# Generalized free energy and thermodynamic phases of black holes in the
gauged Kaluza-Klein theory

Tran N. Hung<EMAIL_ADDRESS>Phenikaa Institute for
Advanced Study and Faculty of Fundamental Sciences, Phenikaa University, Yen
Nghia, Ha Dong, Hanoi 12116, Vietnam  Cao H. Nam  nam.caohoang@phenikaa-
uni.edu.vn Phenikaa Institute for Advanced Study and Faculty of Fundamental
Sciences, Phenikaa University, Yen Nghia, Ha Dong, Hanoi 12116, Vietnam

###### Abstract

In the context of the generalized (off-shell) free energy, we explore the
phase emergence and corresponding phase transitions of charged dilaton AdS
black holes in the gauged Kaluza-Klein (KK) theory where the KK vector field
is gauged such that the fermionic fields are charged under the
U(1)${}_{\text{KK}}$ gauge group. The black hole solutions are asymptotic to
the AdSD geometry and can be realized as the dimensional reduction of the
gauged supergravities on the compact internal manifolds, leading to the
restriction as $4\leq D\leq 7$. By studying the behavior of the generalized
free energy under the change of the ensemble temperature, we determine the
thermodynamic phases and the corresponding phase transitions of black holes.
This is confirmed by investigating the heat capacity at the constant pressure
and the on-shell free energy. In the canonical ensemble, the thermodynamics of
black holes can be classified into three different classes as follows: (i)
$D=4$, (ii) $D=5$, and (iii) $D=6,7$. Whereas, in the grand canonical
ensemble, the thermodynamics of black holes is independent of the number of
spacetime dimensions and the pressure, but depends on the chemical potential
$\Phi$. The thermodynamic behavior of black holes can be classified into three
different classes as follows: (i) $\Phi<1$, (ii) $\Phi>1$, and (iii) $\Phi=1$.

##  I Introduction

The Kaluza-Klein (KK) theory provides a beautiful and elegant way to unify the
gravitational interaction with the non-the gravitational interactions mediated
by the vector fields in a geometric framework Witten1981 ; Bailin1987 ;
Overduin1997 . In this context, the gauge symmetry of the non-gravitational
interactions is the isometry of the compact internal space where the number of
the generators of the gauge symmetry is equal to the number of the Killing
vectors of the internal manifold. The vector fields mediating the non-
gravitational interactions describe the excitations of the off-diagonal
components of the metric of the higher-dimensional spacetime. In the
phenomenological aspect, the KK gauge bosons can lead to the emergence of new
neutral gauge bosons beyond the Standard Model which are being searched for at
the colliders Nam2019 ; Nam2020  as well as the candidate for the dark matter
Nam2023 . Whereas, the radion (or dilaton) fields can be responsible for the
inflaton Cline2000 ; Kolb2003 ; Sundrum2010 ; Trudeau2012 ; Fukazawa2013 ;
Nam2021 . In addition, the KK reduction on a circle can produce the black hole
solutions without the unphysical curvature singularity Bah2021a ; Bah2021b
and the microstates for the Bekenstein-Hawking entropy of the observed black
holes Nam2023b .

The KK theory which is the dimensional reduction of the pure gravity on the
circle is a particularly interesting class of the Einstein-Maxwell-dilaton
(EMD) theories whose black hole solutions can be embedded in string
theory/supergravity Horo1991 ; Dgar1991 ; Behr1999 ; Duff199 ; Lu1999 ;
Cvetic1999 . Interestingly, the KK vector field can be gauged such that the
fermion fields carry the U(1)${}_{\text{KK}}$ charge, resulting in the
presence of the Killing spinor equations in the KK theory Pope2011 . This
gauge process allows us to embed the KK theory in gauged supergravities in
some dimensions. In addition, gauging the KK vector field generates a scalar
potential for the dilaton field which is restricted to exponential terms by
the projected integrability condition of the Killing spinor equations. The
scalar potential which is an extension of the cosmological constant plays a
pivotal role in changing the asymptotic behavior of the black hole solutions.
The black hole solutions in the gauged KK theory are asymptotic to the anti-de
Sitter (AdS) geometry where the curvature radius of asymptotic AdS space is
determined in terms of the KK gauge coupling Liu2012 .

It was shown that new thermodynamic aspects and phase transitions of black
holes emerge as the cosmological constant is realized as the thermodynamic
pressure leading to the extended phase space MTeo2017 . In this context, the
ADM mass of black holes is identified as the enthalpy rather than the internal
energy. New thermodynamic phenomena of black holes emerge in the extended
phase space such as the $P-V$ criticality Mann2012 ; Cai2013 ; Hendi2013 ;
Fernando2016 ; Hendi2017 ; CNam2019 , the reentrant phase transition Mann2013
, the triple point Altamirano2014 ; WeiLiu2014 , the black hole heat engine
Johson2014 ; Bhamidipati2017 ; YLi2018 ; QLan2018 , and the Joule-Thomson
expansion Aydner2017 ; Aydner2018 ; XMo2018 ; CNam2020 . Inspired by the
AdS/CFT correspondence Maldacena1998 , the thermodynamics and phase
transitions of AdS black holes have been extensively investigated to
understand the universal properties of the strongly interacting systems on the
AdS boundary Ahmed2023 ; WCong2023 .

Recently, the generalized free energy extended from the on-shell free energy
has been proposed for study the thermodynamic phases and phase transitions of
black holes JWang2020 ; KZhang2020 ; KZhang2021 ; WeiLiu2022 ; LiWang2022 ;
CLiu2023 ; MSAli2023 ; LiWang2023 ; Ksafir2023 ; WangXu2024 . In this
formalism, the generalized free energy depends on the event horizon radius
which is treated as an order parameter which is analogous to the density
playing the role of the older parameter in the liquid-gas system. In this way,
the ensemble at a specific temperature comprises all black hole states with
various values of event horizon radius from zero to infinity. Note that, the
black hole state with the zero event horizon radius is the thermal AdS space.
By investigating the behavior of the generalized free energy in the event
horizon radius when changing the ensemble temperature, we can determine the
phase emergence of the black holes, their thermodynamic stability, and the
corresponding phase transitions. The extremal points of the generalized free
energy represent the on-shell black hole states which are the solutions of
equations of motion. The local maximum points correspond to the
thermodynamically unstable black holes which have a negative heat capacity.
The local and global minimum points represent black holes corresponding to the
local and global stationary states with a positive heat capacity,
respectively. The thermal fluctuations in the event horizon radius can be
generated in a stochastic way in which the force driving the stochastic
process is encoded in the generalized free energy. As a result, it is
necessary to include the off-shell black hole states which are not the
solutions of equations of motion. The presence of the fluctuating black holes
hence leads to the phase transitions.

In the present paper, we explore the emergence of the thermodynamic phases and
the corresponding phase transitions of charged dilaton AdS black hole in the
gauged KK theory by using the formalism of the generalized free energy. In the
canonical ensemble where the black hole charge is kept fixed, the
thermodynamics and phase transitions of black holes depend on the number of
spacetime dimensions. In this situation, we classify the thermodynamic
behavior of black holes into three different classes corresponding to $D=4$,
$D=5$, and $D=6,7$. In the grand canonical ensemble where the chemical
potential is kept fixed, we show that the thermodynamics and phase transitions
of black holes depend only on the chemical potential $\Phi$ and are classified
in the classes as follows $\Phi<1$, $\Phi>1$, and $\Phi=1$. In addition, we
study the behavior of the heat capacity and the on-shell free energy to
confirm the thermodynamic stability and the phase transitions that are
observed from the behavior of the generalized free energy as changing the
ensemble temperature.

This paper is organized as follows. In Sec. II, we review charged dilaton AdS
black holes in the gauged KK theory, and then we represent their thermodynamic
rules in the extended phase space. In Sec. III, we study the thermodynamics of
black holes in the canonical ensemble where we classify the emergence of the
thermodynamic phases and the corresponding phase transitions based on the
number of spacetime dimensions. In Sec. IV, we represent the thermodynamics of
black holes in the grand canonical ensemble where the classification for the
phase emergence and phase transitions is based on the chemical potential.
Finally, we make conclusions in Sec. V.

##  II Charged dilaton AdS black holes in the gauged KK theory

We consider the $D$-dimensional effective field theory derived from the KK
reduction of $(D+1)$-dimensional pure Einstein gravity on a circle $S^{1}$.
The corresponding action is given by

$\displaystyle S_{D}=\int
d^{D}x\sqrt{-g}\left[\mathcal{R}-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-\frac{1}{4}e^{a\phi}F_{\mu\nu}F^{\mu\nu}-V(\phi)\right],$
(1)

where $g$ denotes the determinant of the $D$-dimensional metric, $\mathcal{R}$
is the Ricci scalar, $\phi$ is the dilaton,
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the field strength
tensor of the KK gauge field $A_{\mu}$, $a=\sqrt{2(D-1)/(D-2)}$ is the
coupling between the dilaton and the KK gauge field, and $V$ is the potential
of the dilaton. In the usual KK reduction, the potential of the dilaton is
zero, i.e. $V(\phi)=0$. The equations of motion for the metric $g_{\mu\nu}$,
the KK gauge field $A_{\mu}$, and the dilaton $\phi$ which are derived from
the variation of the action (1) read

$\displaystyle R_{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\partial_{\mu}\phi\partial_{\nu}\phi-\frac{V(\phi)}{D-2}g_{\mu\nu}-\frac{e^{a\phi}}{2}\left(F_{\mu\rho}{F_{\nu}}^{\rho}-\frac{F_{\rho\lambda}F^{\rho\lambda}}{2(D-2)g_{\mu\nu}}\right),$
(2) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\partial_{\mu}\left(\sqrt{-g}e^{a\phi}F^{\mu\nu}\right),$ (3)
$\displaystyle\nabla^{2}\phi$ $\displaystyle=$ $\displaystyle
V^{\prime}(\phi)-\frac{e^{a\phi}}{4}F_{\mu\nu}F^{\mu\nu}.$ (4)

It is possible to gauge the KK gauge field $A_{\mu}$ such that the fermionic
fields are charged under the U(1)${}_{\text{KK}}$ gauge group. This leads to a
pseudo-supersymmetrization of the $D$-dimensional KK theory where the pseudo-
gravitino and pseudo-dilatino are charged under U(1)${}_{\text{KK}}$.
Interestingly, gauging the KK gauge field allows us to generate a potential
for the dilaton, in analogy to the gauged supergravity Liu2011 . In the
bosonic background given by (1), the pseudo-supersymmetric variation of the
fermionic partners is zero, leading to the following Killing spinor equations
Liu2012

