# Unified Composite Distribution and Its Applications to Double Shadowed
$\alpha-\kappa-\mu$ Fading Channels

Hussien Al-Hmood, and H. S. Al-Raweshidy Manuscript received November 00,
2020.The authors are with the Department of Electronic and Computer
Engineering, College of Engineering, Design and Physical Sciences, Brunel
University London, UK, e-mail<EMAIL_ADDRESS>Hamed.Al-
<EMAIL_ADDRESS>
###### Abstract

In this paper, we propose a mixture Gamma shadowed (MGS) distribution as a
unified composite model via representing the shadowing by an inverse
Nakagami-$m$. Accordingly, the exact expression and the asymptotic behaviour
at high average signal-to-noise ratio (SNR) regime of the fundamental
statistics of a MGS distribution are derived first. These statistics are then
applied to analyze the performance of the wireless communications systems over
double shadowed $\alpha-\kappa-\mu$ fading via providing the outage
probability (OP), the average bit error probability (ABEP), the average
channel capacity (ACC), and the effective capacity (EC). The average area
under the receiver operating characteristics curve (AUC) of energy detection
(ED) is also analyzed. The numerical and simulation results are presented to
verify the validation of our analysis.

###### Index Terms:

Mixture Gamma distribution, average bit error probability, channel capacity,
effective capacity , double shadowed $\alpha-\kappa-\mu$ fading.

##  I Introduction

The wireless communications channel may undergo the effect of the multipath
and shadowing fading simultaneously [1]. Accordingly, many works have been
recently dedicated to analyse the performance of the generalized composite
models, such as, $\kappa-\mu$, $\eta-\mu$, and $\alpha-\mu$ distributions that
are used to model the line-of-sight (LoS), the non-LoS (NLoS), and the non-
linear wireless communication mediums, respectively [2], [3]. These
generalized conditions can provide close results to the practical measurements
and approximately comprise all the classical fading distributions, i.e.,
Rayleigh, Nakagami-$m$, Nakagami-$n$, Nakagami-$q$, one-sided Gaussian,
Weibull, and Gamma. Hence, the probability density function (PDF), the
cumulative distribution function (CDF), and the moment generating function
(MGF) of the composite $\eta-\mu$/Gamma fading models were derived in [4]. The
authors in [5] assumed that both $\kappa-\mu$ and $\eta-\mu$ fading conditions
are shadowed by an inverse Gamma distribution. The fundamental statistics of
the $\kappa-\mu$ shadowed fading in which the shadowing effect is represented
by a Nakagami-$m$ distribution were given in [6] with applications to the
outage probability (OP) and the average bit error probability (ABEP) of
wireless communications systems. The composite of the
$\alpha-\kappa-\mu$/Gamma distribution was analysed in [7]. The non-linear
scenario of the Fisher-Snedecor $\mathcal{F}$ distribution [8], namely,
$\alpha-\mathcal{F}$, was investigated in [9]. The PDF, the CDF, and the MGF
of the $\alpha-\eta-\mathcal{F}$ and the $\alpha-\kappa-\mathcal{F}$ composite
fading conditions were derived in [10]. These models were then unified in a
single distribution which is named $\alpha-\eta-\kappa-\mathcal{F}$ composite
fading [11].

Although different statistical properties have been reported in [4]-[11], they
are derived in mathematically intractable expressions. This is because they
are expressed in terms of either the hypergeometric or the modified Bessel
functions. Therefore, the mathematical intricacy of the performance metrics of
the wireless communications systems is high and this would to unclear insights
into the system behaviour with the fading parameters. Moreover, this
complexity is increased when the generalised multipath fading channels undergo
to double shadowing impacts [12].

To overcome the aforementioned challenges, a mixture Gamma (MG) distribution
has been widely used to approximate with high accuracy most of the composite
generalised/Gamma fading conditions [13]-[16]. For instance, the average area
under the receiver operating characteristics (AUC) curve of the energy
detection (ED) based spectrum sensing was derived in [14]. The average channel
capacity (ACC) of $\alpha-\eta-\lambda-\mu$/Gamma fading and the effective
capacity (EC) of $\alpha-\eta-\mu$/Gamma fading were analyzed in [15] and
[16], respectively, using a MG distribution.

Based on the above observations and motivated by the merits of a MG
distribution, we propose a mixture Gamma shadowed (MGS) as a unified composite
model where the shadowing is represented by an inverse Nakagami-$m$. To this
end, the basic statistics of this distribution are mathematically simple and
tractable. Consequently, a clear insight into the effects of the fading
parameters on the performance of the wireless communications systems can be
deduced when the channel is subjected to double shadowing impacts.

Our main contributions are summarized as follows:

  * •

The exact and the asymptotic expressions at high average signal-to-noise ratio
(SNR) values for novel unified mathematically tractable statistical
characterizations of composite model that is based on MG and inverse
Nakagami-$m$ distributions, namely, MGS distribution, are derived.

  * •

The derived statistics are employed to analyse the performance of the double
shadowed $\alpha-\kappa-\mu$ fading channel in which the first and the second
shadowing impacts are respectively followed the Nakagami-$m$ and the inverse
Nakagami-$m$ distributions. Accordingly, the equivalent parameters of a MG
distribution of composite $\alpha-\kappa-\mu$/Nakagami-$m$ fading channels are
provided.

  * •

Capitalizing on the above, unified closed-form expressions for the ABEP, the
ACC, the EC, and the average AUC are obtained.

##  II MG and Inverse Nakagami-$m$ Distributions

The PDF of a MG distribution is given by [13, eq. (1)]

$\displaystyle f(x)=\sum_{j=1}^{K}\sigma_{j}x^{\beta_{j}-1}e^{-\zeta_{j}x}$
(1)

where $\sigma_{j}$, $\beta_{j}$, and $\zeta_{j}$ are the parameters of $j$th
Gamma component and $K$ that stands for the number of terms is evaluated via
using the mean square error (MSE) method between the exact PDF and its MG
representation [13].

If $\xi$ is an inverse Nakagami-$m$ random variable (RV), its PDF is expressed
as [12, eq. (50)]

$\displaystyle
f_{\xi}(r)=\frac{(m_{s}-1)^{m_{s}}}{\Gamma(m_{s})}r^{m_{s}-1}e^{-\frac{(m_{s}-1)}{r}}.\quad
m_{s}>1$ (2)

where $m_{s}$ refers to the shadowing severity index and $\Gamma(.)$ is the
incomplete Gamma function [17, eq. (8.310.1)].

##  III Statistical Properties of a MGS Distribution

Let $\gamma\sim\text{MGS}(\sigma_{j},\beta_{j},\zeta_{j},m_{s})$ for
$j=1,\cdots,K$ is an MGS-distributed RV. Then, the PDF of $\gamma$ can be
obtained by the product of MG and inverse Nakagami-$m$ RVs as [4, eq. (8)]

$\displaystyle
f_{\gamma}(\gamma)=\int_{0}^{\infty}\frac{1}{z}f\bigg{(}\frac{\gamma}{z}\bigg{)}f_{\xi}(z)dz.$
(3)

Substituting (1) and (2) into (3) and making use of [17, eq. (3.381.4)] with
some mathematical manipulations, this yields

$\displaystyle
f_{\gamma}(\gamma)=(m_{s}-1)^{m_{s}}\sum_{j=1}^{K}\frac{\sigma_{j}(m_{s})_{\beta_{j}}\gamma^{\beta_{j}-1}}{(m_{s}-1+\zeta_{j}\gamma)^{\beta_{j}+m_{s}}}.$
(4)

where $(.)_{n}$ is the Pochhammer symbol.

When $\zeta_{j}\rightarrow 0$ for all $j=1,\cdots,K$, the asymptotic of the
PDF, $f^{\infty}_{\gamma}(\gamma)$, can be expressed as

$\displaystyle
f^{\infty}_{\gamma}(\gamma)\simeq\sum_{j=1}^{K}\frac{\sigma_{j}(m_{s})_{\beta_{j}}\gamma^{\beta_{j}-1}}{(m_{s}-1)^{\beta_{j}}}.$
(5)

Inserting (4) in
$F_{\gamma}(\gamma)=\int_{0}^{\gamma}f_{\gamma}(\gamma)d\gamma$ and recalling
[17, eq. (3.194.1)], the CDF of a MGS distribution can be derived as

$\displaystyle F_{\gamma}(\gamma)=$
$\displaystyle\sum_{j=1}^{K}\frac{\sigma_{j}(m_{s})_{\beta_{j}}{{}_{2}F_{1}}\big{(}\beta_{j}+m_{s},\beta_{j};\beta_{j}+1;-\frac{\zeta_{j}}{m_{s}-1}\gamma\big{)}}{\beta_{j}((m_{s}-1)\gamma^{-1})^{\beta_{j}}}.$
(6)

where ${{}_{2}F_{1}}(.)$ is the Gauss hypergeometric function [17, eq.
(9.14.1)].

Using the fact that ${{}_{2}F_{1}}(.,.;.;0)\simeq 1$ when
$\zeta_{j}\rightarrow 0$ or plugging (5) in
$F_{\gamma}(\gamma)=\int_{0}^{\gamma}f_{\gamma}(\gamma)d\gamma$, the
asymptotic of the CDF, $F^{\infty}_{\gamma}(\gamma)$, can be evaluated as

$\displaystyle
F^{\infty}_{\gamma}(\gamma)\simeq\sum_{j=1}^{K}\frac{\sigma_{j}(m_{s})_{\beta_{j}}\gamma^{\beta_{j}}}{\beta_{j}(m_{s}-1)^{\beta_{j}}}.$
(7)

Using the Laplace transform and invoking [18, eq. (13.2.5)], the MGF of the
MGS distribution can be obtained as

$\displaystyle\mathcal{M}_{\gamma}(s)$
$\displaystyle=\sum_{j=1}^{K}\frac{\sigma_{j}(m_{s})_{\beta_{j}}\Gamma(\beta_{j})U\big{(}\beta_{j};1-m_{s};\frac{m_{s}-1}{\zeta_{j}}s\big{)}}{\zeta^{\beta_{j}}_{j}}.$
(8)

where $U(.)$ is the Tricomi confluent hypergeometric function of the second
kind defined in [17, eq. (9.211.4)].

