# Mechanism of keyhole pore formation in metal additive manufacturing

Lu Wang  Yanming Zhang  Hou Yi Chia  Wentao Yan  Department of Mechanical
Engineering, National University of Singapore, 117575, Singapore
Corresponding author<EMAIL_ADDRESS>
###### Abstract

Metal additive manufacturing has gained extensive attention from research
institutes and companies to fabricate intricate parts and functionally graded
materials. However, the porosity of the as-built part deteriorates the
mechanical property and even hinders the further application of metal additive
manufacturing. Particularly, the mechanisms of keyhole pores associated with
the keyhole fluctuation are not fully understood. To reveal the mechanisms of
the keyhole pores formation, we adopt a multiphysics thermal-fluid flow model
incorporating heat transfer, liquid flow, metal evaporation, Marangoni effect,
and Darcy’s law to simulate the keyhole pore formation process, and the
results are validated with the in-situ X-ray images. The simulation results
present the instant bubble formation due to the keyhole instability and motion
of the instant bubble when it pins on the solidification front. Moreover, the
unevenly distributed recoil pressure on the keyhole surface is an important
factor for keyhole collapse and pentration. Furthermore, comparing the keyhole
pore formation under different laser scanning speeds shows that the keyhole
pore is sensitive to the manufacturing parameters. The keyhole fluctuation
features and energy absorptivity variation on the rear keyhole wall could be
metrics to evaluate the likelihood of the keyhole pore formation.
Additionally, the simulation under a low ambient pressure shows the feaibility
of improving the keyhole stability to reduce and even avoid the formation of
keyhole pores.

###### keywords:

Keyhole pore , Additive manufacturing , Multiphysics thermal-fluid flow model
, Molten pool flow , Recoil pressure , Low ambient pressure

††journal: Journal Name

##  1 Introduction

Metal additive manufacturing (AM) is well known for its ability to fabricate
complex-shaped parts without special tooling [1] and functionally graded parts
[2], shorten the development cycle of products [3], and save the cost of the
material [4]. However, the manufacturing defects [4, 5] are deleterious to the
mechanical property. The porosity of the as-built part, one kind of defects,
decreases the ultimate strength directly and is also a fatal flaw to the
fatigue and fracture strength of the part [6, 7, 8]. The presence of such
defects does not meet the standards required in industry and thus prevents the
adoption of AM technology in these industries. Therefore, massive researches
have been conducted to understand the pore formation mechanisms during AM and
control the as-built part porosity [8, 9, 10, 11, 12, 13]. Among various pore
defects, the porosity under the keyhole mode melting [11, 14] is a ubiquitous
defect in both laser welding [15] and laser powder bed fusion (L-PBF) and has
attracted broad attention [9, 16, 17].

Some progress has been made on the study of the keyhole pore features, keyhole
collapse mechanisms, and keyhole pore formation process. Ex-situ experiments
[4, 5, 11] show that the keyhole pores are usually spherical and concentrate
at the molten pool bottom. However, these experiments have not observed the
keyhole pore formation directly, and cannot present quantitative explanations
for the keyhole pore formation mechanisms. Recently, the in-situ X-ray imaging
of keyhole dynamics [17, 18, 19] were presented, and three types of keyhole
pores were identified: (i) the instant bubble by a ledge on the rear keyhole
wall, (ii) the keyhole pore at the end of the track, and (iii) the keyhole
pore by the keyhole fluctuation. The first type of bubble is eliminated almost
immediately after its formation and is insignificant for the pore formation
mechanism. The pores at the end of the track [18] are not only related to the
keyhole dynamics but also determined by the laser scanning path, while these
pores are usually reduced or eliminated by contour scanning and post-process
polishing. Thus, the keyhole pore by the keyhole fluctuation is the most
significant, which is the focus of this study.

Besides the experimental approaches, numerical simulation of keyhole dynamics
[18, 20, 21, 22, 23] is a complementary, cost-saving and efficient approach to
understand the formation mechanisms of the keyhole pores. Martin et al. [18]
simulated the keyhole pore at the end of the track and developed a strategy to
reduce these pores. The simulation by Lin et al. [24] indicated that adjusting
laser incident angle can reduce the number of keyhole pores during laser
welding. Bayat et al. [25] investigated the relationship between the keyhole
pores and input power, and validated the simulation results with the
experiment results. The simulation by Tang et al. [26] indicated that the
spherical pores at the bottom of the molten pool formed as the energy density
increased during AM. The simulation by Tan et al. [27] showed the keyhole pore
size decreased as the ambient pressure decreased.

Although the in-situ experiments and previous simulations have provided
empirical observations on the keyhole pore formation, the basic principles
like the recoil pressure distribution on the keyhole surface, the relationship
between energy distribution and keyhole fluctuation, and molten pool flow
during the keyhole pore formation remain elusive. In this study, a multi-
physics thermal-fluid flow model [23] incorporating heat transfer, molten pool
flow, Marangoni effect, recoil pressure by metal evaporation, Darcy’s law, and
laser ray-tracing is adopted to simulate the keyhole fluctuation and keyhole
pore formation process. Firstly, we validate the simulation results of instant
bubble formation and pinning on the solidification front against the X-ray
imaging results. Next, the varying trend of the keyhole depth fluctuation,
absorbed energy distribution, keyhole pore size, molten pool flow, and forces
on the keyhole with the increase of laser scanning speed are analyzed to
explain the mechanisms and influence factors of the keyhole pore formation.
Furthermore, we explore the approach to reduce and even eliminate keyhole
pores by simulating the molten pool flow under near-vacuum ambient pressure.

##  2 Multiphysics thermal-fluid flow model

The multiphysics thermal-fluid flow is based on our previous work [23, 28]. In
the model, the liquid phase is assumed to be incompressible Newtonian fluid
with laminar flow. The mass conservation equation is given as follows:

$\nabla\cdot\left(\rho\bm{v}\right)=0$ (1)

where $\rho$ and $\bm{v}$ are the mass density and velocity vector.

The momentum conservation equation is given as follows:

$\rho\frac{\partial\bm{v}}{\partial
t}+\rho\nabla\cdot(\bm{v}\otimes\bm{v})=-\nabla
p+\mu\nabla^{2}\bm{v}+\bm{f}_{b}-\rho D\bm{v}$ (2)

where $\mu$ and $p$ denote the dynamic viscosity and pressure. The buoyancy
force $\bm{f}_{b}$ is accounted for using the Boussinesq approximation

$\bm{f}_{b}=\rho\bm{g}\alpha_{v}(T-T_{ref})$ (3)

where $\bm{g}$, $\alpha_{v}$, and $T_{ref}$ denote the gravitational
acceleration vector, the thermal expansion coefficient, and reference
temperature ( liquidus temperature $T_{l}$ for the current simulation). $D$ is
the Darcy drag force coefficient, which is calculated by the Blake-Kozeny
model [29, 30]

$D=\frac{180\mu}{\rho\lambda_{1}^{2}}\frac{F_{s}^{2}}{(1-F_{s})^{3}}$ (4)

where $\lambda_{1}$ is the characteristic length of mushy zone, taken as the
primary dendrite arm spacing (about $5\;\rm{\mu m}$ for SLM [31]), and $F_{s}$
is the solid fraction.

The energy conservation equation is given as follows:

$\rho\frac{\partial I}{\partial
t}+\rho\nabla\cdot(\bm{v}I)=\nabla\cdot(k\nabla T)+q$ (5)

where $k$ is the thermal conductivity, and $T$ is the temperature. $I=\int
C_{p}\mathrm{d}T+(1-F_{s})L_{m}$ is the specific internal energy, where
$C_{p}$ and $L_{m}$ are the specific heat and specific latent heat of melting.
$q$ is the power absorbed by the material, which is incorporated by using the
ray-tracing method [32] to track the multireflections of laser and calculated
with the Fresnel equation. In the current model, the reflection is assumed to
be specular reflection. The material for simulation in the current work is
Ti-6Al-4V, but titanium’s [33] complex refractive index is used instead due to
the lack of reliable data of Ti-6Al-4V.

The free surface of the molten pool is captured using the volume-of-fluid
(VoF) method [34].

$\frac{\partial F}{\partial t}+\nabla\cdot(F\bm{v})=0$ (6)

where $F$ is the volume fraction.

On the free surface, the normal ($p_{\mathrm{n}}$) and tangent forces
($\tau_{\mathrm{t}}$) incorporate the surface tension, recoil pressure and
Marangoni effect, given as

$\left\\{\begin{aligned} p_{\mathrm{n}}&=\sigma(T)\kappa+P_{rec}(T)\\\
\tau_{\mathrm{t}}&=\sigma_{s}^{T}\left[\nabla T-\bm{n}(\nabla
T\cdot\bm{n})\right]\end{aligned}\right.$ (7)

where $\kappa$ is the curvature of the free surface, and $\bm{n}$ is the
normal vector of the free surface.
$\sigma(T)=\sigma_{0}-\sigma_{s}^{T}(T-T_{l})$ and $P_{rec}(T)$ are the
temperature-dependent surface tension coefficient and recoil pressure [23].
$\sigma_{0}$ and $\sigma_{s}^{T}$ are surface tension coefficient at the
reference temperature $T_{l}$ (liquidus temperature in the current
simulations) and its temperature sensitivity.

For the thermal boundary condition [25], it consists of heat convection , heat
radiation, and heat loss of evaporation

$-k\nabla
T\cdot\bm{n}=h(T-T_{env})+\epsilon\sigma_{s}(T^{4}-T^{4}_{env})+\dot{q}_{evp}$
(8)

where $\sigma_{s}$ and $T_{env}$ are the Stefan-Boltzmann constant
($5.6704\times 10^{-8}\;\rm{W/(m^{2}\cdot K^{4})}$) and the ambient
temperature, respectively. $\dot{q}_{evp}=\dot{m}_{evp}I$ is the heat loss
rate by evaporation, where $\dot{m}_{evp}$ is the mass loss rate by
evaporation and calculated with our previous evaporation model [23].

##  3 Results and discussion

The laser parameters and ambient pressures in the simulation cases are listed
in Table 1, which are the same as the experiments [19] except the Case 5 (the
in-situ experiment under low ambient pressure was not conducted). The laser
spot size is $100\;\mathrm{\mu m}$ for all the simulations, the same as that
in the experiments [19]. The physical properties of Ti-6Al-4V are listed in
Table 4. To rule out the influence of the powder particles, all the
simulations are conducted on bare plates. The mesh size is 4 $\mathrm{\mu m}$
to ensure numerical accuracy. To compare the keyhole pore features, the
physical time of all the simulation cases is $2000\;\mathrm{\mu s}$.

Table 1: Laser scanning parameters and ambient pressures in the simulations

| Laser power (W) | Laser scanning speed ($\mathrm{mm/s}$) | Ambient pressure (atm)
---|---|---|---
Case 1 | 365 | 400 | 1.0
Case 2 | 382 | 525 | 1.0
Case 3 | 382 | 500 | 1.0
Case 4 | 382 | 475 | 1.0
Case 5 | 382 | 500 | $1.0\times 10^{-4}$

Based on our simulation results, the pore formation by keyhole collapse has
two distinct stages: (i) the instant bubble formation due to the keyhole
instability; (ii) the instant bubble pinning on the solidification front. The
keyhole pore formation stages are validated by the X-ray imaging results [19].
Although the bubble can be formed due to the keyhole instability, both the
simulation and experimental results show that the keyhole instability cannot
ensure the instant bubble being captured by the solidification front and
forming a pore in the scanning track.

###  3.1 Instant bubble due to the keyhole instability

Simulation Case 1 is a validation case on the instant bubble formation process
due to the keyhole instability. According to the X-ray imaging results by Zhao
et al. [19], there are four phases of the instant bubble formation by the
keyhole fluctuation:

  1. (1)

A mini-keyhole protrudes on the top of the front keyhole wall with a letter
“J” shape keyhole bottom (Fig. 1 a1).

  2. (2)

An instant bubble is formed with the keyhole collapse (Fig. 1 a2).

  3. (3)

The recoil pressure increases with the formation of a new keyhole. A needle-
like keyhole bottom (NKB) is formed as the keyhole drills down (Fig. 1 a3).