$\displaystyle
D_{\mu}\eta+\frac{W(\phi)}{2\sqrt{2}(D-2)}\Gamma_{\mu}\eta+\frac{ie^{a\phi/2}}{8(D-2)}\left[\Gamma_{\mu}\Gamma^{\nu\rho}-2(D-2)\delta^{\nu}_{\mu}\Gamma^{\rho}\right]F_{\nu\rho}\eta\equiv\hat{D}_{\mu}\eta$
$\displaystyle=$ $\displaystyle 0,$ (5)
$\displaystyle\left[\Gamma^{\mu}\partial_{\mu}\phi+\frac{i}{4}ae^{a\phi/2}\Gamma^{\mu\nu}F_{\mu\nu}-\sqrt{2}W^{\prime}(\phi)\right]\eta$
$\displaystyle=$ $\displaystyle 0,$ (6)

where $D_{\mu}$ is the covariant derivative given by

$\displaystyle
D_{\mu}=\partial_{\mu}+\frac{i}{16}{\omega_{\mu}}^{\alpha\beta}[\Gamma_{\alpha},\Gamma_{\beta}]-\frac{ig(D-3)}{4}A_{\mu},$
(7)

with ${\omega_{\mu}}^{\alpha\beta}$ being the spin connection and $g$ to be
the KK gauge coupling, the matrices $\Gamma_{\mu}$ are related to the
$D$-dimensional gamma matrices $\Gamma_{\alpha}$ as
$\Gamma_{\mu}={e^{\alpha}}_{\mu}\Gamma_{\alpha}$ with ${e^{\alpha}}_{\mu}$ are
the vielbeins, $\Gamma_{\mu\nu}\equiv\frac{i}{4}[\Gamma_{\mu},\Gamma_{\nu}]$,
and $\eta$ is the Killing spinor. Eqs. (5) and (6) corresponds to the pseudo-
supersymmetric transformation rules of the pseudo-gravitino and the pseudo-
dilatino, respectively. Whereas, the pseudo-supersymmetric variation of the
vielbeins, the KK gauge field, and the dilaton automatically vanish in the
bosonic background. Note that, the second term on the left-hand side of Eq.
(5) is inspired by supergravity with $W(\phi)$ referring to the
superpotential. Accordingly, the dilaton $V(\phi)$ potential can be expressed
in terms of the superpotential $W(\phi)$ as follows

$\displaystyle V(\phi)=W^{\prime}(\phi)^{2}-\frac{D-1}{2(D-2)}W(\phi)^{2}.$
(8)

The superpotential $W(\phi)$ is fixed by the projected integrability condition
by acting $\Gamma^{\mu}[\hat{D}_{\mu},\hat{D}_{\mu}]$ on Eq. (5) and
$\Gamma^{\mu}\nabla_{\mu}$ on Eq. (6). The projected integrability condition
gives equations that consist of the terms related to equations of motion
(which automatically vanish) and the unrelated terms that have to vanish.
Then, we determine the superpotential as follows

$\displaystyle
W(\phi)=\frac{(D-3)g}{\sqrt{2}}\left(e^{-a_{1}\phi}+\frac{a_{1}}{a_{2}}e^{a_{2}\phi}\right),$
(9)

where $a_{1}=(D-1)/\sqrt{2(D-1)(D-2)}$ and $a_{2}=(D-3)/\sqrt{2(D-1)(D-2)}$.
Using Eq. (8), we find the dilaton potential as

$\displaystyle
V(\phi)=-(D-1)g^{2}\left[(D-3)e^{-\frac{2a_{1}}{D-1}\phi}+e^{2a_{2}\phi}\right].$
(10)

This potential possesses a maximal point at $\phi=0$ corresponding to a
negative cosmological constant $\Lambda=-(D-1)(D-2)g^{2}/2$.

By solving Eqs. (2), (3), and (4) with the dilaton potential (10), we can find
a charged dilaton AdS black hole solution that is spherically symmetric and
static in the gauged KK theory as follows Liu2012

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-H(r)^{-\frac{D-3}{D-2}}f(r)dt^{2}+H(r)^{\frac{1}{D-2}}\left(\frac{dr^{2}}{f(r)}+r^{2}d\Omega_{D-2}^{2}\right),$
$\displaystyle A_{\mu}$ $\displaystyle=$
$\displaystyle\sqrt{1+\frac{m}{q}}\frac{1}{H(r)}\delta^{t}_{\mu}\equiv\phi(r)\delta^{t}_{\mu},$
(11) $\displaystyle\phi$ $\displaystyle=$ $\displaystyle\frac{a}{2}\log H(r),$

where

$\displaystyle H(r)$ $\displaystyle=$ $\displaystyle 1+\frac{q}{r^{D-3}},$
(12) $\displaystyle f(r)$ $\displaystyle=$ $\displaystyle
1-\frac{m}{r^{D-3}}+g^{2}r^{2}H(r),$ (13)

$m$ and $q$ are the integral constants related to the ADM mass and charge of
black holes, and $d\Omega_{D-2}^{2}$ is the line element of unit
$(D-2)$-sphere. In the limit of $g\rightarrow 0$ corresponding to the usual KK
theory, the black hole solution given in (11) becomes the non-rotating black
hole solution that was found in Gibbons1987 . In addition, one can recover the
Schwarzschild solution when the gauge coupling $g$ goes to zero and the black
hole charge vanishes.

The black hole solution given by Eq. (11) is asymptotic to the AdSD geometry.
Interestingly, this black hole solution has a string/M theory origin as the
dimensional reduction of string/M theory on the compact internal manifolds
Tran1999 . The cases of $D=4$ and $D=7$ can be derived from the
compactification of eleven-dimensional supergravity on $S^{7}$ and $S^{4}$,
respectively. The $D=5$ case can come from the dimensional reduction of IIB
supergravity on $S^{5}$. Whereas, the $D=6$ case can embed in the $F(4)$
gauged supergravity Romans1986 . Motivated by the string/M theory origin, in
this work, we are only interested in studying the emergence of the
thermodynamic phases and corresponding phase transitions of black holes for
$4\leq D\leq 7$.

The horizon radius $r_{h}$ of black holes is given by the root of the
following equation, $f(r_{h})=0$, which leads to

$\displaystyle\left(1+g^{2}r^{2}_{h}\right)r^{D-3}_{h}+g^{2}qr^{2}_{h}-m=0.$
(14)

This equation has a unique solution corresponding to the event horizon, as a
result of the presence of the non-trivial scalar field Cai2021 . The black
hole temperature or the Hawking temperature is computed via the surface
gravity $\kappa=f^{\prime}(r_{h})/(2\sqrt{H(r_{h})})$ at the event horizon as
follows

$\displaystyle
T=\frac{\kappa}{2\pi}=\frac{f^{\prime}(r_{h})}{4\pi\sqrt{H(r_{h})}}.$ (15)

The Bekenstein-Hawking entropy is determined by the area of the event horizon
as follows

$\displaystyle
S=\frac{\mathcal{A}}{4}=\frac{\omega_{D-2}}{4}\sqrt{H(r_{h})}r_{h}^{D-2},$
(16)

where $\omega_{D-2}=2\pi^{(D-1)/2}/\Gamma(\frac{D-1}{2})$ is the surface area
of the unit $(D-2)$-sphere with $\Gamma(z)$ to be the Gamma function.

The ADM mass $M$ and the charge $Q$ of black holes are calculated by using the
conserved charges, leading to Lu2012

$\displaystyle M$ $\displaystyle=$
$\displaystyle\frac{\omega_{D-2}}{16\pi}\left[(D-2)m+(D-3)q\right],$ (17)
$\displaystyle Q$ $\displaystyle=$
$\displaystyle\frac{(D-3)\omega_{D-2}}{16\pi}\sqrt{q(m+q)}.$ (18)

The chemical potential $\Phi$ which is the conjugate quantity of the black
hole charge reads

$\displaystyle\Phi$ $\displaystyle=$ $\displaystyle\phi(\infty)-\phi(r_{h})$
(19) $\displaystyle=$
$\displaystyle\left[1-\frac{1}{H(r_{h})}\right]\sqrt{1+\frac{m}{q}}.$

Additionally, we observe from Eqs. (12) and (13) that the KK coupling constant
$g$ behaves as a negative cosmological constant corresponding to the AdS
curvature radius $L=1/g$. In this way, we define the cosmological constant or
the KK coupling constant as the thermodynamic pressure in the extended phase
space as follows

$\displaystyle P$ $\displaystyle=$ $\displaystyle\frac{(D-1)(D-2)}{16\pi
L^{2}}$ (20) $\displaystyle=$ $\displaystyle\frac{(D-1)(D-2)}{16\pi}g^{2}.$

We can check that the thermodynamic quantities satisfy the first law of the
black hole thermodynamic in the extended phase space as follows

$\displaystyle dM=TdS+\Phi dQ+\mathcal{V}dP,$ (21)

where $\mathcal{V}$ is the thermodynamic volume which is the conjugate
variable of the pressure $P$ and is determined by

$\displaystyle\mathcal{V}$ $\displaystyle=$ $\displaystyle\left(\frac{\partial
M}{\partial P}\right)_{S,Q}$ (22) $\displaystyle=$
$\displaystyle\frac{\omega_{D-2}}{D-3}\left[1+\frac{D-1}{2(D-2)}\frac{q}{r^{D-3}_{h}}\right]r^{D-1}_{h}.$

In addition, we can check that the thermodynamic quantities satisfy the
following Smarr relation

$\displaystyle M=\frac{D-2}{D-3}TS+\Phi Q+\frac{2}{D-3}\mathcal{V}P.$ (23)

In the following, we shall investigate the phase structure of the black hole,
the stability of the thermodynamic phases, and the corresponding phase
transitions, based on the free energy landscape in both the canonical ensemble
(with the black hole charge kept fixed) and the grand canonical ensemble (with
the chemical potential kept fixed).