The asymptotic of the MGF, $\mathcal{M}^{\infty}_{\gamma}(s)$, can be deduced
after applying Laplace transform for (5) and invoking [17, eq. (8.310.1)].
Thus, this yields

$\displaystyle\mathcal{M}^{\infty}_{\gamma}(s)\simeq\sum_{j=1}^{K}\frac{\sigma_{j}(m_{s})_{\beta_{j}}\Gamma(\beta_{j})}{[(m_{s}-1)s]^{\beta_{j}}}.$
(9)

The $n$-th moment, $\boldsymbol{\mu}_{n}$, of the MGS distribution can be
found by using (4) and [17, eq. (3.194.3)] as

$\displaystyle\boldsymbol{\mu}_{n}=\mathbb{E}\\{\gamma^{n}\\}=(m_{s}-1)^{n}\sum_{j=1}^{K}\frac{\sigma_{j}(m_{s})_{\beta_{j}}}{\zeta^{\beta_{j}+n}_{j}}B(\beta_{j}+n,m_{s}-n).$
(10)

where $B(.)$ is the Beta function [17, eq. (8.380.1)].

It is worth interesting that the MGS distribution can used to model the
$\kappa-\mu$/inverse Gamma [5, eq. (6)] with
$\theta_{j}=\frac{e^{-\mu\kappa}\mu^{\mu+2(j-1)}\kappa^{j-1}}{\Gamma(\mu+j-1)\Gamma(j)\bar{\gamma}^{\mu+j-1}}$
which is used to compute $\sigma_{j}$, $\beta_{j}=\mu+j-1$, and
$\zeta_{j}=\frac{\mu(1+\kappa)}{\bar{\gamma}}$. Additionally, the Fisher
Senedcor $\mathcal{F}$ composite fading [8, eq. (6)] can be represented by
(4)-(10) with $K=1$, $\sigma_{1}=\frac{m^{m}}{\Gamma(m)\bar{\gamma}^{m}}$,
$\beta_{1}=m$, and $\zeta_{1}=\frac{m}{\bar{\gamma}}$.

$\displaystyle f(x)=\frac{\alpha\mu
m^{m}\kappa^{\frac{1-\mu}{2}}(1+\kappa)^{\frac{1+\mu}{2}}}{2\Gamma(m)\exp(\mu\kappa)\bar{\gamma}^{\frac{\alpha(1+\mu)}{4}}}x^{\frac{\alpha(1+\mu)}{4}-1}\int_{0}^{\infty}z^{-\frac{\alpha(1+\mu)}{4}+m-1}e^{-\frac{(1+\kappa)\mu
x^{\alpha/2}}{(\bar{\gamma}z)^{\alpha/2}}-mz}I_{\mu-1}\bigg{(}2\mu\sqrt{\frac{\kappa(1+\kappa)x^{\alpha/2}}{(\bar{\gamma}z)^{\alpha/2}}}\bigg{)}dz.$
(14)

##  IV Double Shadowed $\alpha-\kappa-\mu$ Fading Channels

The received signal envelope, $R$, over double shadowed $\alpha-\kappa-\mu$
fading channel can be expressed as

$\displaystyle R^{\alpha}=\xi^{2}\sum_{l=1}^{\mu}(X_{l}+\vartheta
p_{l})^{2}+(Y_{l}+\vartheta q_{l})^{2}$ (11)

where the parameters of (11) are defined as follows:

  1. i)

$\alpha>0$ denotes the non-linearity of the propagation medium.

  2. ii)

$\mu$ is a real-valued extension related to the number of multipath clusters.

  3. iii)

$\xi$ and $\vartheta$ represent the RVs which are responsible for introducing
the shadowing impacts that are modelled by inverse Nakagami-$m$ and
Nakagami-$m$ distributions, respectively, with
$\mathbb{E}[\xi^{2}]=\mathbb{E}[\vartheta^{2}]=1$, where $\mathbb{E}[.]$
stands for the expectation operator. It is worth mentioning that (11) becomes
equivalent to example 1 of the double shadowed $\kappa-\mu$ type I model [12,
eq. (22)], when $\alpha=2$.

  4. iv)

$X_{l}$ and $Y_{l}$ are mutually independent Gaussian random processes with
mean $\mathbb{E}[X_{l}]$ and $\mathbb{E}[Y_{l}]=0$ and variance
$\mathbb{E}[X^{2}_{l}]=\mathbb{E}[Y^{2}_{l}]=\delta^{2}$.

  5. v)

$p_{l}$ and $q_{l}$ are the mean values of the in-phase and quadrature phase
components of the multipath cluster $l$.

From (11), one can note that the PDF of $R$ can be derived by averaging the
PDF of the $\alpha-\kappa-\mu$ fading over the PDF of $\vartheta$ and $\xi$
RVs. However, the PDF of the $\alpha-\kappa-\mu$ fading is included the
modified Bessel function of the first kind, $I_{\upsilon}(.)$ [17, eq.
(8.445)] that would lead to mathematically intractable statistical properties
(please refer to [12]). Therefore, to obtain simple closed-form statistics,
the PDF of inducing the shadowing of the dominant component is approximated by
using a MG distribution whereas the multiplicative shadowing is added by
utilizing a MGS model.

The PDF of the instantaneous SNR, $\gamma$, over double shadowed
$\alpha-\kappa-\mu$ fading is expressed as [7, eq. (4)]

$\displaystyle
f_{\gamma}(r)=\frac{\alpha\mu\kappa^{\frac{1-\mu}{2}}(1+\kappa)^{\frac{1+\mu}{2}}}{2\exp(\mu\kappa)\bar{\gamma}^{\frac{\alpha(1+\mu)}{4}}}r^{\frac{\alpha(1+\mu)}{4}-1}e^{-\frac{(1+\kappa)\mu}{\bar{\gamma}^{\alpha/2}}r^{\alpha/2}}$
$\displaystyle\times
I_{\mu-1}\bigg{(}2\mu\sqrt{\frac{\kappa(1+\kappa)}{\bar{\gamma}^{\alpha/2}}r^{\alpha/2}}\bigg{)}.$
(12)

where $\kappa$ is the ratio between the total powers of the dominant
components and scattered waves and $\bar{\gamma}$ is the average SNR.

The PDF of $\vartheta$ is given by [12, eq. (54)]

$\displaystyle f_{\vartheta}(r)=\frac{m^{m}}{\Gamma(m)}r^{m-1}e^{-mr}.$ (13)

where $m$ is the shadowing severity index of the Nakagami-$m$.

Substituting (12) and (13) into (3), we have (14) shown at the top of this
page.

Using the substitution $y=\frac{(1+\kappa)\mu
x^{\alpha/2}}{(\bar{\gamma}z)^{\alpha/2}}$ in (14), we obtain

$\displaystyle f(x)=\Xi x^{m-1}\int_{0}^{\infty}e^{-y}h(y)dy.$ (15)

where
$\Xi=\frac{m^{m}\mu^{\frac{2}{\alpha}m-\frac{\mu-1}{2}}\kappa^{\frac{1-\mu}{2}}(1+\kappa)^{\frac{2}{\alpha}m}}{\Gamma(m)\exp(\mu\kappa)\bar{\gamma}^{m}}$
and
$h(y)=y^{\frac{\alpha(1+\mu)}{4}-\frac{2}{\alpha}m-1}e^{-\frac{m((1+\kappa)\mu)^{\alpha/2}}{\bar{\gamma}y^{2/\alpha}}x}I_{\mu-1}\big{(}2\sqrt{\kappa\mu
y}\big{)}$.

The integration in (15), $\Phi=\int_{0}^{\infty}e^{-y}h(y)dy$, can be
approximated by using a Gaussian-Laguerre quadrature method, as
$\Phi\approx\sum_{j=1}^{K}w_{j}h(y_{j})$, where $w_{j}$ and $y_{j}$ are the
weight factors and abscissas, respectively, given in [18]. Hence, (15) can be
equivalently expressed by (1) with the following coefficients

$\displaystyle\beta_{j}=m,$
$\displaystyle\quad\zeta_{j}=\frac{m((1+\kappa)\mu)^{2/\alpha}}{\bar{\gamma}y^{2/\alpha}_{j}},\quad\sigma_{j}=\frac{\theta_{j}}{\sum_{l=1}^{K}\theta_{l}\Gamma(\beta_{l})\zeta^{-\beta_{l}}_{l}}$
$\displaystyle\theta_{j}=\Xi
w_{j}y^{\frac{\alpha(1+\mu)}{4}-\frac{2}{\alpha}m-1}_{j}I_{\mu-1}\big{(}2\sqrt{\kappa\mu
y_{j}}\big{)}.$ (16)

From (16), one can see that $\zeta_{j}\rightarrow 0$ when
$\bar{\gamma}\rightarrow\infty$.

##  V Performance Analysis using a MGS Model

###  V-A Outage Probability

The OP is defined as the probability of falling the values of the output SNR
below a predefined threshold value $\varphi$.

The OP, $P_{o}$, can be computed by [1, eq. (1.4)]

$\displaystyle P_{o}=F_{\gamma}(\varphi).$ (17)

where $F_{\gamma}(.)$ is provided in (6).

The asymptotic of the OP, $P^{\infty}_{o}$, can be analysed by (7), i.e.,
$P^{\infty}_{o}=F^{\infty}_{\gamma}(\varphi)$. Furthermore, the
$P^{\infty}_{o}$ may be closely represented as
$P^{\infty}_{o}\simeq\bar{\gamma}^{-G_{d}}$ whereby $G_{d}$ denotes the
diversity gain that demonstrates the increasing in the slope of the OP versus
$\bar{\gamma}$. Hence, plugging $\sigma_{j}$ of (16) in (7), one can notice
that the values of the $G_{d}$ of the MGS and double shadowed
$\alpha-\kappa-\mu$ are proportional to $\beta_{j}$ and $m$, respectively.

$\displaystyle\bar{\mathcal{A}}=1-\sum_{l=0}^{u-1}\sum_{i=0}^{l}{{l+u-1}\choose{l-i}}\frac{2^{-(l+i+u)}}{i!}(m_{s}-1)^{i}\sum_{j=1}^{K}\sigma_{j}(m_{s})_{\beta_{j}}\Gamma(\beta_{j}+i)U\bigg{(}\beta_{j}+i;i-m_{s}+1;\frac{m_{s}-1}{2\zeta_{j}}\bigg{)}.$
(34)

###  V-B Average Bit Error Probability

The ABEP can be evaluated by [1, eq. (9.11)]

$\displaystyle\bar{P}_{e}=\frac{1}{\pi}\int_{0}^{\frac{\pi}{2}}\mathcal{M}_{\gamma}\bigg{(}\frac{\rho}{\sin^{2}\phi}\bigg{)}d\phi.$
(18)

where $\rho=0.5$, $\rho=1$, and $\rho=0.715$ for coherent BFSK, BPSK, and BFSK
with minimum correlation, respectively.