  4. (4)

The instant bubble moves away from the keyhole.

To compare the keyhole dynamics between the X-ray imaging and simulation
results, the starting time point is taken as the mini-keyhole (protrusion)
generation time (Fig. 1 a). The geometrical feature of the keyhole in the
simulation is similar to the experimental results, as presented in Fig. 1 b
and c series in time sequence. The depths of the newborn keyhole and NKB below
the substrate in the experiment are about $99\;\mathrm{\mu m}$ and
$226\;\mathrm{\mu m}$ respectively, while they are $115\;\mathrm{\mu m}$ and
$150\;\mathrm{\mu m}$ in the simulation as shown in Fig. 1 b2 and b3.

Figure 1: Instant bubble formation due to the keyhole instability. Series a is
the X-ray imaging results of the keyhole instability [19]. Series b, c, and d
are the velocity, recoil pressure, and keyhole surface temperature in the
simulation Case 1. The laser position and scanning direction are indicated in
a1 and b1. The arrows in series b represent the velocity directions. The black
dashed circle in b1 shows the protrusion on the front keyhole wall. The black
dash circle in b2 shows the newborn keyhole. The red dashed line in c1
represents the laser position and separates the keyhole as rear and front
part. The white solid lines in b, c, and d series are the contour of the
solidus temperature $T_{s}$. The boiling temperature of Ti-6Al-4V is 3315 K
under common ambient pressure. (Experimental figures are from [19], reprinted
with permission from AAAS.)

The velocity field of the molten pool to form the instant bubble is clearly
described in Fig. 1 b series. In the $1^{\mathrm{st}}$ phase, the velocity
around the middle of the rear keyhole (the rear and front keyhole are defined
in Fig. 1 c1) is higher than other regions of the molten pool as shown in Fig.
1 b1. A protrusion, similar to the mini keyhole in Fig. 1 a1, on the front
wall of the keyhole is generated in Fig. 1 b1, and its velocity increases from
the $1^{\mathrm{st}}$ phase to the $2^{\mathrm{nd}}$ phase (Fig. 1 b1 to b2)
along the negative $z$ direction so that it finally merges with the rear
keyhole wall to form the newborn keyhole. The high speed at the bottom of the
newborn keyhole drives the keyhole penetration further to form a NKB in the
$3^{\mathrm{rd}}$ phase as shown in Fig. 1 b3. The distribution of the recoil
pressure on the keyhole surface presented in Fig. 1 c series can further
explain the instant bubble formation. The temperature on the keyhole surface
above the boiling temperature is shown in Fig. 1 d series, where the
evaporation on the keyhole surface is a local process. Correspondingly, there
is nearly no recoil pressure on the rear keyhole wall (Fig 1 c1), which causes
the collapse of the keyhole. If the surface tension, gravity of liquid,
hydrodynamic pressure on the rear keyhole is unbalanced, the keyhole would
collapse like Fig. 1 c2. When the newborn keyhole wall forms, the evaporation
in Fig. 1 d2 becomes much more violent and the recoil pressure drives the
keyhole tip deeper. Thus, the unevenly distributed recoil pressure is the
driving force for the keyhole collapse and penetration.

Zhao et al. [19] proposed that in the $4^{\mathrm{th}}$ phase a shock wave by
the phase explosion [22, 35] causes the keyhole tip to penetrate the bubble
$\mathbf{P_{0}}$ and splitting it into two bubbles. It should be clarified
that the sharp shock wave cannot be simulated by the current model, because
the phase explosion is a fierce non-linear process that occurs in less than
$1\;\mathrm{\mu s}$ with “water hammer” pressure on $\mathbf{P_{0}}$ at $\sim
140\;\mathrm{MPa}$. More importantly, the shock wave is not the decisive
factor for the pinning of the bubble at the solidification front to form the
keyhole pore. As the bubble is nearly stationary after its formation, the
shock wave triggers and splits the bubble into two, which subsequently escapes
to the surface of the molten pool. Hence, no keyhole pore forms when a shock
wave is encountered. Therefore, more simulation cases are conducted to
investigate how the instant bubble pins on the solidification front and
eventually forms the keyhole pores as observed in experiments [19].

Before further discussion of the keyhole pore formation, the issue below
should be clarified. In the simulation, there is a keyhole pore with the
diameter of $45\;\mathrm{\mu m}$ by two merged bubbles as shown in Fig. 1 b3,
while there is no keyhole pore in the experiment [19]. There could be several
reasons for this difference. The experimental results by Zhao et al. [19] show
that the laser parameters in Case 1 are close to threshold for no keyhole
pore, where the possibility of the keyhole pore formation is around zero.
Meanwhile, the diameter of the laser in the experiment is an approximate
value, and the influence of laser defocusing and plasma are not incorporated
in the simulations. Thus, this kind of random error is acceptable.

###  3.2 Keyhole pore formation

According to the experimental results [19], instant bubbles pin on the
solidification front and form the keyhole pores under the manufacturing
parameters in Case 2-4. The simulation result of Case 2 is taken as an example
to explain the keyhole pore formation process, and the velocity distribution
in the molten pool during the keyhole pore formation is shown in Fig. 2 (a-e).
In Fig. 2 (a) and (b), the instant bubble $\mathbf{b}$ is formed as explained
in Sect. 3.1. A protrusion is formed on the front keyhole wall as shown in the
black dashed circle of Fig. 2 (a), and an instant bubble $\mathbf{b}$ is
generated by the keyhole collapse in Fig. 2 (b). It takes about
$240\;\mathrm{\mu s}$ for the instant bubble $\mathbf{b}$ to be pinned on the
solidification front (Fig. 2 (e)).

Figure 2: Keyhole pore formation in Case 2. (a-e) The simulation results of
the velocity magnitude and direction (arrows) in the molten pool in the
longitudinal section (2D views). (f-i) The keyhole pore formation captured by
X-ray imaging [19]. The black dashed circle in (a) is the mini keyhole. The
bubble $\mathbf{b}$ in (b-e) is the instant bubble by the keyhole collapse.
The white solid lines in (a-e) are the contour of the solidus temperature
$T_{s}$. The instant bubble in the experiment is shown in the yellow circle of
(h). (Experimental figures are from [19], reprinted with permission from
AAAS.)

The bubble motion and pore formation at the bottom of the molten pool in both
the experiment and simulation are similar during the instant bubble pinning
stage. The bubble $\mathbf{b}$ mainly moves along the horizontal direction
without floating up as shown in Fig. 2 (c) and (d), similar to the
experimental result in Fig. 2 (g) and (h). About $20\sim 30\;\mathrm{\mu s}$
after the keyhole collapse, the horizontal distance between the bubble
$\mathbf{b}$ and the keyhole bottom is about $112\;\mathrm{\mu m}$ and
$129\;\mathrm{\mu m}$ in the simulation and experiment, respectively, as shown
in Fig. 2 (c) and (h). Furthermore, by observing the velocity distribution
around the instant bubble $\mathbf{b}$, we find that the liquid velocity below
the bubble is much higher than that above in Fig. 2 (c). Based on the
Bernoulli’s principle, the high velocity below the pore leads to low local
pressure, which impedes the upward motion of the bubble. Eventually the bubble
is caught up by the solidification front. Thus, the velocity distribution at
the bottom of the molten is one of the factors for the formation of the
keyhole pore. This phenomenon also explains larger bubbles bear larger
buoyancy and thus float up from the molten pool bottom with less hindrance in
Fig. 2 (e).

Additionally, the drag force in the mushy zone is acting on the fluctuating
keyhole, especially at the keyhole bottom. The interaction between the mushy
zone and keyhole surface does not happen coincidentally. The molten pool
temperature fields at two different time points in the $1^{\mathrm{st}}$ phase
of the keyhole collapse in simulation Case 2 are given in Fig. 3 (a) and (c).
The regions in the white dashed circle of Fig. 3 (a) and (c) indicate that the
keyhole surface contacts the mushy zone at different times. According to Eq.
4, the drag force in the mushy zone increases hyperbolically as the solid
fraction increases, and the drag coefficient $D$ is larger than $1.2\times
10^{6}\;\mathrm{s^{-1}}$ when the solid fraction is larger than 0.25. Thus,
the front part of the keyhole bottom can hardly expand, while the rear part of
the keyhole bottom expands to form a “J” shape keyhole, as the metal
evaporates. Moreover, the higher amplitude of $z$ direction velocity around
the middle of the molten pool accelerates the keyhole collapse as shown in the
dashed rectangles of Fig. 3 (b) and (c). It should be mentioned that as the
grain morphology in the mushy zone is mostly columnar and rarely equiaxed [4,
36, 37], the drag force should be anisotropic. The drag force by the Blake-
Kozeny model used here is isotropic. Thus, a more accurate Darcy drag force
model with the consideration of anisotropy of the mushy zone morphology is
worthy of investigation in future study.

Figure 3: Influence of mushy zone on the keyhole fluctuation. (a and b) are
the temperature and $z$-direction velocity distribution in the molten pool at
$t=1134\;\mathrm{\mu s}$. (c and d) are the temperature and $z$-direction
velocity distribution in the molten pool at $t=1302\;\mathrm{\mu s}$. The
white solid lines in (a-d) are the contour of the liquidus temperature
$T_{l}$.

Based on the analysis of the instant bubble formation and pinning on
solidification front stages, the schematic of the keyhole pore formation is
presented in Fig. 4. With the unbalanced forces, the keyhole fluctuates up and
down and even collapses to form instant bubbles ($\mathbf{b}_{1}$ and
$\mathbf{b}_{2}$). In the bubble pinning on solidification front stage, the
molten pool bottom has higher velocity than that above the bubble
$\mathbf{b}_{1}$ and lower pressure, which leads to a vertical drag force on
the bubble $\mathbf{b}_{1}$. If the drag force is sufficiently strong, the
bubble $\mathbf{b}_{1}$ would move nearly horizontally and pins on the
solidification front like $\mathbf{b}_{2}$.

Figure 4: Schematic of the keyhole pore formation process. The recoil pressure
($\bm{P}_{rec}$) by metal evaporation, hydrodynamic pressure ($\bm{P}_{l}$),
pressure by surface tension ($\bm{P}_{s}$), and drag force in the mushy zone
($\bm{F}_{D}$) lead to the instability of the keyhole and generate bubbles
$\mathbf{b}_{1}$ and $\mathbf{b}_{2}$. The bubbles are not floating up
directly due to Bernoulli’s principle and are caught by the solidification
front.

###  3.3 Keyhole pore feature

The keyhole and keyhole pore features in simulation Case 2-4 are presented in
Fig. 5 and listed in Table 2. The maximum keyhole pore sizes in both
simulation and experiment decrease as the laser scanning speed decreases (with
the same laser power). At low scanning speeds, the maximum keyhole pore sizes
in simulation and experiment is close to each other with differences lower
than $4\;\mathrm{\mu m}$ as shown in Fig. 5 (d). When the laser scanning speed
reaches the threshold for no keyhole pore (Case 2), the contingency of the
maximum keyhole pore size increases. In simulation Case 2, the bubble
$\mathbf{b}_{1}$ (size of $76\;\mathrm{\mu m}$) is pinned on the
solidification front, while the similar bubble $\mathbf{b}_{1}^{*}$ (size of
$80\;\mathrm{\mu m}$) floats up to the molten pool surface. Nonetheless, the
size of the second maximum keyhole pore $\mathbf{b}_{1}^{\prime}$ is
$36\;\mathrm{\mu m}$ and close to the experimental result of $18\;\mathrm{\mu
m}$. It should be mentioned that the keyhole pore size is sensitive to the
experimental parameters near the threshold for no keyhole pore situations,
such as the laser focusing and defocusing, material purity, and laser
deflection of the vapor plume, etc. While the current simulation model cannot
incorporate all the influence factors, the difference of maximum keyhole pore
size in Case 2 is acceptable. Since the maximum keyhole pore size has certain
randomness in both simulation and experiment, other parameters, such as mean
keyhole pore size, the median value of pore size, and pore distribution, would
be more representative, but were not provided by the experiment [19]. The mean
keyhole pore size in the simulations increases steadily as the laser scanning
speed decreases (Table 2).