##  III The thermodynamic phases in the canonical ensemble

In this section, we study the thermodynamic phases and phase transitions of
black holes in the canonical ensemble with the black hole charge $Q$ kept
fixed. The generalized free energy in this ensemble is given by

$\displaystyle F=M-\tau S=F(r_{h},\tau),$ (24)

where $\tau$ is the ensemble temperature. The generalized free energy $F$ is
an extension of the on-shell free energy with the black hole temperature
replaced by the ensemble temperature. The generalized free energy $F$ is a
function of $r_{h}$ and $\tau$, which is consistent with the fact that the
ensemble comprises all black hole states with the event horizon radius taking
the value from zero to infinity. In this way, the event horizon radius is
realized as an order parameter that characterizes the thermodynamic phases of
black holes. Hence, the value of the event horizon radius can help us to
distinguish the different thermodynamic phases of black holes. The canonical
ensemble includes both on-shell black hole states (which satisfy the equations
of motion) and off-shell (fluctuating) black hole states (which are not the
solutions of the equations of motion) with the charge kept fixed.

In order to study the free energy landscape of black holes in the canonical
ensemble, let us express $m$ and $q$ in terms of the event horizon radius
$r_{h}$, the black hole charge $Q$, and the KK coupling constant $g$. From
Eqs. (14) and (18), we find

$\displaystyle q$ $\displaystyle=$
$\displaystyle\frac{r^{D-3}_{h}}{2}\left(\sqrt{1+\frac{4\hat{Q}^{2}r^{2(3-D)}_{h}}{1+g^{2}r^{2}_{h}}}-1\right),$
(25) $\displaystyle m$ $\displaystyle=$
$\displaystyle\left(1+g^{2}r^{2}_{h}\right)r_{h}^{D-3}+g^{2}r_{h}^{2}q,$ (26)

where $\hat{Q}\equiv 16\pi Q/[(D-3)\omega_{D-2}]$. By substituting these
expressions of $q$ and $m$ into Eq. (17), the ADM mass of black holes becomes

$\displaystyle
M=\frac{\omega_{D-2}}{32\pi}r^{D-3}_{h}\left\\{D-1+(D-2)g^{2}r^{2}_{h}+\left[D-3+(D-2)g^{2}r^{2}_{h}\right]\sqrt{1+\frac{4\hat{Q}^{2}r^{2(3-D)}_{h}}{1+g^{2}r^{2}_{h}}}\right\\}.$
(27)

Analogously, the Bekenstein-Hawking entropy given in Eq. (16) becomes

$\displaystyle
S=\frac{\omega_{D-2}}{4\sqrt{2}}\left[1+\sqrt{1+\frac{4\hat{Q}^{2}r^{2(3-D)}_{h}}{1+g^{2}r^{2}_{h}}}\right]^{\frac{1}{2}}r_{h}^{D-2}.$
(28)

From Eqs. (27) and (28), we can determine the generalized free energy $F$ in
terms of the event horizon radius $r_{h}$, the thermodynamic pressure $P$, and
the black hole charge $Q$.

We can find the value of the free energy of the thermal AdS space which is the
pure radiation without the black hole by taking a limit of the zero event
horizon radius, given by

$\displaystyle\lim_{r_{h}\rightarrow 0}F=Q\neq 0.$ (29)

This value is essential to determine the Hawking-Page (HP) phase transition
between the thermal AdS space and the black hole Page1983 .

The extremal points of the generalized free energy $F$ are determined by

$\displaystyle\frac{\partial F}{\partial r_{h}}=0\ \ \rightarrow\ \
\tau=\left(\frac{\partial M}{\partial S}\right)_{Q,P}.$ (30)

From the first law given in Eq. (21), we see that the ensemble temperature at
the extremal points of $F$ is equal to the Hawking temperature $T$ which is
rewritten by replacing the expressions of $q$ and $m$ given in Eqs. (25) and
(26) into Eq. (15) as follows

$\displaystyle T=\frac{1}{2\sqrt{2}\pi
r_{h}}\left[D-3+g^{2}r^{2}_{h}\left(D-2+\sqrt{1+\frac{4\hat{Q}^{2}r^{2(3-D)}_{h}}{1+g^{2}r^{2}_{h}}}\right)\right]\left[1+\sqrt{1+\frac{4\hat{Q}^{2}r^{2(3-D)}_{h}}{1+g^{2}r^{2}_{h}}}\right]^{-\frac{1}{2}}.$
(31)

In this sense, the black hole solutions which satisfy the equations of motion
given by Eqs. (2), (3), and (4) would correspond to the extremal points of the
generalized free energy.

We depict the behavior of the Hawking temperature in terms of the event
horizon radius for various values of the spacetime dimension and the pressure
in Fig. 1, with $Q$ kept fixed to be a unit (in the following discussions) for
simplicity.

Figure 1: The behavior of the Hawking temperature in terms of the event
horizon radius for the spacetime dimension $D=4$, $5$, $6$, and $7$ with
various values of the pressure.

The behavior of the Hawking temperature $T$ depends strongly on the spacetime
dimension and the pressure, which can be classified into the following three
classes. The first class corresponds to $D=4$ where the isobaric curves
$T-r_{h}$ have only one extremum independent of the value of the pressure. The
second class corresponds to $D=5$ where the Hawking temperature approaches a
constant in the limit of $r_{h}\rightarrow 0$, given by

$\displaystyle\lim_{r_{h}\rightarrow 0}T=\frac{1+8\pi
gQ/\omega_{3}}{4\pi\sqrt{2\pi Q/\omega_{3}}}.$ (32)

In this way, black holes with the event horizon radius which is enough small
have nearly the same temperature with the pressure kept fixed. In addition,
when the pressure is above a critical value $P_{c}$, the isobaric curves
$T-r_{h}$ do not have any extremum. On the contrary, when the pressure is
lower than $P_{c}$, the isobaric curves exhibit some extrema, which implies
the different thermodynamic phases of black holes and the phase transitions
between them as seen later. The third class corresponds to $D=6$ and $D=7$.
The behavior of the isobaric curves in this class is analogous to that of the
first class when the pressure is higher than a critical value. When the
pressure is below the critical value, the isobaric curves exhibit the addition
of two extrema in analogy to the second class.

The critical point of the isobaric curves is determined by

$\displaystyle\frac{\partial T}{\partial
r_{h}}=0,\quad\frac{\partial^{2}T}{\partial r_{h}^{2}}=0.$ (33)

By solving numerically, we find the critical values for the event horizon
radius, the pressure, and the temperature, given in Table 1.

$D$ | $r_{c}$ | $P_{c}$ | $T_{c}$
---|---|---|---
5 | 1.091 | 0.037 | 0.170
6 | 1.199 | 0.116 | 0.329
7 | 1.120 | 0.247 | 0.498
Table 1: Table 1. The numerical values for the critical event horizon radius
$r_{c}$, the critical pressure $P_{c}$, and the critical temperature $T_{c}$
with $5\leq D\leq 7$.

The $P-V$ criticality of black holes has been studied in HaoWei2019 .

###  III.1 The first class: $D=4$

We study the landscape of the generalized free energy $F$ as the function of
the event horizon radius for various values of the ensemble temperature $\tau$
in the case of $D=4$ in Fig. 2.

---
Figure 2: The generalized free energy $F$ is plotted in the event horizon
radius $r_{h}$ with the change of the ensemble temperature $\tau$ for $D=4$
and $P=0.1$. The blue point refers to the inflection point. The orange and red
points represent the extremum.

When the ensemble temperature is lower than a minimal value which is $\approx
0.208$ for $P=0.1$, the generalized free energy $F$ has only one global
minimum with $F=1$ located at the origin $r_{h}=0$. This implies that the
system corresponds to the thermal AdS space. When the ensemble temperature is
above the minimal value, the generalized free energy $F$ exhibits one minimum
and a local maximum corresponding to the thermodynamically unstable and stable
black holes, respectively. We can also point to the thermodynamic stability of
these black hole phases based on investigating the behavior of the heat
capacity at the constant pressure defined by

$\displaystyle C_{P}$ $\displaystyle\equiv$ $\displaystyle
T\left(\frac{\partial S}{\partial T}\right)_{P}=T\left(\frac{\partial
S}{\partial r_{h}}\right)_{P}\left(\frac{\partial T}{\partial
r_{h}}\right)_{P}^{-1}.$ (34)

As depicted in Fig. 3

---
Figure 3: The heat capacity at the constant pressure in terms of the event
horizon radius at $D=4$.

we observe that the large black holes corresponding to the minimum of the
generalized free energy have a positive heat capacity, hence they are
thermodynamically stable. Whereas, the small black holes the local maximum of
the generalized free energy have a negative heat capacity, implying that they
are thermodynamically unstable. This means there is a second-order phase
transition from the small black holes to the large black holes, corresponding
to the divergence of $C_{P}$. We also indicate this phase transition from the
multivalued behavior of the on-sell free energy, as depicted in Fig. 4.

---
Figure 4: The on-shell free energy as a function of the black hole temperature
for $D=4$ and $P=0.1$.

The upper branch represents the small black holes with the free energy that is
higher than that of the large black holes represented by the lower branch. The
intersection of these two branches determines the phase transition between the
small and large black holes.

In addition, from Figs. 2 and 4, we observe that when the temperature is above
a value $\approx 0.221$ (for $P=0.1$), the free energy of the large black
holes is lower than that of the thermal AdS space. On the other hand, the
large black holes correspond to the global minimum of the free energy. As a
result, the large black holes are the dominant contribution to the partition
function, hence they are the thermodynamically preferred states. This
indicates the HP phase transition from the thermal radiation state to the
large black holes. The temperature of the HP phase transition is easily
determined by the intersection between the branch of the large black holes and
the horizontal line $F=1$, in Fig. 4.

###  III.2 The second class: $D=5$

Unlike the case of $D=4$, the behavior of the generalized free energy $F$ for
$D=5$ depends on not only the ensemble temperature $\tau$ but also the
thermodynamic pressure $P$ generated by gauging the KK gauge field. The phase
structure and the phase transitions in $D=5$ thus are richer than those in
$D=4$.

For the pressure $P>P_{c}\approx 0.037$, we depict the behavior of the
generalized free energy $F$ in terms of the event horizon radius for various
values of the ensemble temperature $\tau$ in Fig. 5.

Figure 5: The generalized free energy $F$ versus the event horizon radius
$r_{h}$ for $D=5$ and $P=0.038$.