Substituting (8) into (18) and employing the identity [19, eq.
(07.33.26.0004.01)], we have

$\displaystyle\bar{P}_{e}=\frac{1}{\pi\Gamma(m_{s})}\sum_{j=1}^{K}\frac{\sigma_{j}}{\zeta^{\beta_{j}}_{j}}\int_{0}^{\frac{\pi}{2}}G^{2,1}_{1,2}\bigg{[}\frac{(m_{s}-1)\rho}{\zeta_{j}\sin^{2}\phi}\bigg{|}\begin{matrix}1-\beta_{j}\\\
0,m_{s}\\\ \end{matrix}\bigg{]}d\phi.$ (19)

By employing the definition of the Meijer’s $G$-function [19, eq.
(07.34.02.0001.01)] and $t=\sin^{2}\phi$, (19) becomes

$\displaystyle\bar{P}_{e}$
$\displaystyle=\frac{1}{2\pi\Gamma(m_{s})}\sum_{j=1}^{K}\frac{\sigma_{j}}{\zeta^{\beta_{j}}_{j}}\int_{0}^{1}\frac{1}{\sqrt{(1-t)t}}\frac{1}{2\pi
i}\int_{\mathcal{L}}\Gamma(r)$
$\displaystyle\Gamma(m_{s}+r)\Gamma(\beta_{j}-r)\bigg{(}\frac{(m_{s}-1)\rho}{t\zeta_{j}}\bigg{)}^{-r}drdt.$
(20)

where $i=\sqrt{-1}$ and $\mathcal{L}$ is the suitable contours in the
$r$-plane from $\varrho-i\infty$ to $\varrho+i\infty$ with $\varrho$ is a
constant value.

Changing the order of the integrations of (20) and then using [17, eq.
(3.191.3)] for the linear integral, we obtain

$\displaystyle\bar{P}_{e}=\frac{1}{2\pi\Gamma(m_{s})}$
$\displaystyle\sum_{j=1}^{K}\frac{\sigma_{j}}{\zeta^{\beta_{j}}_{j}}\frac{1}{2\pi
i}\int_{\mathcal{L}}\Gamma(r)\Gamma(m_{s}+r)\Gamma(\beta_{j}-r)$
$\displaystyle
B(r+0.5,0.5)\bigg{(}\frac{(m_{s}-1)\rho}{\zeta_{j}}\bigg{)}^{-r}dr.$ (21)

Recalling the properties [17, eq. (8.384.1)/ eq. (8.338.2)] and using [19, eq.
(07.34.02.0001.01)], (21) is expressed as

$\displaystyle\bar{P}_{e}=\frac{1}{2\sqrt{\pi}\Gamma(m_{s})}\sum_{j=1}^{K}\frac{\sigma_{j}G^{2,1}_{2,2}\bigg{[}\frac{(m_{s}-1)\rho}{\zeta_{j}}\bigg{|}\begin{matrix}1-\beta_{j},1\\\
0,m_{s},0.5\\\ \end{matrix}\bigg{]}}{\zeta^{\beta_{j}}_{j}}.$ (22)

The asymptotic of the ABEP at high $\bar{\gamma}$ regime,
$\bar{P}^{\infty}_{e}$, can be deduced after inserting (9) in (18) and using
$t=\sin^{2}\phi$ as

$\displaystyle\bar{P}^{\infty}_{e}\simeq\frac{1}{\pi}\sum_{j=1}^{K}\frac{\sigma_{j}(m_{s})_{\beta_{j}}\Gamma(\beta_{j})}{[(m_{s}-1)\rho]^{\beta_{j}}}\int_{0}^{1}\frac{t^{\beta_{j}}}{\sqrt{(1-t)t}}dt.$
(23)

The integration of (23) is recorded in [17, eq. (3.191.3)]. Thus, after some
mathematical manipulations, this yields

$\displaystyle\bar{P}^{\infty}_{e}\simeq\frac{1}{2\sqrt{\pi}}\sum_{j=1}^{K}\frac{\sigma_{j}(m_{s})_{\beta_{j}}\Gamma(\beta_{j}+0.5)}{\beta_{j}[(m_{s}-1)\rho]^{\beta_{j}}}.$
(24)

It is evident from (24) that the $G_{d}$ of the MGS model is proportional to
$\beta_{j}$, i.e., $\bar{P}^{\infty}_{e}\simeq\bar{\gamma}^{-\beta_{j}}$.
Hence, the $G_{d}$ for the double shadowed $\alpha-\kappa-\mu$ fading is
proportional to $m$.

###  V-C Average Channel Capacity

According to Shannon’s theorem, the normalized ACC, $\bar{C}$, can be
determined by [8, eq. (26)]

$\bar{C}=\frac{1}{\ln(2)}\int_{0}^{\infty}\text{ln}(1+\gamma)f_{\gamma}(\gamma)d\gamma.$
(25)

Plugging (4) in (25) and making use the identity [19, eq. (01.04.26.0002.01)]
to write the natural logarithm in terms of the Meijer’s $G$-function, we
obtain

$\displaystyle\bar{C}=$
$\displaystyle\frac{(m_{s}-1)^{m_{s}}}{\ln(2)}\sum_{j=1}^{K}\sigma_{j}(m_{s})_{\beta_{j}}$
$\displaystyle\times\int_{0}^{\infty}\frac{\gamma^{\beta_{j}-1}}{(m_{s}-1+\zeta_{j}\gamma)^{\beta_{j}+m_{s}}}G^{1,2}_{2,2}\bigg{[}\gamma\bigg{|}\begin{matrix}1,1\\\
1,0\\\ \end{matrix}\bigg{]}d\gamma.$ (26)

Employing [17, eq. (7.811.5)], (26) can be expressed in exact closed-form as

$\displaystyle\bar{C}=\frac{1}{\ln(2)\Gamma(m_{s})}\sum_{j=1}^{K}\frac{\sigma_{j}}{\zeta_{j}^{\beta_{j}}}G^{2,3}_{3,3}\bigg{[}\frac{m_{s}-1}{\zeta_{j}}\bigg{|}\begin{matrix}1-\beta_{j},1,1\\\
m_{s},1,0\\\ \end{matrix}\bigg{]}.$ (27)

The asymptotic of the ACC for $\bar{\gamma}\rightarrow\infty$,
$\bar{C}^{\infty}$, can be evaluated via [8, eq. (28)]

$\displaystyle\bar{C}^{\infty}\simeq\frac{1}{\ln(2)}\frac{\partial}{\partial
n}\mathbb{E}[\gamma^{n}]\bigg{|}_{n=0}.$ (28)

Substituting (10) into (28), computing the partial derivative, and setting
$n=0$, $\bar{C}^{\infty}$ over MGS model is deduced as

$\displaystyle\bar{C}^{\infty}\simeq\sum_{j=1}^{K}\frac{\sigma_{j}\Gamma(\beta_{j})\bigg{[}\ln\big{(}\frac{m_{s}-1}{\zeta_{j}}\big{)}+\psi(\beta_{j})-\psi(m_{s})\bigg{]}}{\ln(2)\zeta_{j}^{\beta_{j}}}.$
(29)

where $\psi(.)$ is the digamma function [17, eq. (8.360.1)].

###  V-D Effective Capacity

In Shannon’s theorem, the ACC has been measured under perfect quality of
service (QoS). However, in the EC, the constraints of the QoS, such as, system
delay, are taken into consideration [5].

The EC can be calculated by [15, eq. (4)]

$\displaystyle\mathcal{R}=-\frac{1}{A}\text{log}_{2}\bigg{\\{}\int_{0}^{\infty}(1+\gamma)^{-A}f_{\gamma}(\gamma)d\gamma\bigg{\\}}.$
(30)

where $A\triangleq\theta TB/\mathrm{ln}(2)$ with $\theta$, $T$, and $B$ denote
the delay exponent, the time and the bandwidth of the channel.

Inserting (4) in (30) and employing [17, eq. (3.197.1)], the expression of the
EC over MGS distribution is yielded as

$\displaystyle\mathcal{R}$
$\displaystyle=-\frac{1}{A}\text{log}_{2}\bigg{\\{}\sum_{j=1}^{K}\frac{\sigma_{j}(m_{s})_{\beta_{j}}B(\beta_{j},m_{s}+A)}{(m_{s}-1)^{\beta_{j}}}$
$\displaystyle\times{{}_{2}F_{1}}\bigg{(}\beta_{j}+m_{s},\beta_{j};\beta_{j}+m_{s}+A;1-\frac{\zeta_{j}}{m_{s}-1}\bigg{)}\bigg{\\}}.$
(31)

Plugging (5) in (30) and recalling [17, eq. (3.194.3)], the asymptotic of the
EC at $\bar{\gamma}\rightarrow\infty$, $R^{\infty}$, is deduced as

$\displaystyle
R^{\infty}\simeq-\frac{1}{A}\text{log}_{2}\bigg{\\{}\sum_{j=1}^{K}\frac{\sigma_{j}(m_{s})_{\beta_{j}}}{(m_{s}-1)^{\beta_{j}}}B(\beta_{j},A-\beta_{j})\bigg{\\}}.$
(32)

###  V-E Average AUC of Energy Detection

The AUC is a single figure of merit that is suggested as an alternative
performance measure to the ROC curve [14].

The average AUC, $\bar{\mathcal{A}}$, can be computed by [20, eq. (20)/ eq.
(21)]

$\displaystyle\bar{\mathcal{A}}$
$\displaystyle=1-\sum_{l=0}^{u-1}\sum_{i=0}^{l}\frac{{{l+u-1}\choose{l-i}}}{2^{l+i+u}i!}\int_{0}^{\infty}\gamma^{i}e^{-\frac{\gamma}{2}}f_{\gamma}(\gamma)d\gamma.$
(33)

where ${a\choose b}=\frac{a!}{b!(a-b)!}$ is the binomial coefficient.

Plugging (6) in (33) and utilizing [18, eq. (13.2.5)], $\bar{\mathcal{A}}$ is
yielded as in (34) given at the top of this page.

The average AUC at high $\bar{\gamma}$ value, $\bar{\mathcal{A}}^{\infty}$,
can be evaluated after inserting (5) in (33) and invoking [17, eq. (3.381.4)]
as

$\displaystyle\bar{\mathcal{A}}^{\infty}\simeq
1-\sum_{l=0}^{u-1}\sum_{i=0}^{l}\sum_{j=1}^{K}\frac{\sigma_{j}(m_{s})_{\beta_{j}}{{l+u-1}\choose{l-i}}\Gamma(\beta_{j}+i)}{2^{l+u-\beta_{j}}(m_{s}-1)^{\beta_{j}}i!}.$
(35)

##  VI Analytical and Simulation Results

In this section, the numerical and the asymptotic results of the derived
performance measures are presented for different scenarios. To achieve MSE
$\leq 10^{-5}$, we have used $K=15$.