Figure 5: Keyhole and keyhole pore features. Keyhole pores in (a) Case2
($525\;\mathrm{mm/s}$), (b) Case 3 ($500\;\mathrm{mm/s}$), and (c) Case 4
($475\;\mathrm{mm/s}$) in the center plane of the scanning track at
$t=2000\;\mathrm{\mu s}$. (d) Maximum keyhole pore size and (e) mean keyhole
depth between simulation and experiment. (f) Keyhole depth fluctuation with
time. $\mathbf{b}_{1}$, $\mathbf{b}_{2}$, and $\mathbf{b}_{3}$ are the maximum
pore in Case2- 4, respectively. $\mathbf{b}_{1}^{\prime}$ in (a) is the second
largest pore in simulation Case 2, indicated in (e). The case number and laser
scanning speed are shown in (d-e) together for better comparison.

The keyhole pores are concentrated at the bottom of the laser scanning track
with smooth surfaces compared to the lack-of-fusion pores. Further observation
of the keyhole pore distribution in Fig. 5 (a-c) shows that not only does the
keyhole pore size increase sharply as the laser scanning speed decreases, but
also the position and shape of the pores change. In Case 2, the keyhole pores
are spherical and horizontally distributed in the laser scanning track. As the
laser scanning speed decreases, the shape of the keyhole pores becomes
increasingly irregular with a flat or sharp bottom, while the upper surface
remains smooth and spherical. Additionally, the different sizes of the instant
bubbles in the low scanning speed cases also lead to different buoyant forces,
vertical motions to a certain degree and thus non-horizontal distribution of
the pores.

Table 2: Keyhole pore size and depth statistical features

|  |  Case 2 |  Case 3 |  Case 4 |  Case 5
---|---|---|---|---|---
Mean keyhole depth ($\mathrm{\mu m}$) |  Exp.a |  355 |  388 |  456 |  -
Sim.b |  345 |  372 |  388 |  351
Mean keyhole pore sizec ($\mathrm{\mu m}$) |  |  37 |  49 |  82 |  -
Standard deviation of keyhole depth ($\mathrm{\mu m}$) |  |  30 |  29 |  35 |  37
Maximum fluctuation of keyhole depthd ($\mathrm{\mu m}$) |  |  152 |  170 |  156 |  164

  * a

Exp. represents the experiment result.

  * b

Sim. represents the simulation result.

  * c

The pores in the solidified region are considered.

  * d

It is the distance from the peak to the valley of keyhole depth curves.

As the instant bubble formation is related to the keyhole collapse, the
statistical features of the keyhole depth are given in Fig. 5 (e-f) and Table
2. The mean keyhole depths in the simulation cases match those in the
experiments (Fig. 5 (e)), although the relative difference of the keyhole
depth between the simulations and experiments increases from 2.5% to 15% as
the laser scanning speed decreases. This diverging discrepancy can be
explained by several reasons. Firstly, the aforementioned bulk explosion is
fiercer as the energy density increases. Next, the plume in a deeper keyhole
traps more photons, leading to greater energy absorptivity in the material.
This effect is not considered in the current model. Additionally, the laser
diameter varies along the keyhole depth due to defocusing, which influences
the laser reflections and energy absorption. The laser in the current model is
parallel with a constant diameter.

As the laser scanning speed decreases, the keyhole depth curves in Fig. 5 (f)
show an obvious increase, similar to the increasing trend of mean keyhole pore
size. Moreover, the standard deviation of the keyhole depth in Case 4 are
obviously larger than those Case 2 and 3 in Table 2, which indicates that the
keyhole fluctuation in Case 4 is much more violent. Compared to the mean
keyhole depth, the maximum fluctuations of keyhole depth in three cases are
larger than 40% of the mean keyhole depth. The analysis of Sect. 3.1 indicates
that the higher keyhole depth variation between the $1^{\mathrm{st}}$ phase
and the $2^{\mathrm{nd}}$ phase makes the keyhole more unstable and generates
instant bubbles more easily. Thus, there is a higher probability of the
keyhole pore formation in Case 4. Additionally, the X-ray imaging results [17,
19] also show that the shallow keyhole with a large keyhole fluctuation
distance can lead to pores, especially in the case of laser scanning the
powder bed (the spattering particles influence laser absorption and metal
vapor flow). It suggests that the keyhole depth and its fluctuation are two
important parameters for keyhole pore formation and should be considered for
future observations and analysis.

The velocity magnitudes and streamlines of the Case 2-4 share similar
distributions as plotted in Fig. 6. The velocity magnitude is larger around
the keyhole and molten pool bottom than other regions. The streamline
distributions in Fig. 6 (d-f) show that there are two vortices in the molten
pool: a larger stronger clockwise vortex at the front part of the molten pool
and a smaller weaker anticlockwise vortex at the rear part of the molten pool.
The distribution and location of the vortex pair match well with those
observed in the X-ray imaging experiments by Hojjatzadeh et al. [38]. As the
keyhole pores always form at the bottom of the melt track, the velocity
distribution at the bottom of the melt pool is more relevant and influential.
At the molten pool bottom, the liquid flows horizontally to the rear part of
the molten pool and thus drives the instant bubble backward to the
solidification front.

Figure 6: (a-c) Velocity magnitude and (d-f) streamline in the molten pool at
$t=2000\;\mathrm{\mu s}$ in Case2 (a and d), Case 3 (b and e), and Case 4 (c
and f). The white solid curves in (a-c) and gray contours in (d-f) are the
contour of the solidus temperature $T_{s}$.

In Case 2-4, the total recoil force, i.e., the integral of recoil pressure
over the keyhole surface, acts mainly along the $z$ direction as shown in Fig.
7, and the $z$ components of the recoil forces in these cases have higher
amplitudes and are fluctuating around $-1\sim-4\times 10^{-3}\;\mathrm{N}$,
compared to the $x$ components fluctuating around $0\sim-2\times
10^{-3}\;\mathrm{N}$. However, it is difficult to tell the difference of the
recoil forces in the three cases. This means that the keyhole instability and
keyhole pore formation are more related to the locally distributed recoil
force rather than the total force on the whole keyhole surface.

Figure 7: Recoil forces on the keyhole surface in (a) $z$ and (b) $x$
directions in Case2 ($525\;\mathrm{mm/s}$), Case 3 ($500\;\mathrm{mm/s}$), and
Case 4 ($475\;\mathrm{mm/s}$). The keyhole is nearly symmetry along the $y$
direction. Thus, the recoil force along the $y$ direction is around 0 and not
plotted.

Further analysis of the energy absorptivity in the three cases hints the
relationship between the keyhole pore formation and keyhole dynamics. The
total energy absorptivity and that on rear keyhole wall are different from
that on the front keyhole wall (Fig. 8). In simulation Case 4, the total
energy absorptivity and that on rear keyhole wall have fiercer fluctuations
compared to that on the front keyhole wall. As the laser reflection is
determined by the morphology of the keyhole, it indicates that the shape
variation of the rear keyhole wall is larger in Case 4. The mean value and
standard deviation of the absorptivity are listed in Table 3. While the mean
absorptivity on each part of the keyhole surface is similar in the three
cases, the standard deviation on the rear keyhole wall increases as the
scanning speed decreases. This observation implies that the keyhole morphology
is sensitive to the laser scanning speed, especially the rear keyhole wall,
which is closely related to the keyhole fluctuation and instant bubble
formation. In other words, the energy absorptivity fluctuation could be an
effective criterion to evaluate the possibility of the keyhole pore formation.

Figure 8: Energy absorptivity on (a) the whole keyhole surface, (b) front
keyhole wall, and (c) rear keyhole wall in Case2 ($525\;\mathrm{mm/s}$), Case
3 ($500\;\mathrm{mm/s}$), and Case 4 ($475\;\mathrm{mm/s}$).

The above analysis of the keyhole pore features in Case 2-4 shows that the
keyhole pore size is sensitive to the manufacturing parameters. The increase
of laser scanning speed is $50\;\mathrm{mm/s}$, while the mean keyhole pore
size decreases from $82\;\mathrm{\mu m}$ to $37\;\mathrm{\mu m}$. While the
molten pool flow of the cases are similar, the deviation of keyhole depth and
energy absoptivity fluctuation reflect the keyhole instability.

Table 3: Energy absorptivity on the front and rear keyhole wall

Energy absorptivity | Case 2 | Case 3 | Case 4 | Case 5
---|---|---|---|---
Front keyhole wall | Mean (%) | 25.54 | 26.07 | 27.07 | 25.58
SDa (%) | 2.29 | 2.51 | 2.63 | 2.23
Rear keyhole wall | Mean (%) | 50.04 | 49.88 | 49.04 | 50.40
SDa (%) | 3.02 | 3.26 | 4.21 | 2.75

  * a

SD means standard deviation.

###  3.4 Reduce keyhole pores: low ambient pressure

The mechanisms of keyhole pore formation as discussed above shows that the
balance of the forces on the keyhole surface is important to ameliorate the
pores formed by keyhole fluctuation. Although it has been proven that the
porosity of the as-built part under near-vacuum and low ambient pressure can
be reduced during both AM and laser welding [39, 40, 41], the principles on
how the ambient environment influences the keyhole and molten pool dynamics
and reduce the pores during AM are rarely studied yet. To explore the
principles of using low ambient pressure to reduce keyhole pores, a simulation
(Case 5) is conducted under $10^{-4}\;\mathrm{atm}$, in which the other
meanufacturing parameters are kept the same as Case 3.

As presented in Fig. 9, there are no keyhole pores in the simulation domain in
Case 5. The decreasing trend of pore formation under low ambient pressure in
the simulation is similar to the experiments [40, 42], where there are nearly
no pores under the ambient pressure below $100\;\mathrm{Pa}$. The temperature,
velocity distribution, and streamline in Case 5 are similar to those in Case
3. Despite the decrease in ambient pressure, the molten pool shapes are
similar, and the depth increases slightly from $416\;\mathrm{\mu m}$ to
$430\;\mathrm{\mu m}$. A vortex pair is observed, with a larger and stronger
clockwise rotating vortex at the front part of the molten pool. At the bottom
region, the velocity decreases longitudinally from the front to the rear end
of the molten pool.

Figure 9: Simulation results of the molten pool flow under
$10^{-4}\;\mathrm{atm}$ ambient pressure (Case 5) at $t=2000\;\mathrm{\mu s}$.
(a) Temperature distribution, (b) velocity magnitude distribution, and (c)
streamline in the molten pool. The white curve in (a) is the contour of the
liquidus temperature $T_{l}$, and the white curve in (b) and grey contour in
(c) are the contour of the solidus temperature $T_{s}$.

The main differences between Case 3 and 5 are the ambient pressure and
consequently the recoil pressure. The keyhole fluctuation is dynamic, and the
recoil pressure changes with the ambient pressure and keyhole surface
temperature. Thus, further analysis of the recoil pressure on the keyhole
surface is necessary as given in Fig. 10. Calculated by our evaporation model
[23], the recoil pressure-surface temperature curves under different ambient
pressures are different as shown in Fig. 11. Although the amplitude of the
recoil pressure at high temperature is similar, the boiling temperature
decreases as the ambient pressure drops. Therefore, the temperature and recoil
pressure ranges in Fig. 10 for the two cases are different.

Figure 10: (a and b) Temperature field above the boiling temperatures and (c
and d) recoil pressure on the keyhole surface under the common ambient (Case
3) and low ambient pressure (Case 5). (a and c) and (b and d) are the
simulation results of Case 3 and 5, respectively. The white curves in (a) and
(b) are the contour of the solidus temperature $T_{s}$.

In Case 3, the region near the keyhole with the temperature above the boiling
point is small and randomly distributed on the front keyhole wall (Fig. 10
(a)). Correspondingly, the recoil pressure is concentrated on the front
keyhole wall (Fig. 10 (c)). Thus, the rear keyhole wall has weak support from
the recoil pressure and collapses easily to form instant bubbles. In contrast,
the recoil pressure distribution under the low ambient pressure is different.
Although the highest temperature on the keyhole surface in Fig. 10 (b) is
close to that in Fig. 10 (a) at around 4000 K, the temperature of nearly
entire keyhole surface is above the boiling temperature and the temperature
increases continuously from the upper to the bottom region of the keyhole.
This temperature distribution indicates stronger evaporation from keyhole
surface under the low ambient pressure, which follows the trend in previous
experiments [40, 43]. Moreover, the recoil pressure on the keyhole surface is
distributed all over the keyhole surface with the value above 0.1 atm and up
to 5 atm at the bottom of the keyhole. Compared to the recoil pressure
distribution in Case 3 (Fig. 10 (c)), the recoil pressure on the rear keyhole
wall under the low ambient pressure is higher which stabilizes the rear
keyhole wall during fluctuation and thus reduces keyhole pores.