When $\tau$ is lower than a minimum value $\tau_{\text{m}}$ which is about
$0.173$ for $P=0.038$, the generalized free energy possesses only a minimum
with $F=1$ located at $r_{h}=0$, corresponding to the pure radiation state or
the thermal AdS space. Increasing the ensemble temperature such that
$\tau\gtrsim\tau_{\text{m}}$ leads to the occurrence of a global minimum
(represented by the blue dot in Fig. 5) in the curve of the generalized free
energy, corresponding to the presence of the on-shell black hole states.
Because the free energy of black holes is lower than that of the thermal AdS
space and they have a positive heat capacity as seen in the top-left panel of
Fig. 7, black holes are the thermodynamically preferred states.

For the pressure $P<P_{c}$, we plot the generalized free energy $F$ as a
function of the event horizon radius for different values of the ensemble
temperature $\tau$ in Fig. 6.

Figure 6: The behavior of the generalized free energy $F$ in the event horizon
radius $r_{h}$ for $D=5$ and $P=0.036$. $F$ has the shapes of a double well
when $\tau_{1}<\tau<\tau_{2}$ with $\tau_{1}\approx 0.1689$ and
$\tau_{2}\approx 0.1691$, and a single well in other cases. The red points are
the local extremal points that represent the on-shell states. The blue points
represent the inflection point. Two local minima have the same value when
$\tau=\tau_{0}\approx 0.16899$.

In analogy to the case of $P>P_{c}$, the thermal AdS space exits when the
ensemble temperature is lower than $\tau_{\text{m}}$ which is about $0.1683$
for $P=0.036$. However, some extremal points of the generalized free energy
$F$ appear for $\tau>\tau_{\text{m}}$ which corresponds to the on-shell black
hole states.

We observe that there are two critical values of the ensemble temperature
which are $\tau_{1}\approx 0.1689$ and $\tau_{2}\approx 0.1691$ for $P=0.036$.
When $\tau_{1}<\tau<\tau_{2}$, the generalized free energy $F$ possesses two
local minima and one local maximum corresponding to the on-shell black hole
states. The two local minima represent the small and large black holes which
are thermodynamically stable because their heat capacity is positive as seen
in the bottom panel of Fig. 7.

Figure 7: The behavior of the heat capacity at the constant pressure $C_{P}$
in terms of the event horizon radius for three cases: $P>P_{c}$ (top-left),
$P=P_{c}$ (top-right), and $P<P_{c}$ (bottom).

Which one prefers thermodynamically depends on its relative depth in the free
energy landscape. We can find a value of the ensemble temperature $\tau_{0}$
where the free energy of the small and large black holes has the same value.
For $\tau<\tau_{0}$, the free energy of the small black holes is lower than
that of the large black holes, indicating that the small black holes are the
thermodynamically preferred states. On the contrary, for $\tau>\tau_{0}$, the
large black holes are the global stable state and hence they are
thermodynamically preferred. When the ensemble temperature is above
$\tau_{2}$, the generalized free energy only manifests an extremum which is a
global minimum. This means that there is a unique thermodynamic phase and
hence there is no phase transition.

The local maximum of the generalized free energy $F$ represents the
intermediate black holes that have a negative heat capacity (as seen in the
bottom panel of Fig. 7) which implies the thermodynamic instability. As a
result, the intermediate black holes would decay into the small or large black
holes, which leads to the second-order phase transitions. We find these
second-order phase transitions by observing the divergence of the heat
capacity. In the bottom panel of Fig. 7, we observe that there are two
divergences in the behavior of the heat capacity $C_{P}$ which divide the
domain of black holes into three subdomains: the small black holes, the
intermediate black holes, and the large black holes. The divergence with the
small $r_{h}$ indicates the phase transition between the small black holes and
the intermediate black holes, whereas the one with the large $r_{h}$ exhibits
the phase transition between the large black holes and the intermediate black
holes. For $\tau<\tau_{0}$, because the small black holes are preferred
thermodynamically, the rate at which the intermediate black holes decay into
the small black holes would be greater than the rate at which the intermediate
black holes transform into the large black holes. However, this happens
oppositely when $\tau>\tau_{0}$.

We can observe more phase transitions of black holes from the behavior of the
on-shell free energy as depicted in Fig. 8.

Figure 8: The on-shell free energy to the temperature for five-dimensional
black holes in three cases: $P<P_{c}$ (red), $P=P_{c}$ (black), and $P>P_{c}$
(green). The blue dot is the critical point of the Van der Waals-like phase
transition.

The green, black, and red curves represent the on-shell free energy of black
holes for $P>P_{c}$, $P=P_{c}$, and $P<P_{c}$, respectively. The dashed red
line corresponds to the intermediate black holes, while the upper and lower
branches of the solid red curve show the small and large black holes,
respectively. The behavior of the on-shell free energy only reveals the phase
transitions when $P<P_{c}$. We can observe the second-order phase transitions
between the small(large) black holes and the intermediate black holes from the
multivalued behavior of the on-sell free energy. Furthermore, the on-shell
free energy exhibits the behavior of the _swallowtail_ , indicating the first-
order phase transition between the small black holes and the large black holes
because the difference of the entropy in the two phases is non-zero. This
phase transition is analogous to the Van der Waals phase transition between
the liquid and gas.

###  III.3 The third class: $D=6$ and $7$

The generalized free energy $F$ always has a local minimum and a local maximum
which corresponds to the thermodynamically stable and unstable black holes,
respectively, independent of the value of the pressure.

Figure 9: The behavior of the generalized free energy $F$ in the event horizon
radius $r_{h}$ for $D=6$ with $P=0.1$. $F$ has a double well when
$\tau_{1}<\tau<\tau_{2}$ with $\tau_{1}\approx 0.3070$ and $\tau_{2}\approx
0.3113$, and a single well in other cases. The red points are the local
extremal points that represent the on-shell black hole states. The blue points
represent the inflection points. Two local minima have the same value when
$\tau=\tau_{0}\approx 0.3083$.

This particular point is shown by the small panel given in Fig. 9 and it can
also be observed from the behavior of both heat capacity and on-shell free
energy.

Figure 10: The heat capacity $C_{P}$ at the constant pressure as a function of
the event horizon radius for $D=6$ in three cases: $P<P_{c}$ (top-left),
$P=P_{c}$ (top-right), and $P>P_{c}$ (bottom). The small panels are the zoom
of the dashed rectangles.

In Fig. 9, we find that black holes are thermodynamically unstable at the
regime of the relatively small $r_{h}$ because they correspond to the local
maximum of the generalized free energy. This is also observed in Fig. 10 where
these black holes have a negative heat capacity. The thermodynamically
unstable phase is separated from the thermodynamically stable phase by a
divergent line which indicates a second-order phase transition between these
phases.

Figure 11: The behavior of the on-shell free energy in terms of the black hole
temperature for $D=6$ in three cases: $P<P_{c}$ (red), $P=P_{c}$ (black), and
$P>P_{c}$ (green). The dashed curves refer to the unstable phases which have a
negative heat capacity. The blue dot is the critical point of the Van der
Waals-like phase transition.

In Figs. 11, we also observe the presence of these two thermodynamic phases
and the corresponding phase transition with an arbitrary pressure. The
thermodynamically unstable small black holes are represented by the dashed
lines with the on-shell free energy close to one. This branch is connected to
the branch of the thermodynamically stable black holes represented by the
solid curves with the lower on-shell free energy. This connection leads to the
multivalued behavior of the on-sell free energy indicating the second-order
phase transition.

When the pressure is lower than the critical value $P_{c}$, there are two
additional second-order phase transitions between the thermodynamically stable
small(large) black holes and the thermodynamically unstable intermediate black
hole. In addition, there is a first-order phase transition (or the Van der
Waals-like phase transition) between the small black holes and large black
holes. This reveals the presence of two divergent lines in the behavior of the
heat capacity which separate the negative regime from the positive one. The
two thermodynamic phases corresponding to the intermediate and large black
holes are exhibited in the behavior of the generalized free energy $F$ with
the occurrence of additional local extremal points which are the on-shell
black hole states when $P<P_{c}$. These thermodynamic phases and the
corresponding phase transitions are analogous to those in the second class
$D=5$.

##  IV The thermodynamic phases in the grand canonical ensemble

We study the thermodynamic phases and the phase transitions of black holes in
the grand canonical ensemble where the chemical potential $\Phi$ is kept
fixed. From Eqs. (19) and (14), we express the parameters $q$ and $m$ in terms
of $\Phi$ and $r_{h}$ as follows

$\displaystyle q$ $\displaystyle=$
$\displaystyle\frac{r_{h}^{D-3}\Phi^{2}}{1+g^{2}r_{h}^{2}-\Phi^{2}},$ (35)
$\displaystyle m$ $\displaystyle=$
$\displaystyle\frac{r_{h}^{D-3}\left[\left(1+g^{2}r_{h}^{2}\right)^{2}-\Phi^{2}\right]}{1+g^{2}r_{h}^{2}-\Phi^{2}}.$
(36)

Using these equations, we write the generalized Gibbs free energy in the grand
canonical ensemble as follows

$\displaystyle\Omega$ $\displaystyle=$ $\displaystyle M-\tau S-\Phi Q$ (37)
$\displaystyle=$
$\displaystyle\frac{\omega_{D-2}}{16\pi}r^{D-3}_{h}\left[\frac{\Phi^{2}\left(\Phi^{2}-1\right)}{1+g^{2}r^{2}_{h}-\Phi^{2}}+\left(D-2\right)\left(1+g^{2}r^{2}_{h}\right)+\Phi^{2}-4\pi\tau
r_{h}\sqrt{\frac{1+g^{2}r^{2}_{h}}{1+g^{2}r^{2}_{h}-\Phi^{2}}}\right].$

By taking the first-order derivative of the generalized free energy in the
event horizon radius to be zero, we can obtain the Hawking temperature which
is rewritten in the grand canonical ensemble by using Eqs. (35) and (36) as
follows

$\displaystyle T=\frac{D-3+(D-1)g^{2}r_{h}^{2}-(D-3)\Phi^{2}}{4\pi
r_{h}}\sqrt{\frac{1+g^{2}r_{h}^{2}}{1+g^{2}r_{h}^{2}-\Phi^{2}}}.$ (38)

From this expression, we find a constraint on the event horizon radius given
by

$\displaystyle r_{h}^{2}>\frac{\Phi^{2}-1}{g^{2}}.$ (39)

This constraint implies that when $\Phi^{2}>1$, the event horizon radius
$r_{h}$ should have a lower bound. The behavior of the Hawking temperature in
terms of the event horizon radius crucially depends on the chemical potential
$\Phi$, as depicted in Fig. 12.