Figs. 1-4 demonstrate the OP for $\varphi=5$ dB, the ABEP for BPSK modulation,
the comparison between the normalized ACC and the EC for $A=3.5$, and the
average complementary of AUC (CAUC), $1-\bar{\mathcal{A}}$, for $u=3$ versus
$\bar{\gamma}$, respectively. Three cases for the shadowing index $m_{s}$,
namely, heavy ($m_{s}=1.5$), moderate ($m_{s}=5.5$), and light ($m_{s}=50$)
are analyzed.

From the provided results, it can be observed that the performance becomes
better when $\alpha$, $\kappa$ and/or $\mu$ increase. This is because the high
values of $\alpha$, $\kappa$, and $\mu$ indicate that the system tends to be
linear, the total power of the dominant components is less than that of the
scattered waves and large number of multipath clusters arrive at the receiver,
respectively. Additionally, the increasing in $m$ and/or $m_{s}$ means the
impacts of the first and/or the second shadowing on the received signal are
low. However, $m$ has the high effect on the performance when
$\bar{\gamma}\rightarrow\infty$ due to improving in the $G_{d}$.

In Fig. 3, one can note that the ACC is higher than the EC for the same
scenario. This refers to the impact of the system delay on the EC whereas its
ignored in the ACC. Moreover, the EC is related to the average AUC via $u$
[20]. This relationship explains how the low quality of the received signal by
the unlicensed user would reduce the detectability of the ED.

In all figures, a perfect agreement between the exact and the simulation
results as well as the asymptotic at high $\bar{\gamma}$ can be noticed which
proves the validation of our expressions.

Figure 1: OP versus normalized outage threshold for $\varphi=5$ dB. Figure 2:
ABEP for BPSK versus average SNR. Figure 3: Normalized ACC and EC with $A=3.5$
versus average SNR. Figure 4: Complementary AUC for $u=3$ versus average SNR.

##  VII Conclusions

In this paper, a MGS distribution was proposed as highly accurate approximate
unified representation where the shadowing impact was assumed to be an inverse
Nakagami-$m$ RV. This distribution was then applied to the double shadowed
$\alpha-\kappa-\mu$ fading channel. The exact and the asymptotic expressions
of the OP, the ABEP, the ACC, and the ER of the wireless communications
systems and the average AUC of ED over a MGS model were derived. The results
for different values of the fading parameters were presented. The expressions
of this work can be employed for many fading conditions, such as,
$\alpha-\kappa-\mathcal{F}$ fading with $m\rightarrow\infty$.

## References

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# The non-decreasing condition on $\mathbf{g}$-vectors

Mohamad Haerizadeh<EMAIL_ADDRESS>and  Siamak Yassemi<EMAIL_ADDRESS>
(Date: August 28, 2024)

###### Abstract.

The non-decreasing condition on g-vectors is introduced. Our study shows that
this condition is both necessary and sufficient to ensure that the generically
indecomposable direct summands of a given g-vector are linearly independent.
Additionally, we prove that for any finite dimensional algebra $\Lambda$,
under the non-decreasing condition, the number of generically indecomposable
irreducible components that appear in the decomposition of a given generically
$\tau$-reduced component is lower than or equal to $|\Lambda|$. This solves
the conjecture concerning the cardinality of component clusters by Cerulli-
Labardini-Schröer, in a reasonable generality. Lastly, we study numerical
criteria to check the wildness of g-vectors.

###### Key words and phrases:

g-vectors; generic decomposition; non-decreasing condition; generically
$\tau$-reduced component; variety of representations; Grothendieck group

###### Contents

  1. 1 Preliminaries
  2. 2 $\mathrm{g}$-vectors
  3. 3 Generically $\tau$-reduced components
  4. 4 Numerical criteria for tameness

## Introduction

Because of technical reasons concerning the Voigt’s Isomorphism, to generalize
results about general representations of quivers to bound quivers, one should
to work with general representations of the irreducible components of
varieties of representations. In this regard, inspired by the works of Kac
[Kac82] and Schofield [Sch92], Crawley-Boevey and Schröer [CBS02] construct
decomposition for irreducible components of varieties of representation, which
is an analogue of the Krull-Remak-Schmidt decomposition.

There also exists a parallel approach that relies on general presentations of
varieties of presentations. In their paper [DF15], Derksen and Fei show that
general presentations present general representations [DF15, Theorem 2.3].
Moreover, they introduce decomposition for varieties of presentations, which
is nicely related to decomposition of general representations. Indeed, they
introduce canonical decomposition of g-vectors [DF15, Theorem 4.4], which we
currently call generic decomposition.

The main results of our study concern the generic decomposition of g-vectors
and the decomposition of generically $\tau$-reduced components. More
precisely, we generalize some results of [CILFS14] and [AI21], about g-vectors
and generically $\tau$-reduced components by applying results of the papers
[Pla13]. In this paper, Plamondon assigns a generically $\tau$-reduced
component to each g-vector. The process gives us a surjective map from the
Grothendieck group to the set of generically $\tau$-reduced components of
$\operatorname{rep}(\Lambda)$ 3.9.

On the other hand, Cerulli-Irelli, Labardini-Fragoso and Schröer study the
decompositions of generically $\tau$-reduced components [CILFS14, Theorem
1.3]. This theorem together with Lemma 3.17 and Lemma 3.6 reveals that there
is a link between generic decomposition of g-vectors and decomposition of
generically $\tau$-reduced components.

######  Theorem 1 (3.18).

Let $\\{\mathcal{Z}_{1},\mathcal{Z}_{2},\cdots,\mathcal{Z}_{s}\\}$ be a set of
irreducible components of $\operatorname{rep}(\Lambda)$. Then the following
statements are equivalent.

  1. $(1)$

$\mathcal{Z}=\overline{\mathcal{Z}_{1}\oplus\mathcal{Z}_{2}\oplus\cdots\oplus\mathcal{Z}_{s}}$
is a generically $\tau$-reduced component.

  2. $(2)$

Each $\mathcal{Z}_{i}$ is a generically $\tau$-reduced component and

$\min\operatorname{hom}_{\Lambda}(\mathcal{Z}_{i},\tau\mathcal{Z}_{j})=0,$

for all $1\leq i\neq j\leq s$.

  3. $(3)$

Each $\mathcal{Z}_{i}$ is a generically $\tau$-reduced component and

$e(g^{\mathcal{Z}_{i}},g^{\mathcal{Z}_{j}})=0,$

for all $1\leq i\neq j\leq s$.

In this case, $g^{\mathcal{Z}}=g^{\mathcal{Z}_{1}}\oplus\cdots\oplus
g^{\mathcal{Z}_{s}}$ is the generic decomposition of $g^{\mathcal{Z}}$.

This connection leads us to provide an answer for the conjecture 3.14, by
Cerulli-Labardini-Schröer concerning component clusters and thus to generalize
[CILFS14, Theorem 6.1].

######  Theorem 2 (3.21).

Let $\mathcal{Z}$ be a generically $\tau$-reduced component. If
$g^{\mathcal{Z}}$ satisfies the non-decreasing condition (See Definition
2.26), then $|\mathcal{Z}|\leq|\Lambda|$. In particular, if $\Lambda$
satisfies the non-decreasing condition, then for all generically
$\tau$-reduced components $\mathcal{Z}$, we have $|\mathcal{Z}|\leq|\Lambda|$.

######  Theorem 3 (3.22).

Let $\mathcal{Z}$ be a generically $\tau$-reduced component. Then the
following statements are equivalent.

  * $(1)$

$g^{\mathcal{Z}}\in\mathcal{C}^{\circ}(\mathrm{P})$ for a $2$-term silting
complex $\mathrm{P}\in K^{b}(\operatorname{proj}\Lambda)$.

  * $(2)$

$|\mathcal{Z}|=|\Lambda|$ and $g^{\mathcal{Z}}$ satisfies the non-decreasing
condition.

More precisely, we show that under the non-decreasing condition, the
generically indecomposable direct summands of a given g-vector are linearly
independent. Then we apply the connecting theorem 3.18. Furthermore, we prove
that generically indecomposable direct summands in any generic decomposition
are linearly independent, exactly, when $\Lambda$ satisfies the non-decreasing
condition. Due to Proposition 2.27, the following theorem also generalize
[AI21, Proposition 6.7(b)].

######  Theorem 4 (2.30).

Let $g$ be a g-vector. If $g$ satisfies the non-decreasing condition, then
$\operatorname{ind}(g)$ is linearly independent. Moreover, the following
statements are equivalent.

  1. $(1)$

$\Lambda$ satisfies the non-decreasing condition.

  2. $(2)$

$\operatorname{ind}(g)$ is linearly independent, for each g-vector $g$.

Since determining whether a g-vector is tame or wild may be a little
complicated, examining some numerical criteria can be helpful. Therefore, we
will study some criteria in the last section.

######  Theorem 5 (4.3).

Consider a g-vector $g$ with no negative111We say $g$ is positive (negative,
respectively) if $[P^{g-}]=0$ ($[P^{g+}]=0$, respectively). direct summand.
Then $g$ is wild if and only if

$\min\operatorname{hom}_{\Lambda}(\mathcal{Z}_{g},\mathcal{Z}_{g})>\langle
g,d(g)\rangle.$

## Conventions

We consider a basic finite dimensional algebra $\Lambda$ over an algebraically
closed field $k$. So because of Morita equivalence, one can assume
$\Lambda=kQ/I$, where $(Q,I)$ is a bound quiver with $n$ vertices. For
simplicity, we assume that $Q$ is connected. It is well known that, up to
isomorphism, there are exactly $n$ simple modules
$\\{S_{(1)},S_{(2)},\cdots,S_{(n)}\\}$. Moreover, the Grothendieck group of
the category of finitely generated modules $K_{0}(\operatorname{mod}\Lambda)$
is a free abelian group of finite rank $n$ with a basis
$\\{[S_{(1)}],[S_{(2)}],\cdots,[S_{(n)}]\\}$. One can also consider the
Grothendieck group of category of finitely generated projective modules
$K_{0}(\operatorname{proj}\Lambda)$ which is also free abelian group of finite
rank $n$ with a basis $\\{[P_{(1)}],[P_{(2)}],\cdots,[P_{(n)}]\\}$, where
$P_{(i)}$ is a projective cover of $S_{(i)}$, for each $i=1,2,\cdots,n$. We
also fix the set $\\{e_{1},\cdots,e_{n}\\}$ as the complete set of primitive
orthogonal idempotents corresponding to $\\{P_{(1)},\cdots,P_{(n)}\\}$.

##  1\. Preliminaries

In this section, we review some geometric methods in representation theory of
finite dimensional algebras. Nonetheless, we refer non-expert readers to the
Crawley-Boevey’s note [CB20].