Figure 11: Relationship between recoil pressure and surface temperature of
Ti-6Al-4V under different ambient pressures. The boiling temperature varies
with the ambient pressure, so the starting points of the three curves are
different.

To further analyze the stability of the keyhole under different ambient
pressures, the keyhole depth and energy absorptivity of Case 3 and 5 are given
in Fig. 12. The keyhole depth in Case 5 fluctuates more violently than that in
Case 3. The standard deviation of keyhole depth in Case 5 is $37\;\mathrm{\mu
m}$, about 28% higher than that in Case 3 as listed in Table 2. However, the
keyhole depth is not indicative of the keyhole instability under the low
ambient pressure. Further analysis of the energy absorptivity on keyhole
surface from Fig. 12 (b-d) and Table 3 shows that the absorptivity
fluctuations trend under different ambient pressures are similar to each
other, whereas the standard deviations of the energy absorptivity on both the
front and real keyhole wall in Case 5 are lower than those in Case 3 and even
lower than those in Case 2. As analyzed in Sect. 3.3, the keyhole stability is
sensitive to the manufacturing parameters, which also explains why there is no
keyhole pore in Case 5. The energy absorption is calculated via the ray-
tracing method, which is largely determined by the geometric shape of the
keyhole. Therefore, the lower standard deviation of the energy absorptivity is
indicative of a smaller distortion of the keyhole shape, especially the rear
keyhole wall.

Furthermore, the similar energy absorptivity on the rear keyhole wall under
the low ambient leads to higher evaporation and higher recoil pressure than
that under common ambient pressure. The analysis above indicates that the
keyhole depth fluctuation is more volatile under low ambient pressures because
the evaporation is greater compared to that under common ambient pressure.
Simultaneously, the recoil pressure on the rear keyhole wall increases to
maintain the stability of the keyhole shape. Thus, the keyhole is stabler
under low ambient pressure to reduce the porosity in the keyhole melting mode.

Figure 12: Comparison of Case3 ($1\;\mathrm{atm}$) and Case 5
($10^{-4}\;\mathrm{atm}$): (a) keyhole depth, (b) total energy absorptivity,
(c) energy absorptivity on the front keyhole wall, and (d) energy absorptivity
on the rear keyhole wall.

##  4 Conclusions

Keyhole pore formation during metal additive manufacturing has been
investigated in this work with a multiphysics thermal-fluid flow model, and
directly validated with the X-ray imaging results. The following conclusions
can be drawn:

  1. (1)

The keyhole pore formation process has two distinct stages: (i) the instant
bubble formation and (ii) pinning on the solidification front stage. The
instant bubble formation is mainly due to the keyhole instability (unbalanced
forces on the rear keyhole wall). During the bubble pinning on the
solidification front, the high flow speed below the instant bubble generates a
vertical drag force which impedes the bubble from floating up to the molten
pool surface. The bubble is finally caught by the solidification front to form
the keyhole pore.

  2. (2)

The unevenly distributed recoil pressure on the keyhole surface increases the
possibility of keyhole collapse to form keyhole pores. Additionally, the drag
force from the mushy zone is pertinent as it determines the keyhole
fluctuation at the bottom of the melt pool. A Darcy drag force model with the
consideration of grain morphology in AM is required to improve the accuracy of
drag force and keyhole fluctuation calculation.

  3. (3)

The keyhole pore size is sensitive to the manufacturing parameters. As the
laser scanning speed slightly increases, the keyhole pore sizes decrease
significantly, and the shape of the keyhole pore becomes spherical and
horizontally distributed at the molten pool bottom. Moreover, the features of
keyhole fluctuation and energy aborptivity variation could be criterions to
predict the likelihood of keyhole pore formation.

  4. (4)

Low ambient pressure is a feasible way to reduce or even eliminate the
formation of keyhole pores. Compared to the common ambient pressure, the
recoil pressure on the rear keyhole wall under low ambient is larger and
maintains a stable keyhole shape.

## Acknowledgments

This research is supported by A*STAR under its AME IRG Grant (Project No.
A20E5c0091).

##  Appendix A Material Property

Table 4: Material properties of Ti-6Al-4V [23, 25, 28, 44, 45] Property | Ti-6Al-4V
---|---
Solidus temperature ($T_{s}$) | 1878 [K]
Liquidus temperature ($T_{l}$) | 1928 [K]
Boiling temperature ($T_{b}$) | 3315 [K]
Solidus Density ($\rho$) | $4400\rm{[kg/m^{3}]}$
Latent heat of melting ($L_{m}$) |  $2.86\times 10^{5}$ [J/kg]
Latent heat of evaporation ($L_{v}$) |  9.7$\times 10^{6}$ [J/kg]
Saturated vapor pressure ($P_{e}$) |  1.013$\times 10^{5}$ [Pa] ($T_{b}$=3315 K)
Solidus Specific heat ($c_{s}$) |  570 [J/(K$\cdot$kg)]
Liquidus Specific heat ($c_{l}$) |  831 [J/(K$\cdot$kg)]
Thermal conductivity at solidus ($k_{s}$) |  16 [W/(m$\cdot$K)]
Thermal conductivity at liquidus ($k_{l}$) |  32 [W/(m$\cdot$K)]
Surface radiation coefficient ($\epsilon$) | 0.4
Surface tension coefficient ($\sigma_{0}$) | 1.68 [N/m]
Temperature sensitivity of $\sigma$ ($\sigma_{s}^{T}$) | 0.00026
Viscosity ($\mu$) |  0.005 [Pa$\cdot$s]

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# Existence of Nonzero Trace-Zero Idempotents in the Group Algebras of Finite
Groups

Wenhua Zhao and Dan Yan  W. Zhao, Department of Mathematics, Illinois State
University, Normal, IL 61761. Email<EMAIL_ADDRESS>D. Yan, MOE-LCSM, School
of Mathematics and Statistics, Hunan Normal University, Changsha 410081,
China. _Email:_<EMAIL_ADDRESS>
(Date: August 09, 2022)

###### Abstract.

Let $G$ be a finite group and $K$ a splitting field of $G$ of characteristic
$p>0$. Denote by $KG$ the group algebra of $G$ over $K$ and $Z(KG)$ the center
of $KG$. Let $V_{G}$ be the $K$-subspace of trace-zero elements of $KG$. We
give some numerical sufficient and necessary conditions for $V_{G}$ and
$V_{G}\cap Z(KG)$, respectively, to be Mathieu subspaces of $KG$ in terms of
the degrees of irreducible representations of $G$ over $K$. The same numerical
conditions also characterize the finite groups $G$ that $KG$ has no nonzero
trace-zero idempotents and the finite groups $G$ that $KG$ has no nonzero
central trace-zero idempotents, respectively.

###### Key words and phrases:

Mathieu subspace (Mathieu-Zhao space); group algebras of finite groups; trace-
zero idempotents

###### 2000 Mathematics Subject Classification:

16S34, 20C05, 16D99, 20C20

The first author has been partially supported by the Simons Foundation grant
#278638. The second author is supported by the NSF of China (Grant No.
11871241; 11601146), the China Scholarship Council and the Construct Program
of the Key Discipline in Hunan Province.

##  1\. Motivations and the Main Results

Let $G$ be a finite group, $K$ a splitting field of $G$ (unless stated
otherwise) and $KG$ (or $K[G]$) the group algebra of $G$ over $K$. Denote by
$1_{G}$, or simply $1$, the identity element of $G$ and $KG$. We denote by
$Z(KG)$ and $V_{KG}$, or simply by $V_{G}$ if the base field $K$ is clear in
context, the center of $KG$ and the $K$-subspace of $KG$ spanned over $K$ by
all $1\neq g\in G$, respectively. An element $e\in KG$ is said to be an
idempotent if $e^{2}=e$, and a non-trivial idempotent if $e$ is an idempotent
and $e\neq 1$ or $0$.

For each $u\in KG$, denote by $\operatorname{tr}_{G}u$, or
$\operatorname{tr\,}u$ if $KG$ is clear in context, the trace of $u$ (i.e.,
the coefficient of $1_{G}$ in $u$). An element $v\in KG$ is said to be trace-
zero if $\operatorname{tr\,}u=0$ and trace-one if $\operatorname{tr\,}u=1$.
Hence, the subspace $V_{G}$ defined above is just the subspace of trace-zero
elements of $KG$.

Since an element $e$ is a trace-zero idempotent of $KG$, if and only if $1-e$
is a trace-one idempotent of $KG$, all the results derived in this paper on
trace-zero idempotents of $KG$ can be transformed in the obvious ways to the
results on trace-one idempotents of $KG$.

The purpose of this paper is twofold. The first one is to find some numerical
sufficient and necessary conditions in terms of the degrees of irreducible
representations of $G$ over $K$ for that $KG$ has no nonzero trace-zero
idempotents and that $KG$ has no nonzero central trace-zero idempotents,
respectively.

To explain the second purpose of the paper, we need to recall the following
concept that was introduced by the first author in [Z2] and [Z3].

######  Definition 1.1.

Let $R$ be a commutative ring and ${\mathcal{A}}$ a unital $R$-algebra. An
$R$-subspace $V$ of ${\mathcal{A}}$ is said to be a Mathieu subspace (MS) of
${\mathcal{A}}$ if for all $a,b,c\in{\mathcal{A}}$ with $b^{m}\in V$ for all
$m\geq 1$, there exists $N\in{\mathbb{N}}$ (depending $a,b$ and $c$) such that
$ab^{m}c\in V$ for all $m\geq N$.

A Mathieu subspace is also called a Mathieu-Zhao space or Mathieu-Zhao
subspace in the literature (e.g., see [EKC, DEZ]). The introduction of Mathieu
subspaces was directly motivated by the Mathieu Conjecture [Mat] and the Image
Conjecture [Z1], which both imply the famous Jacobian conjecture [Ke, BCW, E].
Note that all ideals are Mathieu subspaces, but not conversely. Therefore, the
concept of Mathieu subspaces can be viewed as a natural generalization of the
concept of ideals.

One of the most fundamental theorems in the theory of Mathieu subspaces is the
Duistermaat and van der Kallen theorem [DK] which was proved in $1998$. It
solves the abelian case of the Mathieu Conjecture. The theorem in terms of
group algebras and Mathieu subspaces can be essentially restated as follows.

######  Theorem 1.2.

Let $G={\mathbb{Z}}^{n}$ (the free abelian group of rank $n$). Then $V_{G}$ is
a MS of $\mathbb{C}G$. In terms of $K$-algebra of Laurent polynomials
${\mathbb{C}}[x^{-1},x]$ in $n$ variables, the subspace $V$ of all Laurent
polynomials in $x$ with no constant term is a MS of ${\mathbb{C}}[x^{-1},x]$.

In order to understand the Duistermaat and van der Kallen theorem in a more
general setting, the following problem was proposed in [ZW, Problem $1.4$].

######  Open Problem 1.3.

Decide commutative rings $R$ and groups $G$ such that $V_{RG}$ is a Mathieu
subspace of the group algebra $RG$ of $G$ over $R$.

The following proposition and theorem among some other results were also
proved [ZW].

######  Proposition 1.4.

Let $G$ be a finite group and $K$ an arbitrary field. Then $V_{G}$ is a MS of
$KG$, if and only if $V_{G}$ does not contain any nonzero idempotent.

######  Theorem 1.5.

Let $G$ be a finite group and $K$ an arbitrary field. Then the following
statements hold:

  1. $i)$

If char. $K=0$ or char. $K=p>|G|$, then $V_{G}$ is a MS of $KG$, where $|G|$
denotes the order of $G$.