Figure 12: The Hawking temperature versus the event horizon radius for $D=4$
with various chemical potential values.

For $\Phi^{2}\neq 1$, the Hawking temperature increases to infinity as $r_{h}$
approaches either the lower bound (which is nonzero for $\Phi^{2}>1$) or the
infinity. In this case, the Hawking temperature possesses a global minimum.
However, in the case of $\Phi^{2}=1$, the Hawking temperature is a
monotonously increasing function of $r_{h}$. When $r_{h}$ goes to zero, the
Hawking temperature approaches a minimum value given by

$\displaystyle T_{\text{min}}=\sqrt{\frac{D-1}{(D-2)\pi}}.$ (40)

Unlike the thermodynamics in the canonical ensemble which depends on the
number of spacetime dimensions and the pressure, the thermodynamics in the
grand canonical ensemble depends on the chemical potential. It is classified
into the following classes: (i) $\Phi^{2}<1$; (ii) $\Phi^{2}>1$; and (iii)
$\Phi^{2}=1$.

Figure 13: The generalized free energy $F$ in terms of the event horizon
radius $r_{h}$ for various values of the ensemble temperature and the chemical
potential with $D=4$ and $P=0.1$. The top, middle, and bottom panels
correspond to $\Phi^{2}=0.9$, $1$, and $1.1$, respectively. The red points
represent the on-shell black hole states corresponding to the black holes. The
blue points represent the inflection point.

The thermodynamics of the $\Phi^{2}<1$ class is the same as that of the $D=4$
class in the canonical ensemble. When the ensemble temperature is above a
minimum value, the generalized free energy $\Omega$ exhibits a local maximum
and global minimum which indicate the on-shell black hole states, as depicted
by the top panel in Fig. 13. The local maximum of the generalized free energy
represents the thermodynamically unstable black holes which would decay into
the thermodynamically stable black holes represented by the global minimum.
This decay corresponds to a second-order phase transition which can also be
pointed by studying the behavior of the heat capacity at the constant pressure
$C_{P}$ and the on-shell free energy $G$ which are given by

$\displaystyle C_{P}$ $\displaystyle=$ $\displaystyle\frac{2\pi
r_{h}^{2}(1+8\pi r_{h}^{2}-\Phi^{2})\left[(3+8\pi r_{h}^{2})^{2}-9(1+4\pi
r_{h}^{2})\Phi^{2}\right]}{(8\pi r_{h}^{2}-1)(3+8\pi
r_{2}^{2})^{2}+2\left[9-4\pi r_{h}^{2}(3+4\pi
r_{h}^{2})\right]\Phi^{2}-9\Phi^{4}}\sqrt{\frac{3+8\pi r_{h}^{2}}{3+8\pi
r_{h}^{2}-3\Phi^{2}}},$ (41) $\displaystyle\Omega$ $\displaystyle=$
$\displaystyle M-TS-\Phi Q$ (42) $\displaystyle=$ $\displaystyle-\frac{2\pi
r_{h}^{3}}{3}+\frac{r_{h}(3+8\pi r_{h}^{2})}{4(3+8\pi r_{h}^{2}-3\Phi^{2})}.$

As seen in Fig. 14, there is a divergence in the behavior of $C_{P}$ for
$\Phi^{2}<1$ (the red curve).

Figure 14: The heat capacity at the constant pressure $C_{P}$ versus the event
horizon radius for various chemical potentials with $D=4$ and $P=0.1$.

This divergence separates the heat capacity into two thermodynamic phases: the
thermodynamically (un)stable phase with the (negative) positive $C_{P}$
corresponds to the (small) large black holes. Also, From in Fig. 15,

Figure 15: The on-shell free energy in terms of the event horizon radius for
$D=4$ and $P=0.1$.

we find that the free energy of the small black holes (shown by the red line
above the horizontal axis) is higher than that of the large black holes (shown
by the remaining branch of the red curve). This means that the small black
holes are less thermodynamic than the large black holes. In addition, the on-
shell free energy shows multivalued behavior, which indicates the phase
transition between the small and large black holes. Furthermore, we find the
HP phase transition between the thermal AdS space and the large black holes
when the global minimum of the generalized free energy lies on the line
$\Omega=0$. The HP temperature can be determined by the intersection between
the lower branch of the red curve (given in Fig. 15) with the line $\Omega=0$.

For the $\Phi^{2}>1$ class, when the ensemble temperature is below a critical
value which is about $0.3$ for $D=4$ and $P=0.1$, the generalized free energy
has a global minimum corresponding to the on-shell black hole states which are
thermodynamically stable or have the positive heat capacity. However, when the
ensemble temperature is above the critical value, we observe from the middle
panel of Fig. 13 that there are one local maximum, one local minimum, and one
global minimum. The local minimum with the small $r_{h}$ represents the
thermodynamic phase with a positive heat capacity as shown by the dashed
purple curve in Fig. 14. The local maximum represents the thermodynamically
unstable black holes with a negative heat capacity as shown by the purple
curve below the horizontal axis in Fig. 14. The global minimum represents the
thermodynamically stable black holes that have a positive heat capacity as
shown by the purple curve above the horizontal axis in Fig. 14. The unstable
black holes would decay into the stable black holes corresponding to the local
minimum and global minimum via some phase transition. However, it should be
noted that the transition between the on-shell black hole states corresponding
to the local minimum and maximum is a crossover, not a phase transition
because the thermodynamic quantities are continuous.

For the $\Phi^{2}=1$ class, from the bottom panel of Fig. 13, we see that when
the ensemble temperature is above a minimum value, the generalized free energy
possesses a unique extremal point, implying that there is a unique
thermodynamic phase. This phase has a positive heat capacity, as shown by the
blue curve in Fig. 14, and hence it is thermodynamically stable. This is
consistent with the fact that the extremal point of the generalized free
energy is a global minimum. On the contrary, when the ensemble temperature is
below this minimum value, the thermodynamic system exits at the thermal AdS
space. In this sense, this minimum value is nothing but the HP phase
transition temperature which is about $0.22$ for $D=4$ and $P=0.1$. We can
determine this temperature from the intersection of the blue curve (given in
Fig. 15) with the horizontal axis.

##  V  Conclusion

Gauging the KK vector field leads to a pseudo-supersymmetrization of the
$D$-dimensional KK theory (derived from the KK reduction of
$(D+1)$-dimensional pure Einstein gravity on a circle $S^{1}$) where the
pseudo-gravitino and pseudo-dilatino are charged under U(1)${}_{\text{KK}}$.
As a result, this gauge generates a potential for the dilaton field. The
charged dilaton AdS black holes in this effective field theory have been found
in Ref. Liu2012  which are asymptotic to the AdSD geometry where the curvature
radius of asymptotic geometry is determined in terms of the KK gauge coupling.
Interestingly, these black holes can be obtained from the dimensional
reduction of supergravity on the compact internal manifolds. This allows us to
restrict the number of spacetime dimensions as $4\leq D\leq 7$.

In the present work, we study the thermodynamic phases and the phase
transitions of charged dilaton AdS black holes in the context of the
generalized (off-shell) free energy which is an extension of the on-shell free
energy. When the ensemble temperature is below a minimal value, the
generalized (off-shell) free energy has a global minimum located at the zero
event horizon radius (without having any extremum point) corresponding to the
free energy of the thermal AdS space. This implies that the thermodynamic
system exits at the phase of the thermal AdS space. However, the generalized
free energy would exhibit local maximum and local/global minimum when
increasing the ensemble temperature above the minimum value. These extremal
points correspond to the on-shell states which satisfy the equations of
motion. The local maximum points represent the thermodynamically unstable
black holes with negative heat capacity. Hence, they would decay into the
thermodynamically stable black holes (with a negative heat capacity)
represented by the local/global minimum points. These decays are performed
through either the phase transition or the crossover. This is confirmed by
studying the divergence of the heat capacity at the constant pressure and the
multivalued behavior of the on-shell free energy. Investigating the
generalized (off-shell) free energy, we also find the HP phase transition
between the thermal AdS space and black holes, and the phase transition
between the small black holes (represented by local minimum) and the large
black holes (represented by global minimum) analogous to the Van der Waals
phase transition between the liquid and gas.

In particular, we have classified the thermodynamics of charged dilaton AdS
black holes in the gauged KK theory in both the canonical ensemble and grand
canonical ensemble. The thermodynamic phases and phase transitions of black
holes in the canonical ensemble are dependent on the number of spacetime
dimensions and the pressure, consequently, the thermodynamic behavior of the
black hole can be classified into three different classes corresponding to
$D=4$, $D=5$, and $D=6,7$. In the grand canonical ensemble, the thermodynamics
of black holes depends only on the chemical potential $\Phi$ and can be
classified into three different classes corresponding to $\Phi<1$, $\Phi>1$,
and $\Phi=1$.

## Acknowledgments

This research is funded by Phenikaa University under grant number
PU2022-1-A-16.

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# Computing the dimension of ideals in group algebras,
with an application to coding theory

Michele Elia, Elisa Gorla Politecnico di Torino, ItalyUniversity of Neuchâtel,
Switzerland.

###### Abstract

We study the problem of computing the dimension of a left/right ideal in a
group algebra ${\mathbb{F}}[{\mathcal{G}}]$ of a finite group ${\mathcal{G}}$
over a field ${\mathbb{F}}$, by relating the dimension to the rank of an
appropriate matrix, originating from a regular right/left representation of
${\mathcal{G}}$. In particular, the dimension of a principal ideal is equal to
the rank of the matrix representing a generator. From this observation, we
establish a bound and an efficient algorithm for the computation of the
dimension of an ideal in a group ring. Since group codes are ideals in finite
group rings, our algorithm allows to efficiently compute their dimension.

### Keywords:

Group algebra, ideal, group code, representation, rank, characteristic
polynomial.