######  Convention 1.1.

All modules will be finitely generated left modules unless stated otherwise.
As well, product of arrows in a quiver algebra is something like the
composition of functions.

######  Definition 1.2.

Let $M$ be a finitely generated module. Then the dimension vector of $M$ is
$(\operatorname{dim}_{k}e_{1}M,\cdots,\operatorname{dim}_{k}e_{n}M)$. The
category of representations (modules, respectively) of dimension vector $d$ is
denoted by $\operatorname{rep}_{d}(\Lambda)$
($\operatorname{mod}_{d}(\Lambda)$, respectively).

######  Convention 1.3.

Let $d$ be dimension vector. $\operatorname{rep}_{d}(\Lambda)$ and
$\operatorname{mod}_{d}(\Lambda)$ are naturally equivalent. So we use them
freely instead of each other. Indeed, we may consider a module as a
representation and vice versa.

######  Notation 1.4.

For modules $X$ and $Y$, we set [style=itemize](2)
$\operatorname{hom}_{\Lambda}(X,Y)=\operatorname{dim}_{k}\operatorname{Hom}_{\Lambda}(X,Y)$
$\operatorname{end}_{\Lambda}(X)=\operatorname{dim}_{k}\operatorname{End}_{\Lambda}(X)$
$\operatorname{ext}^{1}_{\Lambda}(X,Y)=\operatorname{dim}_{k}\operatorname{Ext}^{1}_{\Lambda}(X,Y)$
$\operatorname{ext}^{1}_{\Lambda}(X,X)=\operatorname{dim}_{k}\operatorname{Ext}^{1}_{\Lambda}(X,X)$

######  Definition 1.5.

Let $\mathcal{X}$ be a topological space. A function
$f:\mathcal{X}\rightarrow\mathbb{Z}$ is upper semi-continuous, if for each
$t\in\mathbb{Z}$ the following subset is open in $\mathcal{X}$.

$\\{x\in\mathcal{X}\mid f(x)<t\\}$

######  Varieties of representations 1.6.

For a dimension vector $d$,

$\operatorname{rep}_{d}(Q)=\prod_{\alpha\in
Q_{1}}\operatorname{Hom}_{k}(k^{d_{s(\alpha)}},k^{d_{t(\alpha)}})$

is an affine variety whose points correspond to representations of $Q$.
Moreover, $\operatorname{rep}_{d}(\Lambda)$ is a Zariski-closed sub-variety of
$\operatorname{rep}_{d}(Q)$ that satisfies the relations in the ideal $I$.
As well, the space of representations of dimension $d\in\mathbb{N}$ is an
affine variety.

$\operatorname{rep}_{d}(\Lambda)=\operatorname{Hom}_{k\operatorname{-Alg}}(\Lambda,\operatorname{Mat}_{d}(k))$

######  Lemma 1.7.

Consider dimension vectors (or any positive integers) $d$ and $d^{\prime}$.
The following functions are all upper semi-continuous.

  * •

$\operatorname{hom}_{\Lambda}(-,?),\operatorname{ext}_{\Lambda}^{1}(-,?):\operatorname{rep}_{d}(\Lambda)\times\operatorname{rep}_{d^{\prime}}(\Lambda)\rightarrow\mathbb{Z}$

  * •

$\operatorname{end}_{\Lambda}(-),\operatorname{ext}_{\Lambda}^{1}(-,-):\operatorname{rep}_{d}(\Lambda)\rightarrow\mathbb{Z}$

###### Proof.

See [CBS02, Lemma 4.2]. ∎

######  Remark 1.8.

Each of the upper semi-continuous functions in the above example admits a
minimum value on given sub-varieties
$\mathcal{X}\subseteq\operatorname{rep}_{d}(\Lambda)$ and
$\mathcal{Y}\subseteq\operatorname{rep}_{d^{\prime}}(\Lambda)$.
[style=itemize]
$\min\operatorname{hom}_{\Lambda}(\mathcal{X},\mathcal{Y}):=\min\\{\operatorname{hom}_{\Lambda}(X,Y)\mid
X\in\mathcal{X},y\in\mathcal{Y}\\}$
$\min\operatorname{ext}_{\Lambda}(\mathcal{X},\mathcal{Y}):=\min\\{\operatorname{ext}_{\Lambda}(X,Y)\mid
X\in\mathcal{X},y\in\mathcal{Y}\\}$
$\min\operatorname{end}_{\Lambda}(\mathcal{X}):=\min\\{\operatorname{end}_{\Lambda}(X)\mid
X\in\mathcal{X}\\}$
$\min\operatorname{ext}_{\Lambda}(\mathcal{X}):=\min\\{\operatorname{ext}_{\Lambda}(X)\mid
X\in\mathcal{X}\\}$ So
$\\{(X,Y)\in\mathcal{X}\times\mathcal{Y}\mid\operatorname{hom}_{\Lambda}(X,Y)=\min\operatorname{hom}_{\Lambda}(\mathcal{X},\mathcal{Y})\\}$
is an open subset of $\mathcal{X}\times\mathcal{Y}$. The similar statements
holds for the other upper semi-continuous functions.

######  Definition 1.9.

Let $\mathcal{X}$ be a variety. Then “a general element of $\mathcal{X}$”
means an arbitrary element of a dense open subset of $\mathcal{X}$. For
example, one can say that for a general module
$M\in\operatorname{rep}_{d}(\Lambda)$,
$\operatorname{end}_{\Lambda}(M)=\min\operatorname{end}_{\Lambda}(\mathcal{X})$.

######  Definition 1.10.

A non-empty topological space $\mathcal{Z}$ is said to be irreducible, if it
can not be written as a union of two closed subsets, that is,
$\mathcal{Z}=\mathcal{C}_{1}\cup\mathcal{C}_{2}$ implies that
$\mathcal{Z}=\mathcal{C}_{1}$ or $\mathcal{Z}=\mathcal{C}_{2}$. Equivalently,
$\mathcal{Z}$ is non-empty and any non-empty open subset of $\mathcal{Z}$ is
dense.

######  Remark 1.11.

Let $\mathcal{Y}$ be an irreducible subset of a variety $\mathcal{X}$. Then
the following statements hold.

  * •

$\overline{\mathcal{Y}}$ is irreducible.

  * •

Any non-empty open subset of $\mathcal{Y}$ is irreducible.

  * •

For a morphisms of varieties $f:\mathcal{X}\rightarrow\mathcal{X}^{\prime}$,
$f(\mathcal{Y})$ is irreducible.

######  Definition-Theorem 1.12.

A variety $\mathcal{X}$ can be written as a finite union of maximal
irreducible (closed) subsets, in a unique way. These maximal irreducible
(closed) subsets are called irreducible components.

######  Remark 1.13.

Let $\mathcal{X}$ be a variety. Then any irreducible subset of $\mathcal{X}$
is contained in a (not necessarily unique) irreducible component.

######  Definition 1.14.

Let $f:\mathcal{X}\rightarrow\mathcal{Y}$ is a morphism of varieties. Then $f$
is said to be dominant, if the image is dense in $\mathcal{Y}$.

######  Remark 1.15.

By definition, a composition of dominant morphisms is dominant as well.

######  Definition 1.16.

Let $\mathcal{X}$ be a variety. A constructible subset is a finite union of
locally closed subsets.

######  Lemma 1.17.

Let $\mathcal{X}$ be an irreducible variety and $\mathcal{Y}$ a constructible
subset. If $\overline{\mathcal{Y}}=\mathcal{X}$, then there is an open subset
$\mathcal{U}$ of $\mathcal{X}$ such that $\mathcal{U}\subseteq\mathcal{Y}$.

######  Chevalley’s Constructible Theorem 1.18.

Let $f:\mathcal{X}\rightarrow\mathcal{Y}$ be a morphism of varieties. Then
$f(\mathcal{X})$ is constructible. Therefore, if $\mathcal{Y}$ is irreducible
and $f$ is dominant, then a general element of $\mathcal{Y}$ is the image of a
general element of $\mathcal{X}$.

######  Definition 1.19.

Let $\mathcal{X}$ be a variety. Then the dimension of $\mathcal{X}$ is defined
as follows.

$\displaystyle\operatorname{dim}\mathcal{X}:=\sup\\{l\in\mathbb{N}\mid\textit{$\exists\mathcal{Z}_{0}\subset\mathcal{Z}_{1}\subset\cdots\subset\mathcal{Z}_{l}$
of irreducible closed subsets of $\mathcal{X}$}\\}$

######  Proposition 1.20.

Let $\mathcal{X}$ be a variety and $\mathcal{Y}\subseteq\mathcal{X}$ be
locally closed. Then th following statements hold.

  * •

$\mathcal{X}$ is of finite dimension.

  * •

$\operatorname{dim}\mathcal{Y}\leq\operatorname{dim}\mathcal{X}$. Especially,
if $\mathcal{Y}$ is proper closed and $\mathcal{X}$ is irreducible, then
$\operatorname{dim}\mathcal{Y}<\operatorname{dim}\mathcal{X}$.

  * •

Let $\mathcal{X}$ be irreducible. Then
$\operatorname{dim}\mathcal{X}=\operatorname{dim}\mathcal{Y}$ if and only if
$\overline{\mathcal{Y}}=\mathcal{X}$. In this case, $\mathcal{Y}$ is open (and
thus dense).

######  Remark 1.21.

A finite dimensional vector space $\mathcal{X}$ over $k$ is an irreducible
affine variety and
$\operatorname{dim}\mathcal{X}=\operatorname{dim}_{k}\mathcal{X}$. In
particular, for modules $X$ and $Y$, $\operatorname{Hom}_{\Lambda}(X,Y)$ is an
irreducible affine variety of dimension $\operatorname{hom}_{\Lambda}(X,Y)$.

######  Definition 1.22.

For $l\in\mathbb{N}$, consider the variety of $l\times l$ matrices
$\operatorname{Mat}_{l}(k)$. The general linear group
$\operatorname{GL}_{l}(k):=\\{A\in\operatorname{Mat}_{l}(k)\mid\det A\neq
0\\}$ is an open sub-variety of $\operatorname{Mat}_{l}(k)$. Moreover, for a
dimension vector $d=(d_{1},\cdots,d_{n})$, we define an affine algebraic
group333An algebraic group is a group which is also a variety with the
property that the multiplication and inversion maps are morphisms of
varieties.