  2. $ii)$

Assume that $G$ is abelian and $|G|=p^{r}d$ with $r,d\in{\mathbb{N}}$ and
$p\nmid d$. If char. $K=p>0$ and $K$ has a $d$-th primitive root of unity,
then $V_{G}$ is a MS of $KG$, if and only if $p>d$.

The second purpose of the paper is to find some numerical sufficient and
necessary conditions in terms of the degrees of irreducible representations of
$G$ over $K$ for that $V_{G}$ and $V_{G}\cap Z(KG)$ are Mathieu subspaces of
$KG$, respectively.

The two purposes of the paper actually coincide. More precisely, by
Proposition 1.4 $V_{G}$ is a MS of $KG$, if and only if $KG$ has no nonzero
trace-zero idempotent, and by Lemma 4.1 in Section 4, $V_{G}\cap Z(KG)$ is a
MS of $KG$, if and only if $KG$ has no nonzero central trace-zero idempotent.

To state the two main results of the paper, we denote by $\phi_{i}$ $(1\leq
i\leq s)$ the (non-isomorphic) irreducible representations of $G$ over $K$,
and $n_{i}$ the degree of the representation $\phi_{i}$. Then the first main
theorem of this paper is as follows.

######  Theorem 1.6.

Let $G$ be a finite group and $K$ a splitting field of $G$ of char. $K=p>0$.
Then $V_{G}$ is a MS of $KG$, if and only if the following property holds:

($\ast$) $\displaystyle c_{1}n_{1}+c_{2}n_{2}+\cdots+c_{s}n_{s}\not\equiv
0\quad(\operatorname{mod}\,\,p)$

for all integers $0\leq c_{i}\leq n_{i}$ $(1\leq i\leq s)$ that are not all
zero.

######  Remark 1.7.

Let $\,\overline{\\!{B}}$ be a sequence of elements in ${\mathbb{Z}}_{p}$
formed by $n_{i}$-copies of $n_{i}$ $($mod $p)$ with $1\leq i\leq s$. Then it
is easy to see that the property $(\ast)$ in Theorem 1.6 for integers $n_{i}$
$(1\leq i\leq s)$ is equivalent to that the sequence $\,\overline{\\!{B}}$ is
a so-called zero-sum free sequence $($e.g., see [G]$)$ of elements of
${\mathbb{Z}}_{p}$.

######  Remark 1.8.

Given a finite group $G$ and $K$ an arbitrary field. If char. $K=0$ or char.
$K=p>|G|$, then by Proposition 1.4 and Theorems 1.5 $KG$ contains no nonzero
trace-zero idempotent. If char. $K=p\leq|G|$, let $L$ be a splitting field of
$G$, which contains $K$ (e.g., let $L=K[\xi]$, where $\xi$ is a primitive
$d$-th root of unity and $d$ is the exponent of $G$. By [PD, Theorem 2.7B, p.
$58$] $L$ is a splitting field of $G$). Then we may apply Theorem 1.6 to
decide if $V_{LG}$ is a MS of $LG$. If it is, then by Proposition 1.4 $V_{KG}$
is a MS of $KG$ and $KG$ contains no nonzero trace-zero idempotent.

In order to state our second main result, we need some basic notations and
facts in modular representation theory. Here we freely adapt those fixed and
developed in [PD].

Assume that char. $K=p>0$ and $|G|=p^{r}d$ with $r,d\in{\mathbb{N}}$ and
$p\nmid d$. Let $\widehat{K}$ be a field extension of finite degree of the
$p$-adic number field ${\mathbb{Q}}_{p}$ such that $X^{d}-1$ splits completely
in $\widehat{K}[X]$. Then by Lemma 2.10 proved in Section 2, $\widehat{K}$ is
a spitting field for all subgroups of $G$. Let $A$ be the integral closure of
the $p$-adic integers in $\widehat{K}$, and $\pi A$ the unique maximal ideal
of $A$ (which contains $pA$). Let $K=A/\pi A$. $K$ is a finite field of
characteristic $p>0$. Then by discussion in [PD, Section $2.7$] $\widehat{K}$
and $K$ are splitting fields for all subgroups and quotient groups of $G$. In
particular, by [PD, Theorem $1.7B$, p. $25$] all irreducible
$\widehat{K}G$-modules and irreducible $KG$-modules are absolutely
irreducible.

Let $f_{i}$ $(1\leq i\leq t)$ be the block idempotents of $KG$ and $n_{i,j}$
$(1\leq j\leq s_{i})$ the degrees of all distinct irreducible representations
of $G$ over $\widehat{K}$ which lies in the $i$-th $p$-block of $KG$.

Our second main theorem of the paper is as follows.

######  Theorem 1.9.

Let $G$ be a finite group and $K$ a splitting field of $G$. Then the following
statements hold:

  1. $i)$

For every field $L$ with char. $L=0$, or char. $L=p>|G|$, $V_{LG}\cap Z(LG)$
is a MS of $LG$.

  2. $ii)$

If char. $K=p>0$ and $p\nmid|G|$, then $V_{KG}\cap Z(KG)$ is a MS of $KG$, if
and only if for each $\emptyset\neq J\subseteq\\{1,2,...,s\\}$ we have

(1.1) $\displaystyle\sum_{i\in J}n_{i}^{2}\not\equiv
0\quad(\operatorname{mod\,}p).$

  3. $iii)$

If char. $K=p>0$ and $p\,|\,|G|$, then $V_{KG}\cap Z(KG)$ is a MS of $KG$, if
and only if for all $\emptyset\neq J\subseteq\\{1,2,...,t\\}$, both the
numerator and the denominator of the rational number $\frac{1}{|G|}\sum_{i\in
J}\sum_{j=1}^{s_{i}}n_{i,j}^{2}$ in the reduced form are co-prime to $p$.

Note also that if char. $K=0$ or char. $K=p>0$ with $p\nmid|G|$, then by
Maschke’s Theorem (e.g., see [PD, Theorem $1.3$B]) $KG$ is semi-simple. For
all $1\leq i\leq t=s$, the dimension over $K$ of $i$-th (indecomposable)
$p$-block of $KG$ is equal to $n_{i}^{2}$.

If char. $K=p>0$ with $p\mid|G|$, then for each $1\leq i\leq t$, by [PD,
Theorem $4.2$D, p. $99$] and also the fact mentioned above on the blocks of
$\widehat{K}G$ (noting char. $\widehat{K}=0)$, we see that the dimension of
the $i$-th $p$-block of $KG$ over $K$ is equal to
$\sum_{j=1}^{s_{i}}n_{i,j}^{2}$ (in the same notation as in Theorem 1.9). Then
it is easy to see that the case of Theorem 1.9 over all splitting base fields
can be re-stated in the following more uniform way.

######  Corollary 1.10.

Let $K$ be a splitting field (of arbitrary characteristic) of $G$. Let $t$ be
the number of (indecomposable) blocks of $KG$ and $d_{i}$ $(1\leq i\leq t)$
the dimension over $K$ of the $i$-th block of $KG$. Then $V_{G}\cap Z(KG)$ is
a MS of $KG$, if and only if for all $\emptyset\neq
J\subseteq\\{1,2,...,t\\}$, both the numerator and the denominator of the
rational number $\frac{1}{|G|}\sum_{i\in J}d_{i}$ in the reduced form are co-
prime to $p$.

Note that by Lemma 4.1 in Section 4 the sufficient and necessary conditions in
Theorem 1.9 and Corollary 1.10 are also sufficient and necessary conditions
for the nonexistence of nonzero central trace-zero idempotents in $KG$.

Arrangements We mainly give a proof for Theorem 1.6 in Section 2 and discuss
some applications of Theorem 1.6 in Section 3. In Section 4 we mainly give a
proof for Theorem 1.9.

##  2\. Proof of Theorem 1.6

Throughout this section, $G$ stands for a finite group and $K$ a splitting
field of $G$ of characteristic $p>0$. We denote by $\phi_{i}$ $(1\leq i\leq
s)$ all the (non-isomorphic) irreducible representations of $G$ over $K$ and
$n_{i}$ the degree of the representation $\phi_{i}$ for all $1\leq i\leq s$.
Here we fix the irreducible representation $\phi_{i}$ $(1\leq i\leq s)$ as a
$K$-algebra homomorphism from $KG$ to the $n_{i}\times n_{i}$ matrix algebra
$M_{n_{i}}(K)$. Note that by Burnside’s Theorem (e.g., see [PD, Corollary, p.
$11$]) each $\phi_{i}$ $(1\leq i\leq s)$ is surjective.

Let ${\mathcal{A}}={\bigoplus}_{i=1}^{s}M_{n_{i}}(K)$ and
${\mathcal{A}}_{i}=M_{n_{i}}(K)$ $(1\leq i\leq s)$. For each $1\leq i\leq s$
we define $\operatorname{tr}_{i}:{\mathcal{A}}\to K$ to the $K$-linear
functional whose restriction on the $i$-th component $M_{n_{i}}(K)$ of
${\mathcal{A}}$ is the usual trace function of the $n_{i}\times n_{i}$
matrices, and on $j$-th component of ${\mathcal{A}}$ is equal to $0$ for all
$j\neq i$.

With the notations and settings above we recall the classical Wedderburn
Structure Theorem (see [PD, Theorem $1.3A$]) for $KG$.

######  Theorem 2.1.

Define $\Phi:KG\to{\mathcal{A}}$ be setting
$\Phi(\sigma)=\sum_{i=1}^{s}\phi_{i}(\sigma)$ for all $\sigma\in KG$. Then
$\Phi$ is a surjective homomorphism of $K$-algebras and
$\operatorname{Ker\,}\Phi=J(KG)$, where $J(KG)$ stands for the Jacobson
radical of $KG$.

Next, we recall the following theorem which clarifies all co-dimensional one
MS’ of ${\mathcal{A}}$.

######  Theorem 2.2.

$($[Ko, Corollary $4.5$]$)$ A codimensional one $K$-subspace $V$ of
${\mathcal{A}}$ is a MS of ${\mathcal{A}}$, if and only if
$V=\operatorname{Ker\,}L$ for some $K$-linear functional $L$ of
${\mathcal{A}}$ such that $L=\sum_{i=1}^{s}c_{i}\operatorname{tr}_{i}$, where
the coefficients $c_{i}\in K$ $(1\leq i\leq s)$ satisfy the property: for all
$\emptyset\neq J\subseteq\\{1,2,...,s\\}$ with not all $c_{i}$ $(i\in J)$
being zero, we have $\sum_{i\in J}c_{i}\neq 0$.

The following proposition classifies all the co-dimensional one MS’ $V$ of
$KG$ with $V\supseteq J(KG)$ (the Jacobson radical of $KG$).

######  Proposition 2.3.

Let $V$ be a co-dimensional one $K$-subspace of $KG$ with $V\supseteq J(KG)$,
and $\chi_{i}$ the character of $G$ obtained from the irreducible
representation $\phi_{i}$ $(1\leq i\leq s)$. Then $V$ is a MS of $KG$, if and
only if $V=\operatorname{Ker\,}\ell$, where $\ell=\sum_{i=1}^{s}c_{i}\chi_{i}$
for some $c_{i}\in K$ $(1\leq i\leq s)$ which satisfy the property: for all
$\emptyset\neq J\subseteq\\{1,2,...,s\\}$ with not all $c_{i}$ $(i\in J)$
being zero, we have $\sum_{i\in J}c_{i}\neq 0$.

Proof: Since $\Phi$ by Theorem 2.1 is surjective and $V\supseteq J(KG)$, there
exists a co-dimensional one $K$-subspace $W$ of ${\mathcal{A}}$ such that
$V=\Phi^{-1}(W)$ (Actually, $W=\Phi(V)$). By Theorem 2.2 (or [EKC, Theorem
$5.8.1$]), $W$ is a MS of ${\mathcal{A}}$ iff $W=\operatorname{Ker\,}L$ for
some $K$-linear functional $L$ of ${\mathcal{A}}$ such that
$L=\sum_{i=1}^{s}c_{i}\operatorname{tr}_{i}$ for some $c_{i}\in K$ $(1\leq
i\leq s)$ which satisfy the property in the proposition.

