{i}=\alpha_{i}-\alpha_{i-1},$ with $0\leqslant i\neq s,\,n,\
t_{s}=\alpha_{s}+1-\alpha_{s-1},\ t_{n}=h-\alpha_{n-1},\
\bar{R}_{0}=(t_{1},\ldots,t_{n}),\
\bar{R}_{u}=(t_{1},\ldots,t_{u}-1,\ldots,t_{n}),\ 1\leqslant u\leqslant n.$
From Proposition 3.7 we have

$C_{(u),R}=\begin{cases}ct_{u}\ ,\ &R=\bar{R}_{u},\\\ 0\
,&\text{otherwise.}\end{cases}$

It is easy to verify that

$r((u),\bar{R}_{u})+r+s+1=nh\ (\text{mod}\ 2).$

Theorem 3.2 now is proved by combining the above equalities. ∎

Now we return to the proof of Theorem 3.4. It is proved by the same argument
as given in the proof of Theorem 3.2. We only compute $St^{S,R}V_{2}$.

######  Proposition 3.9.

For $R=(r_{1},\ldots,r_{n})$, $r_{0}=p-r_{1}-\ldots-r_{n}$, we have

$St^{R}V_{2}=\begin{cases}V_{2}^{p^{s}},\hskip 56.9055ptr_{s}=p,\ r_{i}=0,\
i\neq\ s,\\\
\displaystyle{\sum_{s=0}^{n-1}\frac{(p-1)!(r_{s+1}+\ldots+r_{n})}{r_{0}!\ldots
r_{n}!}V_{1}^{|R|+p-p^{s+1}}V_{2}^{p^{s}}},\\\ \hskip 79.6678pt0\leqslant
r_{i}<p,\ 0\leq i\leqslant n,\\\ 0\ ,\hskip
65.44142pt\text{otherwise.}\end{cases}$

###### Proof.

Recall that $V_{2}=y_{2}^{p}-y_{2}y_{1}^{p-1}$. Applying Proposition 1.1 and
Lemma 3.6 with $y=y_{1}$ or $y=y_{2}$ we get

(a) $\displaystyle d_{n}^{*}P_{n}V_{2}$
$\displaystyle=\sum_{s=0}^{n}(-1)^{n+s}Q_{n,s}^{p}y_{2}^{p^{s+1}}$
$\displaystyle\quad-(-1)^{n}\sum_{u=0}^{n}\sum_{R^{\prime}}(-1)^{u+r(\emptyset,R^{\prime})}\binom{p-1}{R^{\prime}}Q_{n,u}\tilde{Q}^{{R^{\prime}}^{*}_{2(p-1)}}y_{1}^{|R^{\prime}|+p-1}y_{2}^{p^{u}}$
$\displaystyle=(-1)^{n}\sum_{s=0}^{n}(-1)^{s}Q_{n,s}^{p}\big{(}y_{2}^{p^{s+1}}-y_{2}^{p^{s}}y_{1}^{(p-1)p^{s}}\big{)}$
$\displaystyle\quad-(-1)^{n}\sum_{u=0}^{n}\sum_{R}(-1)^{r(\emptyset,R)}\binom{p-1}{R_{u}}\tilde{Q}^{R^{*}_{2p}}y_{1}^{|R|+p-p^{u}}y_{2}^{p^{u}}\Big{)}.$

Here the last sum runs over all $R=(r_{1},\ldots,r_{n})$ with $0\leqslant
r_{i}<p,\ 0\leqslant i\leqslant n,\ R_{0}=R,\
R_{u}=(r_{1},\ldots,r_{u}-1,\ldots,r_{n}),\ 1\leqslant u\leqslant n$.

Let $v$ be the greatest index such that $r_{v}>0$. A simple computation shows

(b) $\displaystyle
y_{1}^{|R|+p-p^{u}}y_{2}^{p^{u}}=-y_{1}^{|R|+p-p^{u}-p^{v}}[u,v]+y_{1}^{|R|+p-p^{v}}y_{2}^{p^{v}}.$

Combining (a), (b), Lemma 3.8 and the fact that
$\sum_{u=0}^{n}\binom{p-1}{R_{u}}=0$ we obtain

$d_{n}^{*}P_{n}V_{2}=\mu(2p)^{n}\Big{(}\sum_{s=0}^{n}(-1)^{s}Q_{n,s}V_{2}^{p^{s}}\\\
+\sum_{R}(-1)^{r(\emptyset,R)}\tilde{Q}^{R^{*}_{2p}}\sum_{s=0}^{n}\sum_{u=s+1}^{n}\binom{p-1}{R_{u}}V_{1}^{|R|+p-p^{s+1}}V_{2}^{p^{s}}\Big{)}.$

The proposition now follows from this equality and Theorem 1.2. ∎

## References

  * [1] H.E.A. Campbell, Upper triangular invariants, Canad. Math. Bull 28 (1985), 243-248, MR0782784.
  * [2] L.E. Dickson, A fundamental system of invariants of the general modular linear group with a solution of the form problem, Trans. Amer. Math. Soc. 12 (1911), 75-98, MR1500882.
  * [3] H. Mùi, Modular invariant theory and the cohomology algebras of symmetric groups, J. Fac. Sci. Univ. Tokyo Sec. IA Math. 22 (1975), 319-369, MR0422451.
  * [4] H. Mùi, Cohomology operations derived from modular invariants, Math. Z. 193 (1986), 151-163, MR852916.
  * [5] I. Madsen, On the action of the Dyer-Lashof algebra in H*(G), Pacific J. Math. 60 (1975), 235-275, MR0388392.
  * [6] I. Madsen and R.J. Milgram, The classifying spaces for surgery and cobordism of manifolds, Ann. of Math. Studies, No. 92 Princeton Univ. Press, 1979, MR0548575.
  * [7] J. Milnor, Steenrod algebra and its dual, Ann. of Math. 67 (1958), 150-171, MR0099653.
  * [8] N.N. Hải and N.H.V. Hưng, Steenrod operations on mod 2 homology of the iterated loop spaces, Acta Math. Vietnam., 13 (1988), 113-126, MR1023710.
  * [9] N.H.V. Hưng, The action of the Steenrod squares on the modular invariants of linear groups, Proc. Amer. Math. Soc. 113 (1991), 1097-1104, MR1064904.
  * [10] N.H.V. Hưng and P.A. Minh, The action of the mod p Steenrod operations the modular invarints of linear groups, Vietnam J. Math. 23 (1995), 39-56, MR1367491.
  * [11] N. Sum, On the action of the Steenrod-Milnor operations on the modular invariants of linear groups, Japan. J. Math., 18 (1992), 115-137, MR1173832.
  * [12] L. Smith and R. Switzer, Realizability and non-realizability of Dickson algebras as cohomology rings, Proc. Amer. Math. Soc. 89 (1983), 303-313, MR712642.
  * [13] N.E. Steenrod and D.B.A. Epstein, Cohomology operations, Ann. of Math. No. 50, Princeton University Press, 1962, MR0145525.
  * [14] C.W. Wilkerson, A primer on the Dickson invariants, Contemporary Mathematics, Amer. Math. Soc. 19 (1983), 421-434, MR711066.

Department of Mathematics

Quinhon Pedagogic University

Dai hoc Su Pham Quinhon

170 Nguyen Hue Quinhon, Vietnam