Mathematics Subject Classification (2010): 20C05, 16D25, 16S34

##  1 Introduction and preliminaries

Let $\mathcal{G}=\\{g_{1},g_{2},\ldots,g_{n}\\}$ be a finite multiplicative
group of order $n=|\mathcal{G}|$, with neutral element $g_{1}=1$. Let
$\mathbb{F}$ be a field of characteristic $p$. Finite fields of order
$q=p^{m}$ are denoted as $\mathbb{F}_{q}$. The group algebra
$\mathbb{F}[\mathcal{G}]$ of $\mathcal{G}$ over $\mathbb{F}$ consists of the
formal sums

$\sum_{j=1}^{n}\alpha_{i}g_{i}~{}~{}~{}~{}\alpha_{i}\in\mathbb{F},~{}~{}g_{i}\in\mathcal{G},$
(1)

where the $\alpha_{i}$’s are called coefficients. The sum in
${\mathbb{F}}[\mathcal{G}]$ is defined coefficientwise, that is the
coefficients of the same $g_{j}$ are added according to the addition in
$\mathbb{F}$. The product is performed by applying the distributive law, and
the group elements are multiplied according to the rule in $\mathcal{G}$. The
group algebra $\mathbb{F}[\mathcal{G}]$ is a vector space of dimension $n$
over the field $\mathbb{F}$, and has the structure of an associative ring with
identity. It is commutative if and only if ${\mathcal{G}}$ is commutative. It
turns out that the structure of the ideals depends on the group $\mathcal{G}$
and the field characteristic. If the field characteristic does not divide the
group order, or is $0$, the group ring is semisimple by Maschke’s Theorem (see
[8]). In addition, every ideal is principal and generated by an idempotent,
[3]. If the field characteristic divides the group order, then the group ring
is not semisimple in general [10].

A group code of length $n$ is a linear code which is the image of an ideal
$I\subseteq{\mathbb{F}}[{\mathcal{G}}]$ via an isomorphism
$\phi:{\mathbb{F}}[{\mathcal{G}}]\rightarrow{\mathbb{F}}^{n}$. In other words,
an ideal of ${\mathbb{F}}[{\mathcal{G}}]$ is a group code in
${\mathbb{F}}^{n}$, for a given choice of a basis of
${\mathbb{F}}[{\mathcal{G}}]$. Denote by $C=\phi(I)$ the group code
corresponding to the ideal $I$, then $C$ is an $[n,k]$-code, where
$k=\dim_{{\mathbb{F}}}I$ (see [12] for more details). In this paper, we study
the problem of how to efficiently compute the dimension of $I$. Our approach
uses some elementary tools from representation theory.

Every representation $D:{\mathcal{G}}\longrightarrow GL_{m}({\mathbb{K}})$ of
$\mathcal{G}$ over an extension field $\mathbb{K}$ of $\mathbb{F}$ induces a
representation of the group algebra $\mathbb{F}[\mathcal{G}]$, which we denote
again by $D$

$\begin{array}[]{rcl}D:{\mathbb{F}}[{\mathcal{G}}]&\longrightarrow&{\mathcal{M}}_{m}({\mathbb{K}})\\\
\sum_{j=1}^{n}\alpha_{j}g_{j}&\longmapsto&\sum_{j=1}^{n}\alpha_{j}D(g_{j}).\end{array}$
(2)

Here ${\mathcal{M}}_{m}({\mathbb{K}})$ is the ring of $m\times m$ matrices
with entries in ${\mathbb{K}}$. In particular a regular representation of
${\mathcal{G}}$ induces a representation of ${\mathbb{F}}[{\mathcal{G}}]$ over
$\mathbb{F}$.

The main results of the paper are contained in Section 2: In Proposition 1 we
relate the dimension of a proper left/right ideal
$I\subset{\mathbb{F}}[{\mathcal{G}}]$ to the rank of a matrix constructed
using a regular right/left representation of ${\mathcal{G}}$. In Proposition 4
and Theorem 5 we discuss how to compute an idempotent generator of a given
left/right ideal and show that the dimension of the left/right ideal that it
generates can be obtained from the characteristic polynomial of the matrix
from Proposition 1. In Corollary 7 we reduce the computation of the dimension
of a principal left/right ideal to the computation of a characteristic
polynomial, even in the case when the ideal is not generated by an idempotent.
In Section 3 we discuss the applications to coding theory and present some
examples.

##  2 Ideal dimension

Computing the dimension of a left/right ideal in $\mathbb{F}[\mathcal{G}]$ has
both theoretical relevance and practical applications. The solution that we
propose is based on the representation of $\mathbb{F}[\mathcal{G}]$ induced by
a regular representation of $\mathcal{G}$. In the next proposition, we relate
the dimension of a left/right ideal to the rank of a matrix built using the
regular right/left representation of ${\mathcal{G}}$. In the rest of the
section, we discuss how to efficiently compute this rank. In particular, we
relate it to the characteristic polynomial of a suitable matrix.

######  Proposition 1.

Let ${\mathcal{G}}=\\{g_{1},\ldots,g_{n}\\}$ be a group, ${\mathbb{F}}$ be a
field, and let $A={\mathbb{F}}[{\mathcal{G}}]$. Let $f_{1},\ldots,f_{t}\in A$,
$f_{i}=\sum_{j=1}^{n}\alpha_{ij}g_{j}$ with $\alpha_{ij}\in{\mathbb{F}}$. Let
$I=Af_{1}+\ldots+Af_{t}$ and $J=f_{1}A+\ldots+f_{t}A$. Let
$\rho(f_{1}),\ldots,\rho(f_{t})$ be the matrices corresponding to
$f_{1},\ldots,f_{t}$ in the representation of $A$ induced by the regular right
representation $\rho$ of ${\mathcal{G}}$. Let
$\lambda(f_{1}),\ldots,\lambda(f_{t})$ be the matrices corresponding to
$f_{1},\ldots,f_{t}$ in the representation of $A$ induced by the regular left
representation $\lambda$ of ${\mathcal{G}}$. Define block matrices

$\rho(I)=\left[\begin{array}[]{c}\rho(f_{1})\\\ \vdots\\\
\rho(f_{t})\end{array}\right]\;\;\;\mbox{ and
}\;\;\;\lambda(J)=\left[\begin{array}[]{c}\lambda(f_{1})\\\ \vdots\\\
\lambda(f_{t})\end{array}\right].$

Then

$\dim_{{\mathbb{F}}}I=\operatorname{rank}\rho(I)\;\;\;\mbox{ and
}\;\;\;\dim_{{\mathbb{F}}}J=\operatorname{rank}\lambda(J).$

###### Proof.

For each $f=\sum_{i=1}^{n}a_{i}g_{i}\in A$, define
$f^{*}=\sum_{i=1}^{n}a_{i}g_{i}^{-1}.$ We have
$\rho(f^{*})=(m_{ij})\in{\mathcal{M}}_{n}({\mathbb{F}})$, where

$g_{i}f=m_{i1}g_{1}+\cdots+m_{in}g_{n}.$

In other words, the entries in the i-th row of $\rho(f_{k}^{*})$ are the
coefficients of $g_{i}f_{k}$ with respect to the ${\mathbb{F}}$-basis
$g_{1},\ldots,g_{n}$ of $A$. The elements $g_{i}f_{k}$ for $i=1,\ldots,n$ and
$k=1,\ldots,t$ generate $I$ as ${\mathbb{F}}$-vector space, hence

$\dim_{{\mathbb{F}}}I=\operatorname{rank}\left[\begin{array}[]{c}\rho(f_{1}^{*})\\\
\vdots\\\ \rho(f_{t}^{*})\end{array}\right].$

The permutation of $G$ that exchanges $g_{i}$ and $g_{i}^{-1}$ for each $i$
induces a permutation of the columns of $\rho(f^{*})$ for all $f\in A$, which
sends $\rho(f^{*})$ to $\rho(f)$. Hence

$\dim_{{\mathbb{F}}}I=\operatorname{rank}\left[\begin{array}[]{c}\rho(f_{1}^{*})\\\
\vdots\\\
\rho(f_{t}^{*})\end{array}\right]=\operatorname{rank}\left[\begin{array}[]{c}\rho(f_{1})\\\
\vdots\\\ \rho(f_{t})\end{array}\right]=\operatorname{rank}\rho(I).$

Similarly, for the right ideal $J=f_{1}A+\cdots+f_{t}A$ we consider the
regular left representation $\lambda$ of ${\mathcal{G}}$. The entries in the
i-th row of $\lambda(f_{k})$ are the coefficients of $f_{k}g_{i}$ with respect
to the ${\mathbb{F}}$-basis $g_{1},\ldots,g_{n}$. Since the elements
$f_{k}g_{i}$ generate $J$ as an ${\mathbb{F}}$-vector space, the rank of
$\lambda(J)$ equals the dimension of $J$. ∎

The computation of characteristic polynomials is straightforward, and can be
used to compute or bound the rank of a matrix. In this context, Proposition 1
has interesting consequences. We start by establishing a preliminary result.
This result is essentially contained in Theorem 24.2 of [3], but we present it
here in the form that we will need.

######  Lemma 2.

Let $A$ be a semisimple ring. Let $I\subset A$ be a proper left (resp., right)
ideal. Then $I=Ae$ (resp., $I=eA$), where $e\in A$ is an idempotent and $a=ae$
(resp. $a=ea$) for all $a\in I$. Further, any idempotent generator of $I$ is
of the form $e=1-\epsilon$, where $\epsilon$ is an idempotent and
$e\epsilon=\epsilon e=0$.

If in addition $I$ is a two-sided ideal, then $I=(e)$ for some $e\in A$
idempotent. Moreover $A=I\oplus J$ where $J=0:_{A}I$ is the annihilator of
$I$, and $J=(\epsilon)$.

###### Proof.

We give the proof for the case of left ideals. The proof for right ideals is
analogous. Since $A$ is semisimple, $A=I\oplus J$, where $J\subset A$ is a
left ideal. Write $1=e+\epsilon$, where $e\in I$ and $\epsilon\in J$.
Multiplying the identity on the left by $e$, and using the fact that $J$ is a
left ideal, we obtain that

$e\epsilon=e-e^{2}\in I\cap J=0.$

Therefore $e=e^{2}\in I$ is an idempotent, and the same holds for $\epsilon\in
J$.

Clearly $Ae\subseteq I$. In order to show the reverse inclusion, observe that
for any $a\in I$ we have $a\epsilon=a-ae\in I\cap J=0$, therefore $a=ae\in
Ae.$ It follows that $I=Ae$ and $a=ae$ for all $a\in I$. This also shows that
$J=A\epsilon$ and $a\epsilon=a$ for all $a\in J$.