Let $\Phi^{*}$ be the dual map of $\Phi$, i.e., $\Phi^{*}(\psi)=\psi\circ\Phi$
for all $K$-linear functional $\psi$ of ${\mathcal{A}}$. Then by definition of
characters we have $\Phi^{*}(\operatorname{tr}_{i})=\chi_{i}$ for all $1\leq
i\leq s$. Since $V=\Phi^{-1}(W)$, $W=\Phi(V)$ and $W=\operatorname{Ker\,}L$,
we have $V=\operatorname{Ker\,}\ell$ with
$\ell=\phi^{*}(L)=\sum_{i=1}^{s}c_{i}\chi_{i}$. Since by Theorem 2.1 and [Z3,
Proposition $2.7$], $V$ is a MS of $KG$, if and only if $W$ is a MS of
${\mathcal{A}}$, the proposition follows. $\Box$

Now, we can prove Theorem 1.6 for the semisimple case.

######  Lemma 2.4.

Assume that $p\nmid|G|$. Then

  1. $i)$

$\Phi(V_{G})=\operatorname{Ker\,}F$, where
$F=\sum_{i=1}^{s}n_{i}\operatorname{tr}_{i}$.

  2. $ii)$

$V_{G}=\operatorname{Ker\,}\ell$, where
$\ell=\Phi^{*}(F)=\sum_{i=1}^{s}n_{i}\chi_{i}$.

  3. $iii)$

$V_{G}$ is a MS of $KG$, if and only if $\sum_{i=1}^{s}c_{i}n_{i}\neq 0$ for
all
$\vec{0}\neq(c_{1},...,c_{s})\in\\{0,1,...,n_{1}\\}\times\cdots\times\\{0,1,...,n_{s}\\}$.
In other words, Theorem 1.6 holds in this case.

Proof: Note first that, since $p\nmid|G|$, by Maschke’s Theorem (e.g., see
[PD, Theorem $1.3B$]) the group algebra $KG$ is semisimple, i.e., $J(KG)=0$.
Then $\Phi$ in Theorem 2.1 is a $K$-algebra isomorphism.

$i)$ Let $1_{G}\neq g\in G$ and $u=\Phi(g)$. Write $u=\sum_{i=1}^{s}B_{i}$
with $B_{i}\in{\mathcal{A}}_{i}\\!:=M_{n_{i}}(K)$ for all $1\leq i\leq s$. To
show $u\in\operatorname{Ker\,}F$, we consider the trace of the regular action
of $g$ on $KG$ and the trace of the regular action of $u$ on ${\mathcal{A}}$.
Since $\Phi$ in this case is an isomorphism of $K$-algebras, we know that
these two traces are equal to one another.

First, by choosing elements of $G$ as a basis of $KG$, it is easy to see that
the trace of the regular action of $g$ on $KG$ is equal to $0$.

Second, for each $1\leq i\leq s$, since the matrix $B_{i}$ annihilates all
elements of ${\mathcal{A}}_{j}$ for all $j\neq i$, the trace of the regular
action of $B_{i}$ on ${\mathcal{A}}$ is equal to the trace of the regular
action of $B_{i}$ on ${\mathcal{A}}_{i}=M_{n_{i}}(K)$, which can be readily
checked to be $n_{i}\operatorname{tr}_{i}(B_{i})$ by choosing the matrices
$E_{k,\ell}$’s (with $1$ as the $(k,\ell)$-entry and zero elsewhere) as a
basis of $\mathcal{A}_{i}=M_{n_{i}}(K)$. Hence the trace of the regular action
of $u$ on ${\mathcal{A}}$ is equal to
$\sum_{i=1}^{s}n_{i}\operatorname{tr}_{i}(B_{i})$. Therefore
$\sum_{i=1}^{s}n_{i}\operatorname{tr}_{i}(B_{i})=0$, i.e.,
$u\in\operatorname{Ker\,}F$, whence
$\Phi(V_{G})\subseteq\operatorname{Ker\,}F$ for $V_{G}$ is spanned over $K$ by
all non-identity elements of $G$. Since $\Phi(V_{G})$ and
$\operatorname{Ker\,}F$ are both co-dimensional one subspaces of
${\mathcal{A}}$, we have $\Phi(V_{G})=\operatorname{Ker\,}F$, and statement
$i)$ follows.

$ii)$ follows immediately from $i)$, and the fact that
$\Phi^{*}(\operatorname{tr}_{i})=\chi_{i}$ for all $1\leq i\leq s$. $iii)$
follows immediately from $ii)$, Proposition 2.3 above and the fact that
$n_{i}\neq 0$ for all $1\leq i\leq s$. $\Box$

Next we shall give a proof of Theorem 1.6 for the case $p\,|\,|G|$. In order
to do so we need first to prove some lemmas, for which we do not assume
$p\,|\,|G|$.

######  Lemma 2.5.

Assume that $KG$ has $k$ nonzero (pairwise) orthogonal idempotents $e_{i}$
$(1\leq i\leq k)$ with $k\geq p$. Then $V_{G}$ is not a MS of $KG$.

Proof: For each $1\leq j\leq k$ set $s_{j}=\sum_{i=1}^{j}e_{i}$. Then it is
easy to see that $s_{j}$ $(1\leq j\leq k)$ are all nonzero idempotents. By
[Pa, Theorem $3.5$ in p. $48$], which was first proved by Zalesskii [Za] in
$1972$,
$\operatorname{tr\,}s_{j}\in{\mathbb{Z}}_{p}={\mathbb{Z}}/p{\mathbb{Z}}$ for
all $1\leq j\leq k$. Since $k\geq p$, there exists either $1\leq j_{0}\leq k$
such that $\operatorname{tr\,}s_{j_{0}}=0$, or $a<b$ such that
$\operatorname{tr\,}s_{a}=\operatorname{tr\,}s_{b}$, whence
$\operatorname{tr\,}(s_{b}-s_{a})=0$. Since $s_{b}-s_{a}=e_{a+1}+\cdots+e_{b}$
is also a nonzero idempotent, $V_{G}$ in either case contains a nonzero trace-
zero idempotent, namely, $s_{j_{0}}$ or $s_{b}-s_{a}$. Then by Proposition 1.4
$V_{G}$ is not a MS of $KG$. $\Box$

######  Lemma 2.6.

Assume that $V_{G}$ is a MS of $KG$. Then

  1. $i)$

$p>\sum_{i=1}^{s}n_{i}$.

  2. $ii)$

$G$ has a normal Sylow $p$-subgroup.

Proof: $i)$ Let $m=\sum_{i=1}^{s}n_{i}$ and assume $p\leq m$. We view
${\mathcal{A}}$ as a $K$-subalgebra of the matrix algebra $M_{m}(K)$ by
putting $M_{n_{i}}(K)$ $(1\leq i\leq s)$ as the $(i,i)$-th block on the
diagonal. Let $f_{i}$ $(1\leq i\leq m)$ be the $m\times m$ diagonal matrix
with all entries being zero except the $(i,i)$-th entry being $1$. Hence
$f_{i}$ $(1\leq i\leq m)$ are orthogonal idempotents of ${\mathcal{A}}$ such
that $\sum_{i=1}^{m}f_{i}=1_{\mathcal{A}}$. Note that by Theorem 2.1,
$\operatorname{Ker\,}\phi=J(KG)$ and hence is nilpotent. Then by [Pa, Lemma
$3.7$ (i), p. $49$] there exist nonzero orthogonal idempotents $e_{i}\in KG$
$(1\leq i\leq m)$ such that $\Phi(e_{i})=f_{i}$. Hence, by Lemma 2.5, $V_{G}$
is not a MS of $KG$. Contradiction.

$ii)$ By $i)$ we have $p>n_{i}$ for each $1\leq i\leq s$, whence $p\nmid
n_{i}$. It is well known that $n_{i}$ $(1\leq i\leq s)$ are also the degrees
of irreducible Brauer characters of $G$ (by the very definition of the
irreducible Brauer characters, e.g., see [N1, p. $17$]). Then by [Mi2, Theorem
$2.4$] 111The proof of [Mi2, Theorem $2.4$] used the classification of finite
simple groups. For a proof of this theorem without using the classification,
see [Man, Theorem on p. $121$]. For a short proof for a special case of the
“only if” part of the theorem, see [R].  (which is a modular version of the
Itô-Michler Theorem [I, Mi2], see also [N2]) $G$ has a normal Sylow
$p$-subgroup. $\Box$

Recall that an element $g\in G$ is said to be a $p$-singular element, or
simply a $p$-element of $G$ if the order of $g$ is a power of $p$. $g$ is said
to be a $p$-regular element, or simply a $p^{\prime}$-element of $G$ of the
order of $g$ is co-prime to $p$. The identity element $1_{G}$ is the only both
$p$-singular and $p$-regular element of $G$.

It is easy to check that every element $g\in G$ can be written as the product
of a $p$-element and $p^{\prime}$-element, which commute with one another. We
denote by $G^{\circ}$ the set of all $p^{\prime}$-elements of $G$.

######  Lemma 2.7.

Let $e$ be an idempotent of $KG$ and $h\in G\backslash G^{\circ}$, i.e., $h$
is not a $p^{\prime}$-element. Then the sum of the coefficients in $e$ of the
elements in the conjugacy class ${\mathcal{C}}$ of $h$ is equal to $0$.

Proof: Write $|G|=p^{r}d$ for some $r,d\in{\mathbb{N}}$ with $p\nmid d$. Then
it is easy to see that $g^{p^{r}}\in G^{\circ}$ for all $g\in G$. Write the
idempotent $e$ as

(2.2) $\displaystyle e=\sum_{a\in G\backslash G^{\circ}}c_{a}\,a+\sum_{b\in
G^{\circ}}c_{b}\,b$

with $c_{a},c_{b}\in K$ $(a\in G\backslash G^{\circ}$, $b\in G^{\circ})$.

By [Pa, Lemma $3.1$, p. $45$] there exists $\beta$ in the commutator subspace
$[KG,KG]$ of $KG$ such that

$\displaystyle e=e^{p^{r}}=\sum_{a\in G\backslash
G^{\circ}}c_{a}^{p^{r}}a^{p^{r}}+\sum_{b\in
G^{\circ}}c_{b}^{p^{r}}b^{p^{r}}+\beta.$

Since $a^{p^{r}}\in G^{\circ}$ for all $a\in G\backslash G^{\circ}$, we
further have

(2.3) $\displaystyle e=e^{p^{r}}=\sum_{b\in G^{\circ}}d_{b}b+\beta$

for some $d_{b}\in K$ $(b\in G^{\circ})$.

Then by eqs. (2.2) and (2.3) we get

(2.4) $\displaystyle\sum_{a\in G\backslash G^{\circ}}c_{a}\,a+\sum_{b\in
G^{\circ}}(c_{b}-d_{b})\,b=\beta\in[KG,KG].$

Since $h\in G\backslash G^{\circ}$, its conjugacy class ${\mathcal{C}}\subset
G\backslash G^{\circ}$. Then by [Pa, Lemma 3.2, p. $46$] and the equation
above we have
$\left(\sum_{a\in\mathcal{C}}c_{a}\right)^{p^{r}}=\sum_{a\in\mathcal{C}}c_{a}^{p^{r}}=0$,
whence the lemma follows. $\Box$

The next lemma will play a crucial role in the proof of Theorem 1.6.

######  Lemma 2.8.

Let $H$ be a normal $p$-subgroup of $G$. Then $V_{G}$ is a MS of $KG$, if and
only if $V_{G/H}$ is a MS of the group algebra $K[G/H]$ of $G/H$ over $K$.

Proof: Let $\rho:K[G]\to K[G/H]$ be the $K$-algebra homomorphism induced by
the quotient map from $G$ to $G/H$. Then by [Pa, Lemma 1.8, p. $10$]
$\operatorname{Ker\,}\rho=\omega(KH)KG$, where $\omega(KH)$ is the
augmentation ideal of $KH$ which is spanned by $(h-1)$ $(h\in H)$ over $K$.
Therefore $K[G/H]\simeq KG/\omega(KH)KG$.