$\operatorname{GL}_{d}(k):=\prod_{i=1}^{n}\operatorname{GL}_{d_{i}}(k).$

######  Actions by conjugation 1.23.

Consider a dimension vector $d$. $\operatorname{GL}_{d}(k)$ acts on
$\operatorname{rep}_{d}(\Lambda)$ by conjugation as follows:
For
$\mathfrak{g}=(\mathfrak{g}_{1},\cdots,\mathfrak{g}_{n})\in\operatorname{GL}_{d}(k)$
and $M\in\operatorname{rep}_{d}(\Lambda)$,

  * •

$(\mathfrak{g}\cdot M)_{i}=M_{i}$, for each vertex $i$ of $Q$, and

  * •

$(\mathfrak{g}\cdot M)_{\alpha}=\mathfrak{g}_{t(\alpha)}\circ
M_{\alpha}\circ\mathfrak{g}_{s(\alpha)}^{-1}$, for each arrow $\alpha$ of $Q$.

Now, consider representations $M$ and $M^{\prime}$ with dimension vectors $d$
and $d^{\prime}$, respectively.
$\operatorname{GL}_{d}(k)\times\operatorname{GL}_{d^{\prime}}(k)$ acts on
$\operatorname{Hom}_{\Lambda}(M,M^{\prime})$ by conjugation as follows:
Consider
$\mathfrak{g}=(\mathfrak{g}_{1},\cdots,\mathfrak{g}_{n})\in\operatorname{GL}_{d}(k)$,
$\mathfrak{g}^{\prime}=(\mathfrak{g}_{1}^{\prime},\cdots,\mathfrak{g}_{n}^{\prime})\in\operatorname{GL}_{d^{\prime}}(k)$
and $a=(a_{1},\cdots,a_{n})\in\operatorname{Hom}_{\Lambda}(M,M^{\prime})$.
Then we define

$((\mathfrak{g},\mathfrak{g}^{\prime})\cdot
a)_{i}=\mathfrak{g^{\prime}}_{i}\circ a_{i}\circ\mathfrak{g}_{i}^{-1}$

for each $i\in Q_{0}$.

######  Orbits 1.24.

Let $\mathfrak{G}$ be an (affine) algebraic group and $\mathcal{X}$ be a
variety. Assume that $\mathfrak{G}$ acts on $\mathcal{X}$. For
$x\in\mathcal{X}$, the following statements hold.

  * •

The orbit $\mathcal{O}_{x}$ is locally closed and irreducible subset of
$\mathcal{X}$.

  * •

The stabilizer
$\operatorname{Stab}_{\mathfrak{G}}(x):=\\{\mathfrak{g}\in\mathfrak{G}\mid\mathfrak{g}\cdot
x=x\\}$ is a closed subset of $\mathfrak{G}$.

  * •

$\operatorname{dim}\mathcal{O}_{x}=\operatorname{dim}\mathfrak{G}-\operatorname{dim}\operatorname{Stab}_{\mathfrak{G}}(x)$.

Consider a dimension vector $d=(d_{1},\cdots,d_{n})$ and the action of
$\operatorname{GL}_{d}(k)$ on $\operatorname{rep}_{d}(\Lambda)$ by
conjugation. Then for a representation $M\in\operatorname{rep}_{d}(\Lambda)$,

$\operatorname{dim}\mathcal{O}_{M}=\operatorname{dim}\operatorname{GL}_{d}(k)-\operatorname{dim}\operatorname{Aut}_{\Lambda}(M).$

But
$\operatorname{Aut}_{\Lambda}(M)=\operatorname{GL}_{\mathbf{d}}(k)\cap\operatorname{End}_{\Lambda}(M)\subseteq\operatorname{Mat}_{\mathbf{d}}(k)$,
where $\mathbf{d}=\Sigma_{i=1}^{n}d_{i}$. Hence
$\operatorname{Aut}_{\Lambda}(M)$ is an open sub-variety of
$\operatorname{End}_{\Lambda}(M)$. Thus
$\operatorname{dim}\operatorname{Aut}_{\Lambda}(M)=\operatorname{end}_{\Lambda}(M)$.
Therefore,

$\operatorname{dim}\mathcal{O}_{M}=\operatorname{dim}\operatorname{GL}_{d}(k)-\operatorname{end}_{\Lambda}(M)=\langle
d,d\rangle-\operatorname{end}_{\Lambda}(M).$

So by upper semi-continuity of $\operatorname{end}_{\Lambda}(-)$, it is easy
to see that

$\operatorname{codim}\mathcal{O}_{-}:\operatorname{rep}_{d}(\Lambda)\rightarrow\mathbb{Z}$

is upper semi-continuous. Immediately, for an irreducible component
$\mathcal{Z}$ of $\operatorname{rep}_{d}(\Lambda)$,

$\\{X\in\mathcal{Z}\mid\operatorname{codim}_{\mathcal{Z}}\mathcal{O}_{X}=\min_{M\in\mathcal{Z}}\operatorname{codim}_{\mathcal{Z}}\mathcal{O}_{M}\\}$

is an open dense subset of $\mathcal{Z}$.

######  Theorem 1.25.

Consider a dimension vector (or any positive integer) $d$. All irreducible
components of $\operatorname{rep}_{d}(\Lambda)$ are
$\operatorname{GL}_{d}(k)$-stable, that is, they are closed under the action
of $\operatorname{GL}_{d}(k)$ on $\operatorname{rep}_{d}(\Lambda)$.

###### Proof.

Consider an irreducible component $\mathcal{Z}$ of
$\operatorname{rep}_{d}(\Lambda)$. Since $\operatorname{GL}_{d}(k)$ and
$\mathcal{Z}$ are irreducible, the image of the morphism
$\operatorname{GL}_{d}(k)\times\mathcal{Z}\rightarrow\operatorname{rep}_{d}(\Lambda)$
is irreducible, as well. So it must be contained in an irreducible component
of $\operatorname{rep}_{d}(\Lambda)$. Therefore,
$\operatorname{GL}_{d}(k)\cdot\mathcal{Z}=\mathcal{Z}$. ∎

######  Corollary 1.26.

Consider a dimension vector (or any positive integer) $d$.
$\operatorname{rep}_{d}(\Lambda)$ is a union of finite number of
$\operatorname{GL}_{d}(k)$-stable irreducible components.

##  2\. $\mathrm{g}$-vectors

In this section, we introduce the non-decreasing condition on g-vectors and
finite dimensional algebras. We prove that under the non-decreasing condition
on $g$, the number of the generically indecomposable direct summands of $g$
never decreases, after multiplications. Then we will see that if $g$ satisfies
the non-decreasing condition, then $\operatorname{ind}(g)$ is linearly
independent. Moreover, we show that the converse of the above statements also
hold, when $\operatorname{ind}(tg)$ is linearly independent, for each
$t\in\mathbb{N}$.

######  Convention 2.1.

To simplify some expressions, in this paper, we always assume that
$K_{0}(\operatorname{proj}\Lambda)=K_{0}(K^{b}(\operatorname{proj}\Lambda))$.
By abuse of notation, the elements of
$K_{0}(\operatorname{mod}\Lambda)_{\mathbb{R}}$ and
$K_{0}(\operatorname{proj}\Lambda)_{\mathbb{R}}$ are also assumed to be
vectors in $\mathbb{R}^{n}$.

######  Definition 2.2.

An element of $K_{0}(\operatorname{proj}\Lambda)$ is called g-vector. Consider
a complex $\mathrm{P}=\cdots\rightarrow P^{-1}\rightarrow P^{0}\rightarrow
P^{1}\rightarrow\cdots\in K^{b}(\operatorname{proj}\Lambda)$. We set

$g^{\mathrm{P}}:=\sum_{i\in\mathbb{Z}}(-1)^{i}[P^{i}]\in
K_{0}(\operatorname{proj}\Lambda)\cong\mathbb{Z}^{n}$

as the g-vector of $\mathrm{P}$. Moreover, if $\mathrm{P}=P^{-1}\rightarrow
P^{0}$ is a minimal projective presentation of a module $M$, then we assign a
g-vector to $M$ as follows:

$g^{M}:=g^{\mathrm{P}}\in K_{0}(\operatorname{proj}\Lambda)$

######  Remark 2.3.

Let $M$ be a module and $g^{M}=(g_{1},g_{2},\cdots,g_{n})$. Then for each
$1\leq i\leq n$,

$g_{i}=\operatorname{hom}_{\Lambda}(M,S_{(i)})-\operatorname{ext}^{1}_{\Lambda}(M,S_{(i)}).$

Moreover, by [GLFS23, Corollary 1.3],
$-g_{i}:\operatorname{rep}_{d}(\Lambda)\rightarrow\mathcal{Z}$ is upper semi-
continuous, for any dimension vector (or any positive integer) $d$.

######  Convention 2.4.

For projective modules $P^{-1}$ and $P^{0}$, one can consider each morphism in
$\operatorname{Hom}_{\Lambda}(P^{-1},P^{0})$ as a two term complex in
$K^{b}(\operatorname{proj}\Lambda)$.

######  Remark 2.5.

Let $g$ be a g-vector. Then there exist unique finitely generated projective
modules $P^{g+}$ and $P^{g-}$ without any non-zero common direct summands such
that $g=[P^{g+}]-[P^{g-}]$.

######  Notation 2.6.

Let $g$ be a g-vector. $\operatorname{Hom}_{\Lambda}(g)$ stands for
$\operatorname{Hom}_{\Lambda}(P^{g-},P^{g+})$.

E-invariant firstly defined for finite dimensional decorated representations
of Jacobian algebras [DWZ10] to solve several conjectures in cluster theory.
Then it was generalized for finite dimensional modules in [AIR14] and used to
generalized some results of the preceding paper. As well, in order to
construct decomposition of presentation spaces, Derksen and Fei [DF15]
introduced E-invariants for g-vectors.

######  Definition 2.7.

Let $g$ and $h$ be g-vectors. The E-invariant of
$(a,b)\in\operatorname{Hom}_{\Lambda}(g)\times\operatorname{Hom}_{\Lambda}(h)$
is defined as follows:

$e(a,b):=\operatorname{hom}_{K^{b}(\operatorname{proj}\Lambda)}(a,b[1])$

Moreover, the E-invariant of $(g,h)$ is defined as follows:

$e(g,h):=\min\\{e(a,b)\mid
a\in\operatorname{Hom}_{\Lambda}(g),b\in\operatorname{Hom}_{\Lambda}(h)\\}$

Since
$e(?,-):\operatorname{Hom}(g)\times\operatorname{Hom}(h)\rightarrow\mathbb{Z}$
is upper semi-continuous [DF15, Corollary 3.7], $e(g,h)=e(a,b)$ for a general
element $(a,b)\in\operatorname{Hom}(g)\times\operatorname{Hom}(h)$.