If in addition $I$ is a two-sided ideal, then $A=I\oplus J$ where $J$ is also
a two-sided ideal. Hence $I=(e)$ and $J=(\epsilon)$ with $e\epsilon=\epsilon
e=0$. Therefore $IJ=JI=0$ and

$J\subseteq 0:_{A}I=\\{a\in A\mid aI=Ia=0\\}.$

Conversely, let $a\in 0:_{A}I$. Then $a=ae+a\epsilon=a\epsilon\in J$.
Therefore $J=0:_{A}I$, as claimed. ∎

######  Remarks 3.

  1. 1.

Since in general a semisimple ring has elements which are not idempotents,
there exist ideals which are generated by an element which is not an
idempotent. However, the lemma implies that they have another generator, which
is an idempotent.

E.g., let $A={\mathbb{F}}_{5}[S_{3}]$ and let $f=1+(12)\in A$. Then
$f^{2}=2f\neq f$, however $Af=A(3f)$ and $e=3f$ is idempotent.

  2. 2.

An idempotent matrix $M\in{\mathcal{M}}_{n}({\mathbb{F}})$ has minimal
polynomial $z,z-1$ or $z^{2}-z$. Hence its characteristic polynomial has the
form $z^{k}(z-1)^{n-k}$ for some $0\leq k\leq n$.

In the previous example, following the notation of Proposition 1 we have

$\rho(f)=\left[\begin{array}[]{cccccc}1&1&0&0&0&0\\\ 1&1&0&0&0&0\\\
0&0&1&1&0&0\\\ 0&0&1&1&0&0\\\ 0&0&0&0&1&1\\\ 0&0&0&0&1&1\end{array}\right].$

The matrix $\rho(f)$ is not idempotent, and has characteristic polynomial
$z^{3}(z-2)^{3}$. Notice that $2\in{\mathbb{F}}_{5}^{*}$ has order $4$, and
$\rho(f)^{4}=3\rho(f)$ has characteristic polynomial $z^{3}(z-1)^{3}$.

Remark 3.1 raises the question of how to compute an idempotent generator of a
given ideal $I$. E.g. let $I=Af$ be a left ideal with a given generator $f\in
A$. Then $e-1\in\\{a\in A\mid fa=0\\}$, but the right annihilator of $f$ is
not always easy to compute. However, the problem has a simple solution if we
consider a representation of $A$, since it reduces to a simple linear algebra
problem. In the next proposition we discuss the case of left ideals. Right
ideals can be treated similarly.

######  Proposition 4.

Let $D$ be a representation of ${\mathcal{G}}=\\{g_{1},\ldots,g_{n}\\}$ such
that $D(g_{1}),\ldots,D(g_{n})$ are linearly independent. Then $D$ induces an
injective representation $D$ of $A={\mathbb{F}}[{\mathcal{G}}]$, and the right
annihilators of $f\in A$ corresponds via $D$ to the solutions over
${\mathbb{F}}$ of the linear system

$D(f)\left(\sum_{i=1}^{n}x_{i}D(g_{i})\right)=0$

in the variables $x_{1}\ldots x_{n}$. In other words, solutions
$(a_{1},\ldots,a_{n})\in{\mathbb{F}}^{n}$ of the system corresponds to right
annihilators $a=a_{1}g_{1}+\ldots+a_{n}g_{n}$ of $f$.

###### Proof.

A representation $D:{\mathcal{G}}\rightarrow GL_{m}({\mathbb{F}})$ induces a
representation $D:A\rightarrow{\mathcal{M}}_{m}({\mathbb{F}})$. $D$ is
injective since $D(g_{1}),\ldots,D(g_{n})$ are linearly independent. Hence $D$
is an isomorphism between $A$ and its image, in particular $fa=0$ iff
$D(fa)=D(f)D(a)=0$. Let $a=a_{1}g_{1}+\ldots+a_{n}g_{n}$, then
$D(a)=a_{1}D(g_{1})+\ldots+a_{n}D(g_{n})$ and $D(f)D(a)=0$ iff
$(a_{1},\ldots,a_{n})\in{\mathbb{F}}^{n}$ is a solution of the linear system

$D(f)\left(\sum_{i=1}^{n}x_{i}D(g_{i})\right)=0.$

∎

We can now state the first consequence of Proposition 1. Notice that if
$p\nmid n=|{\mathcal{G}}|$, then ${\mathbb{F}}[{\mathcal{G}}]$ is semisimple
by Maschke’s Theorem (see [8]). In particular, by Lemma 2 every left/right
ideal of $A$ has an idempotent generator, which one can compute using
Proposition 4. In that case, the next result allows us to reduce the
computation of the dimension of a left/right ideal to the computation of a
characteristic polynomial.

######  Theorem 5.

Let ${\mathcal{G}}=\\{g_{1},\ldots,g_{n}\\}$ be a group, ${\mathbb{F}}$ be a
field, and let $A={\mathbb{F}}[{\mathcal{G}}]$. Let
$f=\alpha_{1}g_{1}+\ldots+\alpha_{n}g_{n}\in A$ and let $I=Af$ be a proper
left ideal (resp., $I=fA$ be a proper right ideal) of $A$. Let $F=\rho(f)$ be
the matrix associated to $f$ in the regular right representation $\rho$ of
${\mathcal{G}}$ (resp., let $F=\lambda(f)$ be the matrix associated to $f$ in
the regular left representation $\lambda$ of ${\mathcal{G}}$). Let $z^{k}g(z)$
be the characteristic polynomial of $F$, where $z\nmid g(z)$. Then

$n-k\leq\dim_{{\mathbb{F}}}I\leq n-1.$

If in addition $f$ is idempotent, then $\dim_{{\mathbb{F}}}I=n-k$.

###### Proof.

It follows from Proposition 1 that

$\dim_{{\mathbb{F}}}I=\operatorname{rank}(F)\geq n-k.$

Since $I$ is proper, $\dim_{{\mathbb{F}}}I\leq n-1$.

If in addition $f$ is idempotent, then $F$ is an idempotent matrix, hence

$\ker F=\ker F^{m}\;\;\mbox{for any $m\geq 1$.}$

Therefore, the algebraic and geometric multiplicities of $0$ as an eigenvalue
of $F$ coincide, and

$\dim_{{\mathbb{F}}}I=\operatorname{rank}(F)=n-k,$

where the first equality again follows from Proposition 1. ∎

In the case that $p\mid n$, there may be ideals of
${\mathbb{F}}[{\mathcal{G}}]$ which have no idempotent generator. If $f$ is
not idempotent, it is possible that $\operatorname{rank}(F)>n-k$. Next we give
some examples where $A$ is commutative and $I=(f)$ has
$\dim_{{\mathbb{F}}}I=\operatorname{rank}(F)>n-k$. In particular, we provide
examples where $n-k=0$ and $\dim_{{\mathbb{F}}}I=\ell$ for any $\ell$ which
divides $n/p$, $p=\operatorname{char}{\mathbb{F}}$.

######  Example 6.

We follow the notation of Theorem 5. Let ${\mathcal{G}}$ be a cyclic group of
order $n=\ell m$ and let $p$ be a prime, $p\mid m$. Let $H$ be a subgroup of
${\mathcal{G}}$ of order $p$, and let

$f=\sum_{h\in H}h\in{\mathbb{F}}[{\mathcal{G}}]$

where ${\mathbb{F}}$ is a field. For a suitable reordering of the elements of
${\mathcal{G}}$, the matrix $F\in{\mathcal{M}}_{n}({\mathbb{F}})$ associated
to $f$ in a regular representation of ${\mathcal{G}}$ is the block matrix

$F=\left.\left[\begin{array}[]{ccccc}1&0&0&\cdots&0\\\ 0&1&0&\cdots&0\\\
\vdots&\ddots&\ddots&\ddots&\vdots\\\ 0&\cdots&0&1&0\\\
0&0&\cdots&0&1\end{array}\right]\right\\}\mbox{$\ell$ blocks}$

where $0$ and $1$ denote the matrix of size $m\times m$ filled with $0$ and
$1$ respectively. If $I=(f)\subset A$, then

$\dim_{{\mathbb{F}}}I=\operatorname{rank}(F)=\ell$

by Proposition 1. It is easy to check that the characteristic polynomial of
$F$ is $z^{n-\ell}(z-m)^{\ell}$. Hence $k=n-\ell$ if
$\operatorname{char}{\mathbb{F}}\neq p$, and $k=n$ if
$\operatorname{char}{\mathbb{F}}=p$. In particular, if
$\operatorname{char}{\mathbb{F}}=p\mid n$, then

$\dim_{{\mathbb{F}}}I=\ell>n-k=0.$

Combining Proposition 1 with a result by Mulmuley [9], we can reduce the
computation of the dimension of a principal ideal to the computation of a
characteristic polynomial, even in the case that the generator of the ideal is
not nilpotent.

######  Corollary 7.

Let $f\in A={\mathbb{F}}[{\mathcal{G}}]$ and let $I=Af$ (resp., $I=fA$). Let
$F$ be the matrix associated to $f$ in the regular right representation
(resp., in the regular left representation) of ${\mathcal{G}}$. Let

$M=F\;\;\;\mbox{if ${\mathcal{G}}$ is commutative,}$

and let

$M=\left[\begin{array}[]{cc}0&F\\\ F^{t}&0\end{array}\right]\;\;\;\mbox{if
${\mathcal{G}}$ is not commutative.}$

Let $x$ be a transcendental element over ${\mathbb{F}}$ and let $X$ be the
diagonal matrix with eigenvalues $1,x,\ldots,x^{m-1}$, where $m$ is the size
of the matrix $M$. Let $z^{k}g(z,x)$ be the characteristic polynomial of the
matrix $XM$, where $z\nmid g(z,x)$. Then

$\dim_{{\mathbb{F}}}I=\left\\{\begin{array}[]{ll}\deg_{z}g(z,x)&\mbox{if
${\mathcal{G}}$ is commutative,}\\\ \deg_{z}g(z,x)/2&\mbox{if ${\mathcal{G}}$
is not commutative.}\end{array}\right.$

######  Remark 8.

If in Corollary 7 we let $x$ be a random element in ${\mathbb{F}}$ (or a
suitable algebraic extension field), we obtain a faster randomized algorithm
to compute the dimension of $I$. Such an approach works well in practice, and
we can increase the probability of obtaining the correct result by repeating
the computation for different random values of $x$.