Let $g_{i}$ $(1\leq i\leq k\\!:=|G/H|)$ with $g_{1}=1_{G}$ be the
representatives of the distinct (right) cosets of $H$ in $G$. Then for every
idempotent $e\in KG$, there exist some $a_{i}\in KH$ $(1\leq i\leq k)$ such
that

(2.5) $\displaystyle e=a_{1}1_{G}+a_{2}g_{2}+\cdots+a_{k}g_{k}.$

Hence, the coefficient of $1_{G}$ in $e$ is equal to the coefficient of
$1_{H}$ in $a_{1}$, i.e., $\operatorname{tr}_{G}e=\operatorname{tr}_{H}a_{1}$.
Furthermore, since $H\unlhd G$, the $G$-conjugacy class of each $h\in H$ is
contained in $H$. Since $H$ is a $p$-group, by Lemma 2.7 the sum of the
coefficients in $a_{1}$ over the $G$-conjugacy class of each $1_{H}\neq h\in
H$ is equal to zero, whence $\rho(a_{1})=\operatorname{tr}_{H}a_{1}$.

On the other hand, by eq. (2.5) and the definition of $\rho$ we also have
$\operatorname{tr}_{G/H}\rho(e)=\rho(a_{1})$. In summary, the following
equations hold:

(2.6)
$\displaystyle\operatorname{tr}_{G}e=\operatorname{tr}_{H}a_{1}=\rho(a_{1})=\operatorname{tr}_{G/H}\rho(e).$

$(\Rightarrow)$ Assume that $V_{G}$ is a MS of $KG$ and $V_{G/H}$ is not a MS
of $K[G/H]$. Then by Proposition 1.4 there exists a nonzero idempotent $u\in
V_{G/H}$ (with $\operatorname{tr}_{G/H}u=0$). Since $H\unlhd G$, it follows
easily from [Pa, Lemma 1.6, p. $70$] that $\operatorname{Ker\,}\rho$
($=\omega(KH)KG$) is a nilpotent ideal of $KG$. Then by [Pa, Lemma 3.7 (i), p.
$49$] there exists an idempotent $e\in KG$ such that $\rho(e)=u$. By eq. (2.6)
$\operatorname{tr}_{G}e=\operatorname{tr}_{G/H}u=0$, i.e, $e\in V_{G}$. Since
$V_{G}$ is MS of $KG$, by Proposition 1.4 we have $e=0$, whence $u=0$.
Contradiction.

$(\Leftarrow)$ Assume that $V_{G/H}$ is a MS of $K[G/H]$ and $V_{G}$ is not a
MS of $KG$. Then by Proposition 1.4 there exists a nonzero idempotent $e\in
V_{G}$ (with $\operatorname{tr}_{G}e=0$). Note that $\rho(e)\neq 0$, for
$\operatorname{Ker\,}\rho=\omega(KH)KG$ is nilpotent (as pointed out above).
Therefore $\rho(e)$ is a nonzero idempotent of $K[G/H]$, and by eq. (2.6)
$\operatorname{tr}_{G/H}\rho(e)=0$, whence $\rho(e)\in V_{G/H}$. Since
$V_{G/H}$ is a MS of $K[G/H]$, by Proposition 1.4 $\rho(e)=0$. Contradiction.
$\square$

######  Remark 2.9.

We will show later in Proposition 3.3 that the $p$-subgroup condition on $H$
for the “only if” part of Lemma 2.8 can actually be dropped.

Now we can prove Theorem 1.6 as follows.

Proof of Theorem 1.6: $(\Rightarrow)$ Assume that $V_{G}$ is a MS of $KG$.
Then by Lemma 2.6 $ii)$ $G$ has a normal Sylow $p$-subgroup $H$. By Lemma 2.8
$V_{G/H}$ is a MS of $K[G/H]$. Since $p\nmid|G/H|$, it follows from Lemma 2.4
$iii)$ that the property $(\ast)$ in Theorem 1.6 holds for $K[G/H]$.

On the other hand, by [Pa, Proposition $4.7$, p. 67] the Jacobson radical
$J(KG)$ of $KG$ is equal to the ideal $I_{H}$ of $KG$ generated by $(h-1)$
$(h\in H)$. Since by Theorem 2.1 $K[G/H]\simeq KG/I_{H}=KG/J(KG)$, the degrees
of irreducible representations of $KG$ over $K$ (or equivalently, irreducible
representations of $G$ over $K$) are the same as the degrees of irreducible
representations of $K[G/H]$ over $K$ (or equivalently, irreducible
representations of $G/H$ over $K$). Therefore, the property $(\ast)$ in
Theorem 1.6 holds also for $KG$.

$(\Leftarrow)$ Assume that the property $(\ast)$ in Theorem 1.6 holds for
$KG$. If $p\leq n_{i_{0}}$ for some $1\leq i_{0}\leq s$, then the property
$(\ast)$ fails for the coefficients $c_{i_{0}}=p$ and $c_{i}=0$ $(i\neq
i_{0})$. Therefore, $p>n_{i}$, and hence $p\nmid n_{i}$, for all $1\leq i\leq
s$. Then by [Mi2, Theorem $2.4$] or [Man, Theorem on p. $121$] or the theorem
in the short note [R], $G$ has a normal Sylow $p$-subgroup $H$. Since as
pointed above, the degrees $n_{i}$ $(1\leq i\leq s)$ of irreducible
representations of $G$ over $K$ are the same as the degrees of irreducible
representations of $G/H$ over $K$, the property $(\ast)$ in Theorem 1.6 holds
for $K[G/H]$. Since $p\nmid|G/H|$, by Lemma 2.4 $iii)$ $V_{G/H}$ is a MS of
$K[G/H]$, and by Lemma 2.8 $V_{G}$ is a MS of $KG$. $\Box$

To end this section we point out that Theorem 1.5, $ii)$ also follows from
Theorem 1.6. First, by the lemma given below, the condition $p>d$ implies that
$K$ is a splitting field of $G$. Second, it is well known (e.g., see [PD,
Corollary on p. $19$]) that all irreducible representations of an abelian
group are of dimension one, i.e., $n_{i}=1$ for all $1\leq i\leq s$. Then it
is easy to see that the property $(\ast)$ in Theorem 1.6 holds, if and only if
$p>s$.

On the other hand, by [PD, Theorem $1.5$] the number of irreducible
representations of $G$ over $K$ is equal to the number of conjugacy classes of
$p^{\prime}$-elements of $G$, i.e., $s=d$ (since $G$ is abelian). Then by the
facts above and also the lemma below, Theorem 1.5 $ii)$ follows from Theorem
1.6.

######  Lemma 2.10.

Assume that $|G|=p^{r}d$ for some $r,d\in{\mathbb{N}}$ such that $p\nmid d$.
Let $L$ be a field of characteristic $p>0$, which contains a primitive $d$-th
root of unity. Then $L$ is a splitting field of $G$.

Proof: Since $L$ contains a primitive $d$-th root of unity, the polynomial
$X^{d}-1$ splits into the product of some degree one polynomials in $L[X]$.
Let $n=|G|$. Since $X^{n}-1=X^{p^{r}d}-1=(X^{d}-1)^{p^{r}}$, the polynomial
$X^{n}-1$ also splits into the product of degree one polynomials in $L[X]$,
hence so does $X^{m}-1$, where $m$ is the exponent of $G$. Then by [PD,
Theorem 2.7B, p. $58$] $L$ is a splitting field of $G$. $\square$

##  3\. Some Applications of Theorem 1.6

Throughout this section $G$ stands for a finite group with $|G|=p^{r}d$ and
$p\nmid d$, and $K$ a splitting filed of $G$ of characteristic $p>0$. We will
also freely use the notation fixed in Sections 1 and 2, e.g., $n_{i}$ $(1\leq
i\leq s)$ stand for the degrees of all irreducible representations of $G$ over
$K$.

######  Proposition 3.1.

Assume that $|G|=p^{r}d$ with $p\nmid d$. If $p>\sum_{i=1}^{s}n_{i}^{2}$, then
$V_{G}$ is a MS of $KG$. In particular, if $G$ has a normal Sylow $p$-subgroup
and $p>d$, then $V_{G}$ is a MS of $KG$.

Proof: First, it is easy to see that the condition $p>\sum_{i=1}^{s}n_{i}^{2}$
implies the property $(\ast)$ in Theorem 1.6, and hence by the same theorem
$V_{G}$ is a MS of $KG$.

Now assume that $G$ has a normal Sylow $p$-subgroup $H$. Then, as pointed out
in the proof of Lemma 2.8 in Section 2, $n_{i}$ $(1\leq i\leq s)$ are also the
degrees of all irreducible representations of $G/H$ over $K$. Since $p\nmid
d=|G/H|$, by Maschke’s Theorem (e.g., see [PD, Theorem $1.3B,\,ii)$] $K[G/H]$
is semi-simple. In particular, we have $\sum_{i=1}^{s}n_{i}^{2}=d$ by Theorem
2.1, and hence $p>\sum_{i=1}^{s}n_{i}^{2}$. Then by the first part of the
proposition proved above, $V_{G}$ is a MS of $KG$. $\square$

######  Remark 3.2.

Let $G$ be a $p$-group, then $V_{G}$ by the proposition above is a MS of $KG$.
Actually, by [Pa, Proposition $4.7$, p. 67] $KG$ is a local $K$-algebra and
hence, does not have any non-trivial idempotents. Then by [Z3, Theorem $4.2$]
every $K$-subspace $V$ of $KG$ such that $1\not\in V$ is a MS of $KG$.

The next two propositions claim that the set of all finite groups $G$ over
some fixed splitting fields $K$ is closed under taking subgroups, and taking
quotient groups as well as taking semi-product with $p$-groups.

######  Proposition 3.3.

Let $Q$ be a subgroup of $G$ or a quotient group of $G$. Assume that $K$ is
also a splitting field of $Q$ and $V_{G}$ is a MS of $KG$. Then $V_{Q}$ is a
MS of $KQ$.

Proof: The case that $Q$ is a subgroup can be seen easily from the assumption
and Proposition 1.4. So we assume that $Q=G/H$ for some $H\unlhd G$.

Let $\rho:KG\to K[G/H]$ be the $K$-algebra homomorphism induced by the
quotient map from $G$ to $G/H$ and $\psi_{i}:K[G/H]\to M_{n_{i}}(K)$ $(1\leq
i\leq t)$ all (non-isomorphic) irreducible representations of $G/H$ over $K$.
Then by Burnside’s Theorem (e.g., see [PD, Corollary, p. $11$]) each
$\psi_{i}$ is surjective. Since $\pi$ is surjective, so is the composition
$\phi_{i}\\!:=\psi_{i}\circ\rho$. Hence $\phi_{i}$ $(1\leq i\leq t)$ are non-
isomorphic irreducible representations of $G$ over $K$ of degree $n_{i}$.
Therefore, the condition $(\ast)$ in Theorem 1.6 for the degrees of
irreducible modules of $G$ over $K$ implies the same condition $(\ast)$ for
the degrees of irreducible modules of $G/H$ over $K$ (by letting $c_{i}=0$ for
all $t+1\leq i\leq s$). Then by applying Theorem 1.6 to $G/H$ the proposition
follows. $\Box$

A special case of the proposition above for quotient groups of $G$ follows
also directly from Proposition 1.4 and the lemma below which has some
interests on its own.

######  Lemma 3.4.

Let $H\unlhd G$ such that $p\nmid|H|$ and $\rho:KG\to K[G/H]$ be the
$K$-algebra homomorphism induced by the quotient map from $G$ to $G/H$. Then
for every idempotent $e\in K[G/H]$ there exists an idempotent $u\in KG$ such
that $\rho(u)=e$ and
$\operatorname{tr}_{G}u=\frac{1}{|H|}\operatorname{tr}_{G/H}e$.