######  Direct sums of g-vectors 2.8.

Consider g-vectors $g$ and $h$. Then $e(g,h)=0=e(h,g)$, if and only if a
general element $a\in\operatorname{Hom}_{\Lambda}(g+h)$ can be written as a
direct sum $a=b\oplus c$, where $b\in\operatorname{Hom}_{\Lambda}(g)$ and
$c\in\operatorname{Hom}_{\Lambda}(h)$. In this case, we say $g$ and $h$ are
direct summands of $g+h$, and denote $g+h=g\oplus h$.

###### Proof.

It follows from [DF15, Corollary 4.2]. ∎

######  Definition 2.9.

Let $g$ be a g-vector. It is said to be tame, if $2g=g\oplus g$. Otherwise, it
is called wild.

It is easy to see that g-vectors of $\tau$-rigid modules are tame. Moreover,
by [PYK21, Proof of Theorem 3.8], if $\Lambda$ is tame, then it is E-tame,
that is, all g-vectors are tame [AI21, Definition 6.3].

######  Remark 2.10.

Let $g$ and $h$ be g-vectors. If $g+h=g\oplus h$, then so is $tg+sh=tg\oplus
sh$, for $t,s\in\mathbb{N}$. In particular, if $g$ is tame, then so is $tg$
for all $t\in\mathbb{N}$.

######  Remark 2.11.

Let $g$ and $h$ be g-vectors. If $g$ is positive or $h$ negative, then
$e(g,h)=0$. In particular, negative and positive g-vectors are tame.

######  Proposition 2.12.

Consider g-vectors $g$ and $h$. $g+h=g\oplus h$ if and only if there exist
$a\in\operatorname{Hom}_{\Lambda}(g)$ and
$b\in\operatorname{Hom}_{\Lambda}(h)$ such that $e(a,b)=0=e(b,a)$. In
particular, $g$ is tame if and only if there exist non-zero
$a,a^{\prime}\in\operatorname{Hom}(g)$ with
$e(a,a^{\prime})=0=e(a^{\prime},a)$.

###### Proof.

Assume that $g+h=g\oplus h$. Then one can find
$a^{\prime\prime}\in\operatorname{Hom}_{\Lambda}(g)$ and
$b\in\operatorname{Hom}_{\Lambda}(h)$ with $e(b,a^{\prime\prime})=0$. By
[AI21, Lemma 3.10], the following subsets of $\operatorname{Hom}_{\Lambda}(g)$
are open.

$\mathcal{U}=\\{a^{\prime}\in\operatorname{Hom}_{\Lambda}(g)\mid\operatorname{Coker}(b)\in\prescript{\perp}{}{\operatorname{Ker}(\nu
a^{\prime})}\\},$
$\mathcal{V}=\\{a^{\prime}\in\operatorname{Hom}_{\Lambda}(g)\mid\operatorname{Ker}(\nu
b)\in\operatorname{Coker}(a^{\prime})^{\perp}\\}$

Since $\operatorname{Hom}_{\Lambda}(g)$ is irreducible, there is
$a\in\mathcal{U}\cap\mathcal{V}$. So by [AI21, Proposition 3.11],
$e(a,b)=0=e(b,a)$. ∎

######  Remark 2.13.

Assume that for g-vectors $g$ and $g^{\prime}$, $g+g^{\prime}=g\oplus
g^{\prime}$. Then for a g-vector $h$, we have

$e(g\oplus g^{\prime},h)=e(g,h)+e(g^{\prime},h)$ $e(h,g\oplus
g^{\prime})=e(h,g)+e(h,g^{\prime})$

Therefore, one can see that $g\oplus g^{\prime}$ is tame if and only if both
of $g$ and $g^{\prime}$ are tame.

######  Definition 2.14.

Consider a g-vector $g$. It is said to be generically indecomposasble, if a
general element of $\operatorname{Hom}_{\Lambda}(g)$ is indecomposable.

######  Generic decompositions of g-vectors 2.15.

Let $g$ be a g-vector. Then there exist unique (up to reordering) generically
indecomposable g-vectors $g_{1},g_{2},\cdots,g_{s}$ such that

$g=g_{1}\oplus g_{2}\oplus\cdots\oplus g_{s}.$

This decomposition is called the generic decomposition of $g$.

###### Proof.

See [DF15, Theorem 4.4] or [Pla13, Theorem 2.7]. ∎

######  Remark 2.16.

By the above assumption, for a general $a\in\operatorname{Hom}_{\Lambda}(g)$,
there exist general (indecomposable) projective presentations
$a_{i}\in\operatorname{Hom}_{\Lambda}(g_{i}),1\leq i\leq s$, such that
$a=a_{1}\oplus\cdots\oplus a_{s}$.

To study algebras through g-vectors, we concentrate on the building blocks of
g-vectors, namely, generically indecomposable direct summands. So in the rest
of this section, we study some conditions on g-vectors and generic
decompositions of g-vectors.

######  Notation 2.17.

Let $g$ be a g-vector. The set of generically indecomposable direct summands
of $g$ is denoted by $\operatorname{ind}(g)$. As well, we set
$\operatorname{smd}(g)$ (respectively, $\operatorname{add}(g)$) for the set of
all direct summands (respectively, the set of all direct summands of direct
sums) of $g$.

######  Remark 2.18.

Let $g$ be a tame g-vector and $g=g_{1}\oplus g_{2}\oplus\cdots\oplus g_{s}$
be its generic decomposition. Then it is easy to see that $tg=tg_{1}\oplus
tg_{2}\oplus\cdots\oplus tg_{s}$ is also the generic decomposition of $tg$,
for all $t\in\mathbb{N}$. Therefore,
$|\operatorname{ind}(g)|=|\operatorname{ind}(tg)|$, for all $t\in\mathbb{N}$.
Furthermore, by uniqueness of generic decompositions,
$\\{g_{1},\cdots,g_{s}\\}$ is linearly independent [AI21, Proposition 6.7(b)].

######  Remark 2.19.

Let $\mathrm{P}\in K^{b}(\operatorname{proj}\Lambda)$ be a $2$-term pre-
silting complex. Then we set $\mathcal{C}^{\circ}(\mathrm{P})$ (respectively,
$\mathcal{C}(\mathrm{P})$), for the open cone (respectively, closed cone)
generated by the generically indecomposable direct summands of
$g^{\mathrm{P}}$. Now, consider the wall and chamber structure of $\Lambda$
defined in [BST19, Definition 3.3]. By [Asa21, Theorem 3.17],

$K_{0}(\operatorname{proj}\Lambda)_{\mathbb{Q}}\setminus\operatorname{Walls}=\coprod_{{\mbox{$\displaystyle\mathrm{P}\in\operatorname{2-silt}\Lambda$}}}\mathcal{C}^{\circ}(\mathrm{P})_{\mathbb{Q}}.$

  * •

If a g-vector $g\in\mathcal{C}(\mathrm{P})$ for some $2$-term silting complex
$\mathrm{P}$, then according to [AI12, Theorem 2.27], it is tame and thus
$\operatorname{ind}(g)$ is linearly independent. Moreover, if
$g\in\mathcal{C}^{\circ}(\mathrm{P})$, then $|\operatorname{ind}(g)|=n$.

  * •

In the case that

$g\in\operatorname{Walls}\setminus\coprod_{{\mbox{$\displaystyle\mathrm{P}\in\operatorname{2-silt}\Lambda$}}}\mathcal{C}(\mathrm{P})$

we don’t know if $\operatorname{ind}(g)$ is linearly independent, in general.
In addition, it is probable for $g$ to lose wildness or even it may happen
that $|\operatorname{ind}(tg)|\neq|\operatorname{ind}(g)|$, for some
$t\in\mathbb{N}$. In [DF15, Question 4.7], it was asked that if there is a
generically indecomposable wild g-vector $g$ such that $tg$ is not so, for
some $t\in\mathbb{N}$.

Hence, it is reasonable to study about the following questions.

  1. (1)

Is there any generically indecomposable wild g-vector $g$ such that $tg$ is
not indecomposable, for some $t\in\mathbb{N}$?

  2. (2)

Is there any wild g-vector $g$ such that $tg$ is tame, for some
$t\in\mathbb{N}$?

For the first question, in [AI21, Example 5.9], there is an example of
generically indecomposable wild g-vector $g$ such that $tg$ is not
indecomposable for all integer $t\geq 2$. So
$|\operatorname{ind}(g)|<|\operatorname{ind}(tg)|$, for all $t\geq 2$. But
there is no answer for the second question, up to our knowledge.

######  Definition 2.20.

Let $g$ be a g-vector.

  * •

[AI21, Definition 5.1] $g$ satisfies the ray condition, if $th$ is generically
indecomposable (thus wild), for all $t\in\mathbb{N}$ and wild
$h\in\operatorname{ind}(g)$.

  * •

$g$ satisfies the regularity condition, if $th$ is wild, for all
$t\in\mathbb{N}$ and wild $h\in\operatorname{ind}(g)$.

We say $\Lambda$ satisfies the ray condition (regularity condition,
respectively) if all g-vectors satisfy the ray condition (regularity
condition, respectively).

######  Remark 2.21.

Let $g$ be a g-vector. Then the following statements are equivalent.

  1. $(1)$

$g$ satisfies the ray condition.

  2. $(2)$

$|\operatorname{ind}(h)|=|\operatorname{ind}(th)|$, for all
$h\in\operatorname{smd}(g)$ and $t\in\mathbb{N}$.

In particular, $\Lambda$ satisfies the ray condition if and only if
$|\operatorname{ind}(g)|=|\operatorname{ind}(tg)|$, for all g-vector $g$ and
$t\in\mathbb{N}$.

######  Definition 2.22.

Consider a g-vector $g$. We define

  * •

$D_{g}:=\\{h\in K_{0}(\operatorname{proj}\Lambda)\mid\exists
t\in\mathbb{N},g+th=g\oplus th\\}$

  * •

$D_{\mathbb{N}g}:=\bigcup_{r\in\mathbb{N}}D_{rg}$.

######  Remark 2.23.

Let $g$ be a g-vector. Then $g\in D_{\mathbb{N}g}$ if and only if $tg$ is tame
for some $t\in\mathbb{N}$. In particular, if $g$ is tame, then $g\in D_{g}$.

######  Remark 2.24.

For a g-vector $g$, the following conditions are equivalent.

  1. $(1)$

$g$ satisfies the regularity condition.

  2. $(2)$

$h\notin D_{\mathbb{N}h}$, for all $t\in\mathbb{N}$ and wild
$h\in\operatorname{smd}(g)$.

In particular, $\Lambda$ satisfies the regularity condition if and only if
$g\notin D_{g}$ for all wild g-vector $g$.

######  Proposition 2.25.

Let $g$ be a g-vector. Then the following statements hold.