##  3 Application to coding theory and examples

A group code is an ${\mathbb{F}}_{q}$-linear code which is image of an ideal
$I\subseteq{\mathbb{F}}_{q}[{\mathcal{G}}]$ via the isomorphism
$\phi:{\mathbb{F}}[{\mathcal{G}}]\rightarrow{\mathbb{F}}^{n}$ which sends
$g_{i}\in{\mathcal{G}}$ to the $i$th element of the canonical basis of
${\mathbb{F}}^{n}$. The length of $\phi(I)$ is equal to $n$, the cardinality
of ${\mathcal{G}}$, and the dimension of the code is the dimension of $I$ over
${\mathbb{F}}_{q}$. The method that we propose therefore allows us to
efficiently compute the dimension of group codes. See [2, 6] for an overview
of group code properties, and [4] for an efficient encoding, and corresponding
syndrome decoding.
Our method will be illustrated by two simple examples. We observe that the
right regular representation of a group
${\mathcal{G}}=\\{g_{1},\ldots,g_{n}\\}$ can be easily obtained from a
modified Cayley table: The element in position $(i,j)$ in the table is
$g_{i}^{-1}g_{j}$. Letting $g_{1}$ be the neutral element, the entries on the
diagonal are all equal to $g_{1}$. For a given $f=\sum_{i=1}^{n}x_{i}g_{i}\in
A=\mathbb{F}[\mathcal{G}]$, the matrix $\rho(f)$ associated to $f$ in the
right regular representation $\rho$ of $A$ is obtained by substituting $g_{i}$
by $x_{i}$ in the Cayley table for $i=1,\ldots,n$.

######  Example 9.

Consider the Klein group
$\mathcal{K}_{4}=\mathbb{Z}_{2}\times\mathbb{Z}_{2}=\\{e,\alpha,\beta,\alpha\beta\\}$,
where $e$ is the neutral element, $\alpha^{2}=\beta^{2}=1$, and
$\alpha\beta=\beta\alpha$. Let $a,b,c,d\in\mathbb{F}$, then the matrix
associated to
$f=ae+b\alpha+c\beta+d\alpha\beta\in{\mathbb{F}}[{\mathcal{K}}_{4}]$ in the
right regular representation $\rho$ of ${\mathbb{F}}[{\mathcal{K}}_{4}]$ is

$\rho(f)=\left[\begin{array}[]{cccc}a&b&c&d\\\ b&a&d&c\\\ c&d&a&b\\\
d&c&b&a\\\ \end{array}\right].$

If $\operatorname{char}({\mathbb{F}})\neq 2$, then $\rho(f)$ is diagonalizable

$\rho(f)\sim\left[\begin{array}[]{cccc}a+b+c+d&0&0&0\\\ 0&a-b-c+d&0&0\\\
0&0&a+b-c-d&0\\\ 0&0&0&a-b+c-d\\\ \end{array}\right].$

In particular, the characteristic polynomial of $\rho(f)$ is

$\phi_{f}(z)=(z-a-b-c-d)(z-a+b+c-d)(z-a-b+c+d)(z-a+b-c+d)=z^{k}g(z)$

with $g(0)\neq 0$, and

$4-k=\deg g(z)=\operatorname{rank}\rho(f)=\dim_{{\mathbb{F}}}(f).$

Notice that every $f\in{\mathbb{F}}[{\mathcal{K}}_{4}]$ is idempotent. In this
case ${\mathbb{F}}[{\mathcal{K}}_{4}]$ is semisimple, and the diagonal form of
$\rho(f)$ corresponds to the decomposition of
${\mathbb{F}}[{\mathcal{K}}_{4}]$ as a direct sum of ideals

${\mathbb{F}}[{\mathcal{K}}_{4}]=(e+\alpha+\beta+\alpha\beta)\oplus(e-\alpha-\beta+\alpha\beta)\oplus(e+\alpha-\beta-\alpha\beta)\oplus(e-\alpha+\beta-\alpha\beta),$

where each of the ideals appearing as direct summands is minimal and
idempotent.

If $\operatorname{char}({\mathbb{F}})=2$, then $\rho(f)$ can be brought into
the form

$\rho(f)\sim\left[\begin{array}[]{cccc}a+b+c+d&0&0&0\\\ b&a+b+c+d&0&0\\\
c&c+d&a+d&b+c\\\ d&c+d&b+c&a+d\\\ \end{array}\right].$

The characteristic polynomial of $\rho(f)$ is

$\phi_{f}(z)=(z+a+b+c+d)^{4}.$

If $f$ is not invertible, then $a+b+c+d=0$, in particular $\phi_{f}(z)=z^{4}$.
Hence $\operatorname{rank}\rho(f)\in\\{0,1,2\\}$. In particular,
$\operatorname{rank}\rho(f)=0$ if and only if $f=0$. Moreover
$\operatorname{rank}\rho(f)=1$ if and only if $a=b=c=d\neq 0$. In this case we
have $(f)=(e+\alpha+\beta+\alpha\beta)$ and $\dim_{{\mathbb{F}}}(f)=1$.
Finally, if $a,b,c,d$ are not all equal and either $b\neq 0$ or $b\neq c$ or
$c\neq d$, then $\operatorname{rank}\rho(f)=2$. In this case
$f\in\\{e+\alpha,e+\beta,e+\alpha\beta\\}$ and $\dim_{{\mathbb{F}}}(f)=2$.
Notice that if $\operatorname{char}({\mathbb{F}})=2$, then
${\mathbb{F}}[{\mathcal{K}}_{4}]$ is not semisimple and every ideal equals its
annihilator.

######  Example 10.

Consider the symmetric group $S_{3}=\\{1,(12),(13),(23),(123),(132)\\}.$
Each element $g\in{\mathbb{F}}[S_{3}]$ is of the form

$g=a1+b(12)+c(13)+d(23)+e(123)+f(132)~{}~{}~{}\mbox{where}~{}~{}a,b,c,d,e,f,\in{\mathbb{F}},$

and is represented by the matrix

$\rho(g)=\left[\begin{array}[]{cccccc}a&b&c&d&e&f\\\ b&a&e&f&c&d\\\
c&f&a&e&d&b\\\ d&e&f&a&b&c\\\ f&c&d&b&a&e\\\ e&d&b&c&f&a\\\
\end{array}\right]~{}~{},$

where $\rho$ denotes the right regular representation of ${\mathcal{G}}$. If
$\operatorname{char}({\mathbb{F}})\neq 2,3$, then $\rho(f)$ can be brought in
the following form

$\rho(g)\sim\left[\begin{array}[]{cccccc}a+b+c+d+e+f&0&0&0&0&0\\\ 0&a-b-
c-d+e+f&0&0&0&0\\\ 0&0&a-f&e-f&-c+d&b-c\\\ 0&0&-e+f&a-e&b-d&c-d\\\
0&0&-c+d&b-c&a-f&e-f\\\ 0&0&b-d&c-d&-e+f&a-e\\\ \end{array}\right].$ (3)

Therefore $A={\mathbb{F}}[S_{3}]=I_{1}\oplus I_{2}\oplus I_{3}=I_{1}\oplus
I_{2}\oplus J_{1}\oplus J_{2}$, where $I_{1},I_{2},I_{3}$ are minimal two-
sided ideals

$I_{1}=A(1+(12)+(13)+(23)+(123)+(132)),~{}~{}I_{2}=A(1-(12)-(13)-(23)+(123)+(132)),$
$I_{3}=A(1-(123))+A((12)-(23))+A((13)-(23))+A((123)-(132))=A(2\cdot
1-(123)-(132)).$

We have $\dim_{{\mathbb{F}}}I_{1}=\dim_{{\mathbb{F}}}I_{2}=1$ and
$\dim_{{\mathbb{F}}}I_{3}=4$. Moreover $I_{3}$ can be decomposed as the direct
sum of two left ideals: $I_{3}=J_{1}\oplus J_{2}$ with

$J_{1}=A(1+(12)-(23)-(123)),~{}~{}J_{2}=A(1-(12)+(23)-(132))$

and $\dim_{{\mathbb{F}}}J_{1}=\dim_{{\mathbb{F}}}J_{2}=2$. Using (3) it is
easy to check that in each case the dimension of the ideal is as predicted by
Theorem 5. It is also easy to check that $I_{1},I_{2},J_{1},J_{2}$ are minimal
left ideals, hence any nonzero $I=Af\subseteq A$ is the sum or one or more
among them. In particular $\dim_{{\mathbb{F}}}I$ is the sum of the dimensions
of the corresponding ideals, again as predicted by Theorem 5.

If $\operatorname{char}({\mathbb{F}})=2$, then

$\rho(g)\sim\left[\begin{array}[]{cccccc}a+b+c+d+e+f&0&0&0&0&0\\\
0&a+e+f&a+b+c+d+e+f&0&0&0\\\ 0&a+b+c+d+e+f&a+e&e+f&b+c&c+d\\\
0&0&e+f&a+f&b+d&b+c\\\ 0&0&b+c&c+d&a+e&e+f\\\ 0&0&b+d&b+c&e+f&a+f\\\
\end{array}\right].$

Denoting by $I_{1},I_{2},I_{3},J_{1},J_{2}$ the same ideals as before, we have
$I_{1}=I_{2}$ and an easy computation involving the above matrix yields

$\dim_{{\mathbb{F}}}I_{1}=1,~{}~{}\dim_{{\mathbb{F}}}I_{3}=4,~{}~{}\dim_{{\mathbb{F}}}J_{1}=\dim_{{\mathbb{F}}}J_{2}=2.$

Notice that if $\operatorname{char}({\mathbb{F}})=2$, then
${\mathbb{F}}[S_{3}]$ is no longer semisimple.

If $\operatorname{char}({\mathbb{F}})=3$, then $I_{1},I_{2}\subset I_{3}$
where

$I_{3}=A(1-(123))+A((12)-(23))+A((13)-(23))+A((123)-(132))=A(1+(123)+(132))$

and

$\dim_{{\mathbb{F}}}I_{1}=\dim_{{\mathbb{F}}}I_{2}=1,~{}~{}\dim_{{\mathbb{F}}}I_{3}=2.$

Again ${\mathbb{F}}[S_{3}]$ is not semisimple.

## Acknowledgments

The authors are grateful to David Conti for several useful discussions during
the preparation of this paper. The second author was partially supported by
the Swiss National Science Foundation under grant no. 123393.

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