Proof: Let $k=|G/H|$ and $g_{i}$ $(1\leq i\leq k)$ be the representatives of
all the cosets of $H$ in $G$ with $g_{1}=1_{G}$. For each $g\in G$ we set also
$\bar{g}=\rho(g)$. Write $e=\sum_{i=1}^{k}c_{i}\bar{g}_{i}$ for some $c_{i}\in
K$ $(1\leq i\leq k)$. Let $E_{H}=\frac{1}{|H|}\sum_{h\in H}h$ (i.e., the
principal idempotent of the group algebra $KH$ of $H$). Then it is well known
(and also easy to check) that we have

(3.7) $\displaystyle E_{H}^{2}=E_{H}\quad\text{and}\quad E_{H}h=E_{H}$

for all $h\in H$. Furthermore, by using the assumption $H\unlhd G$ we have
also that $E_{H}g=gE_{H}$ for all $g\in G$, whence $E_{H}\in Z(KG)$. Set

(3.8) $\displaystyle v\\!:=\sum_{i=1}^{k}c_{i}g_{i}\quad\text{and}\quad
u\\!:=E_{H}v.$

Then $\rho(v)=e$, $\rho(E_{H})=1$, and hence $\rho(u)=e$. Consequently,
$v^{2}-v\in\operatorname{Ker\,}\rho$, for
$\rho(v^{2})=\rho(v)^{2}=e^{2}=e=\rho(v)$, and $v^{2}=v+w$ for some
$w\in\operatorname{Ker\,}\rho$. Furthermore, by [Pa, Lemma $1.8$, p. $10$]
there exist some $\alpha_{h}\in KG$ $(h\in H)$ such that $w=\sum_{h\in
H}(h-1)\alpha_{h}$. Now, by eq. (3.7) and the fact $E_{H}\in Z(KG)$ we have

$\displaystyle u^{2}$
$\displaystyle=E_{H}vE_{H}v=E_{H}^{2}v^{2}=E_{H}v^{2}=E_{H}(v+w)=E_{H}(v+\sum_{h\in
H}(h-1)\alpha_{h})$ $\displaystyle=E_{H}v+\sum_{h\in
H}(E_{h}h-h)\alpha_{h}=E_{H}v=u.$

Therefore $u$ is an idempotent of $KG$ with $\rho(u)=e$, and by eq. (3.8)
$\operatorname{tr}_{G}u=\frac{1}{|H|}\operatorname{tr}_{G/H}e$, as desired.
$\Box$

######  Proposition 3.5.

Let $P$ be a $p$-group and $\widetilde{G}=P\rtimes G$. Then $V_{G}$ is a MS of
$KG$, if and only if $V_{\widetilde{G}}$ is a MS of $K\widetilde{G}$.

Proof: Since $\tilde{G}/P\cong G$ and $P$ is a normal $p$-subgroup of
$\widetilde{G}$, the proposition follows immediately from Lemma 2.8. $\Box$

######  Proposition 3.6.

Assume that $K$ is splitting field for all subgroups and quotient groups of
$G$ and $V_{G}$ is a MS of $KG$. Then for every subgroup or quotient $Q$, the
following statements hold:

  1. $i)$

the degree of every irreducible module of $Q$ over $K$ is less than $p$.

  2. $ii)$

the number of $p$-blocks of $KQ$ is less than $p$.

Proof: By Proposition 3.3 we may assume $Q=G$. Then $i)$ follows from Lemma
2.6 $i)$. Since the number of $p$-blocks of $KG$ is equal to the number of
(orthogonal and primitive) block idempotents of $KG$, statement $ii)$ follows
immediately from Lemma 2.5. $\Box$

Next, we consider the extreme case that char. $K=2$.

######  Proposition 3.7.

Let $L$ be a field of characteristic $p=2$, which is a splitting field of all
subgroups of $G$ (e.g., when the polynomial $X^{d}-1$ splits completely in
$L[X]$, see Lemma 2.10). Then the following statements are equivalent:

  1. $i)$

$V_{G}$ is a MS of $LG$.

  2. $ii)$

$LG$ does not contain any non-trivial idempotent.

  3. $iii)$

$G$ is a $2$-group.

Proof: Let $e$ be an idempotent of $LG$. Then by [Pa, Theorem $3.5$ in p.
$48$], which was first proved by Zalesskii [Za] in $1972$,
$\operatorname{tr\,}e=0$, or $1$. Hence the equivalence of statements $i)$ and
$ii)$ follows immediately from Proposition 1.4. The $iii)\Rightarrow i)$ part
follows from Remark 3.2. So it suffices to show $i)\Rightarrow iii)$.

Assume that $i)$ holds but $iii)$ fails. By Cauchy’s theorem there exists
$g\in G$ of odd order $k\geq 3$. Let $H$ be the cyclic subgroup of $G$
generated by $g$. Then by Proposition 3.3, $V_{H}$ is a MS of $LH$, and the
condition $(\ast)$ in Theorem 1.6 holds for the degrees of irreducible
representations of $H$, which together with the assumption $p=2$ implies that
$H$ has only one irreducible representation, namely, the trivial one.

On the other hand, since $2\nmid k=|H|$, by [PD, Theorem $1.5$] the number of
irreducible modules of $H$ over $K$ is equal to the number of conjugacy
classes of odd order elements of $H$, which is $k$ since $H$ is abelian.
Therefore $k=1$, contradiction. $\Box$

Finally, from Theorem 1.6 and Lemma 2.6 we see that, for the degrees $n_{i}$
$(1\leq i\leq s)$ of irreducible representations of $G$ over $K$, the property
$(\ast)$ in Theorem 1.6 implies that $p>\sum_{i=1}^{s}n_{i}$. Actually, this
implication holds for all finite sequences of positive integers, as we will
show in the next corollary.

######  Corollary 3.8.

Let $n_{i}$ $(1\leq i\leq s)$ be arbitrary positive integers which satisfy the
property $(\ast)$ in Theorem 1.6. Then $p>\sum_{i=1}^{s}n_{i}$.

Proof: Let $\,\overline{\\!{B}}$ be a sequence of elements in
${\mathbb{Z}}_{p}$ formed by $n_{i}$-copies of $n_{i}$ $($mod $p)$ with $1\leq
i\leq s$. Then as pointed out in Remark 1.7 the property $(\ast)$ in Theorem
1.6 for integers $n_{i}$ $(1\leq i\leq s)$ is equivalent to that
$\,\overline{\\!{B}}$ is a zero-sum free sequence of elements of
${\mathbb{Z}}_{p}$. Since the sequence $\,\overline{\\!{B}}$ is of length
$\sum_{i=1}^{s}n_{i}$, the corollary follows from [G, Theorem $10.2$] for the
abelian group ${\mathbb{Z}}_{p}$. $\Box$

The corollary above can also be proved directly by applying similar arguments
as those given in the proof of Lemma 2.5 to the sequence of partial sums of
$\,\overline{\\!{B}}$ in the proof above.

##  4\. Proof of Theorem 1.9

Throughout this section $G$ stands for a finite group and $K$ a field (of
arbitrary characteristic). Let $f_{i}$ $(1\leq i\leq t)$ be all the (central)
block idempotents of $KG$. The notations fixed in the previous sections are
still in force, e.g., $|G|=p^{r}d$ with $p\nmid d$.

In this section we mainly give a proof for Theorem 1.9.

######  Lemma 4.1.

The following statements are equivalent:

  1. $i)$

$V_{G}\cap Z(KG)$ is a MS of $KG$;

  2. $ii)$

$KG$ does not contain any nonzero central trace-zero idempotent.

  3. $iii)$

$\sum_{i\in J}\operatorname{tr\,}f_{i}\neq 0$ for all non-empty
$J\subseteq\\{1,2,...,t\\}$.

Proof: $i)\Rightarrow ii)$ Assume that $KG$ contains a nonzero central
idempotent $e$ with $\operatorname{tr\,}e=0$. Let $g\in G$ such that the
coefficient $c_{g}$ of $g$ in $e$ is nonzero. Then
$\operatorname{tr\,}(g^{-1}e)=c_{g}\neq 0$, whence $g^{-1}e\not\in V_{G}\cap
Z(KG)$. Since $KG$ is of finite dimension over $K$ and hence algebraic over
$K$, by [Z3, Theorem $4.2$] $V_{G}\cap Z(KG)$ is not a MS of $KG$.
Contradiction.

The $ii)\Rightarrow i)$ part follows immediately from [Z3, Theorem $4.2$].

$ii)\Leftrightarrow iii)$ Since $f_{i}$ $(1\leq i\leq t)$ are all primitive
idempotents of $Z(KG)$, by [N1, Theorem $3.11$, p. $55$] every idempotent of
$Z(KG)$ has the form $\sum_{i\in J}f_{i}$ for some non-empty
$J\subseteq\\{1,2,...,t\\}$. Hence statements $ii)$ and $iii)$ are equivalent
to one another. $\square$

######  Remark 4.2.

By a similar argument as the proof of [ZW, Proposition $3.1$], one can
actually show that, replacing the base field $K$ by any commutative base ring
$R$, $V_{RG}\cap Z(RG)$ is a MS of $RG$, if and only if every $\beta\in
V_{RG}\cap Z(RG)$ with $\beta^{m}\in V_{RG}\cap Z(RG)$ for all $m\geq 1$ is
nilpotent.

######  Corollary 4.3.

If $V_{G}$ is a MS of $KG$, then so does $V_{G}\cap Z(KG)$.

Proof: Since $V_{G}$ is a MS of $KG$, by Proposition 1.4 $V_{G}$ does not
contain any nonzero idempotent of $KG$. Hence, neither does $V_{G}\cap Z(KG)$.
Then by Lemma 4.1, $V_{G}\cap Z(KG)$ is a MS of $KG$. $\Box$

######  Corollary 4.4.

Assume that char. $K=p>0$ with $p\mid|G|$, and $V_{G}\cap Z(KG)$ is a MS of
$KG$. Then every block idempotent $f_{i}$ $(1\leq i\leq t)$ is of full defect.

Proof: For each $1\leq i\leq t$, let $r_{i}$ be the defect of the block
idempotent $f_{i}$. Then by [Mi1, Theorem $2.1\,(b)$]
$\operatorname{tr\,}f_{i}=p^{r-r_{i}}\frac{x}{y}$ (as elements of $K$) for
some $x,y\in{\mathbb{N}}$ which are co-prime to $p$. If $r_{i}<r$ for some
$1\leq i\leq t$, then statement $iii)$ in Lemma 4.1 fails for the subset
$J=\\{i\\}$, and by the same lemma, $V_{G}\cap Z(KG)$ is not a MS of $KG$.
Contradiction. $\Box$

Now we can give a proof for Theorem 1.9. We will freely use the notation and
conventions introduced in Section 1 related with Theorem 1.9 and Corollary
1.10.

Proof of Theorem 1.9: $i)$ By Theorem 1.5 $i)$, $V_{G}$ is a MS of $LG$, and
by Corollary 4.3, $V_{G}\cap Z(LG)$ is a MS of $LG$.

$ii)$ Assume that char. $K=p>0$ with $p\nmid|G|$, then by [PD, Theorem
$3.5,\,ii)$, p. $75$], $s_{i}=1$ and $n_{i,1}=n_{i}$ for all $1\leq i\leq t$.
Consequently, we have $t=s$, and by [PD, eq. $(8)$, p. 93] and [PD, Theorems
$3.5$ and $4.2$D] $\operatorname{tr\,}f_{i}=\frac{1}{|G|}n_{i}^{2}$ (as an
element of $K$) for all $1\leq i\leq s=t$. Then by Lemma 4.1 $iii)$,
$V_{G}\cap Z(KG)$ is a MS of $KG$, if and only if for each non-empty
$J\subseteq\\{1,2,...,s\\}$ we have

$\frac{1}{|G|}\sum_{i\in J}n_{i}^{2}\neq 0_{K}.$

Since $|G|\neq 0_{K}$ and $n_{i}^{2}\in{\mathbb{N}}$ for all $1\leq i\leq s$,
we see that the inequality above holds, if and only if $\sum_{i\in
J}n_{i}^{2}\not\equiv 0$ (mod $p$). Hence statement $ii)$ follows.

$iii)$ By [PD, Theorem $4.2D$, p. 99] (noting that the field $K$ there is our
$\widehat{K}$),
$\operatorname{tr\,}\\!f_{i}=\frac{1}{|G|}\left(\sum_{i=1}^{s_{i}}n_{i,j}^{2}\right)$
(as an element of $K$) for all $1\leq i\leq t$. Then statement $iii)$ follows
immediately from Lemma 4.1 $iii)$. $\Box$

Acknowledgments The authors are grateful to an anonymous referee for some
valuable suggestions on an earlier version of the paper. The first author also
thanks Sunil Chebolu for some personal communications.

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