  1. $(1)$

If $g$ is wild and contained in $D_{g}$, then $|\operatorname{ind}(g\oplus
tg)|>|\operatorname{ind}(t(g\oplus tg))|$ for some $t\in\mathbb{N}$.

  2. $(2)$

If $|\operatorname{ind}(tg)|<|\operatorname{ind}(g)|$, for some
$t\in\mathbb{N}$, then there is a wild g-vector $h\in\operatorname{ind}(g)$
with $h\in D_{h}$.

###### Proof.

(1) Let $t=\min\\{r\in\mathbb{N}\mid(1+r)g=g\oplus rg\\}$. One can see that
$tg$ is tame and so by Remark 2.18, $|\operatorname{ind}(t(g\oplus
tg))|=|\operatorname{ind}((1+t)tg)|=|\operatorname{ind}(tg)|$. But
$|\operatorname{ind}(g\oplus tg)|\geq|\operatorname{ind}(tg)|$. If the
equality holds, then $\operatorname{ind}(g)\subseteq\operatorname{ind}(tg)$.
In this case, $g$ is a direct summand of $tg$ and it contradicts the
minimality of $t$. Therefore, $|\operatorname{ind}(g\oplus
tg)|>|\operatorname{ind}(t(g\oplus tg))|$.

(2) Consider the generic decomposition $g=h_{1}\oplus\cdots\oplus h_{r}\oplus
g_{1}\oplus\cdots\oplus g_{s}$, where $h_{i}$ is tame and $g_{i}$ is wild.
Suppose that for each $1\leq i\leq s$, there exits
$g^{\prime}_{i}\in\operatorname{ind}(tg_{i})\setminus\operatorname{ind}(tg-
tg_{i})$. Then

$h^{\prime}=h_{1}\oplus\cdots\oplus h_{r}\oplus
g^{\prime}_{1}\oplus\cdots\oplus g^{\prime}_{s}\in\operatorname{smd}(tg).$

So
$|\operatorname{ind}(g)|=|\operatorname{ind}(h^{\prime})|\leq|\operatorname{ind}(tg)|$.
It contradicts the assumption of $(2)$. Hence, there exists $1\leq i\leq s$
such that $\operatorname{ind}(tg_{i})\subseteq\operatorname{ind}(tg-tg_{i})$.
So $g_{i}\in D_{g_{i}}$. ∎

######  Definition 2.26.

Let $g$ be a g-vector. Then $g$ satisfies the non-decreasing condition, if
$h\notin D_{h}$, for all wild $h\in\operatorname{ind}(g)$. We say $\Lambda$
satisfies the non-decreasing condition, if all g-vectors satisfy the non-
decreasing condition.

######  Proposition 2.27.

The ray condition implies the regularity condition and the regularity
condition implies the non-decreasing condition. In particular, tame g-vectors
satisfy all of the three conditions.

###### Proof.

Consider a g-vector $g$. Let $h\in\operatorname{ind}(g)$ be a wild g-vector.
If $th$ is tame, then $2th=th\oplus th$. Hence, $2th$ is not generically
indecomposable. Therefore, the ray condition implies the regularity condition.
The other parts are also clear. ∎

######  Question 2.28.

Is there any finite dimensional algebra that does not satisfy the non-
decreasing condition?

Proposition 2.25 and the following observation are our main reasons to call
this condition “non-decreasing”.

######  Theorem 2.29.

$\Lambda$ satisfies the non-decreasing condition if and only if

$|\operatorname{ind}(g)|\leq|\operatorname{ind}(tg)|,$

for all g-vector $g$ and $t\in\mathbb{N}$. Indeed, for a g-vector $g$, the
following statements are equivalent.

  1. $(1)$

$tg$ satisfies the non-decreasing condition, for all $t\in\mathbb{N}$.

  2. $(2)$

$|\operatorname{ind}(h)|\leq|\operatorname{ind}(th)|$, for all
$t\in\mathbb{N}$ and wild $h\in\operatorname{add}(g)$.

###### Proof.

It is a direct consequence of Proposition 2.25. ∎

######  Theorem 2.30.

Let $g$ be a g-vector. If $g$ satisfies the non-decreasing condition, then
$\operatorname{ind}(g)$ is linearly independent and thus
$|\operatorname{ind}(g)|\leq n$. Furthermore, the following statements are
equivalent.

  1. $(1)$

$\operatorname{ind}(tg)$ is linearly independent, for all $t\in\mathbb{N}$.

  2. $(2)$

$tg$ satisfies the non-decreasing condition, for all $t\in\mathbb{N}$.

In particular, $\Lambda$ satisfies the non-decreasing condition if and only if
$\operatorname{ind}(g)$ is linearly independent, for each g-vector $g$.

###### Proof.

$(2)\Rightarrow(1):$ Consider the generic decomposition $g=g_{1}\oplus
g_{2}\oplus\cdots\oplus g_{s}$. We may assume that $g_{i}\neq g_{j}$, for all
$1\leq i\neq j\leq s$. Suppose that there exist non-empty subsets $I,J$ of
$\\{1,2,\cdots,s\\}$ with an empty intersection, such that

$\bigoplus_{i\in I}a_{i}g_{i}=\bigoplus_{j\in J}b_{j}g_{j}$

for some $a_{i},b_{j}\in\mathbb{N}$. Since the right hand side is in
$D_{g_{i}}$ for $i\in I$, we conclude that $g_{i}\in D_{g_{i}}$. So $g_{i}$ is
tame. Similarly, for all $l\in I\cup J$, $g_{l}$ is tame. Therefore, by the
uniqueness of generic decompositions, we deduce that $\\{g_{i}\mid i\in
I\\}=\\{g_{j}\mid j\in J\\}$. It contradicts $I\cap J\neq\emptyset$. Thus
$\operatorname{ind}(g)$ is linearly independent.

$(1)\Rightarrow(2):$ Suppose that $(2)$ does not hold. We may assume that
there is a wild g-vector $h\in\operatorname{ind}(g)$ with $h\in D_{h}$. Set

$t=\min\\{t^{\prime}\in\mathbb{N}\mid h+t^{\prime}h=h\oplus t^{\prime}h\\}.$

Because of the minimality of $t$, one can check that $h$ is not a direct
summand of $th$. Consider a generic decomposition $th=h_{1}\oplus
h_{2}\oplus\cdots\oplus h_{s}$. Then $h+th=h\oplus h_{1}\oplus\cdots\oplus
h_{s}$ is a generic decomposition. But
$\operatorname{ind}(h+th)=\\{h,h_{1},\cdots,h_{s}\\}$ is not linearly
independent. Therefore, $\operatorname{ind}((1+t)g)$ is not linearly
independent. It is a contradiction. ∎

######  Problem 2.31.

Let $g$ be a g-vector. Is it possible to remove “t” from the statements $(1)$
and $(2)$ in Theorem 2.30?666This problem was posed by Toshiya Yurikusa in our
email-discussion.

##  3\. Generically $\tau$-reduced components

In this section, we work on direct sums of generically $\tau$-reduced
components and especially we study the relation between generic decompositions
of g-vectors and decompositions of generically $\tau$-reduced components. At
the first step, we need to look into Crawley-Boevey-Schroer’s decomposition
theorem [CBS02, Theorem 1.1]. Then, we will peruse its $\tau$-reduced version
proved in [CILFS14].

######  Proposition 3.1.

[Gab75, Corollary 1.4] Consider $d\in\mathbb{N}$. Assume that for a dimension
vector $\mathbf{d}=(d_{1},\cdots,d_{n})$, $d=\Sigma_{i=1}^{n}d_{i}$. Then
$\operatorname{rep}_{\mathbf{d}}(\Lambda)$ is a connected component of
$\operatorname{rep}_{d}(\Lambda)$. Moreover, all connected components of
$\operatorname{rep}_{d}(\Lambda)$ arise in this way.

Therefore, rather than varieties of representations of the same dimension, we
work on varieties of representations with the same dimension vectors.

######  Decomposition of irreducible components 3.2.

For each $1\leq i\leq s$, let
$\mathcal{Z}_{i}\subseteq\operatorname{rep}_{d_{i}}(\Lambda)$ be a
$\operatorname{GL}_{d_{i}}(k)$-stable subset. Consider the map

$\operatorname{GL}_{d}(k)\times\mathcal{Z}_{1}\times\cdots\times\mathcal{Z}_{s}\rightarrow\operatorname{rep}_{d}(\Lambda),$
$(\mathfrak{g},M_{1},\cdots,M_{s})\mapsto\mathfrak{g}.(M_{1}\oplus
M_{2}\oplus\cdots\oplus M_{s})$

where $d=d_{1}+d_{2}+\cdots+d_{s}$. The image of the map is denoted by

$\mathcal{Z}_{1}\oplus\mathcal{Z}_{2}\oplus\cdots\oplus\mathcal{Z}_{s}.$

If $\mathcal{Z}_{i}$ is an irreducible locally closed subset, for each $1\leq
i\leq s$, then the Zariski-closure
$\overline{\mathcal{Z}_{1}\oplus\mathcal{Z}_{2}\oplus\cdots\oplus\mathcal{Z}_{s}}$
is an irreducible closed subset of $\operatorname{rep}_{d}(\Lambda)$.
Moreover, if $\mathcal{Z}_{i}$ is an irreducible component, for each $1\leq
i\leq s$, then the following statements are equivalent.

  1. $(1)$

$\overline{\mathcal{Z}_{1}\oplus\mathcal{Z}_{2}\oplus\cdots\oplus\mathcal{Z}_{s}}$
is an irreducible component of $\operatorname{rep}_{d}(\Lambda)$

  2. $(2)$

$\min\operatorname{ext}_{\Lambda}^{1}(\mathcal{Z}_{i},\mathcal{Z}_{j})=0,$ for
all $1\leq i\neq j\leq s$

An irreducible component $\mathcal{Z}$ is said to be generically
indecomposable, if a general module in $\mathcal{Z}$ is indecomposable. Each
irreducible component $\mathcal{Z}$ can be written as a direct sum of
generically indecomposable irreducible components
$\mathcal{Z}=\overline{\mathcal{Z}_{1}\oplus\mathcal{Z}_{2}\oplus\cdots\oplus\mathcal{Z}_{s}}$,
in a unique way. In this case, we set
$|\mathcal{Z}|=|\\{\mathcal{Z}_{1},\cdots,\mathcal{Z}_{s}\\}|$.

######  Lemma 3.3.

Consider a dimension vector $d$ and an irreducible component $\mathcal{Z}_{i}$
of $\operatorname{rep}_{d_{i}}(\Lambda)$, for $1\leq i\leq s$. Assume that
$\mathcal{Z}=\overline{\mathcal{Z}_{1}\oplus\mathcal{Z}_{2}\oplus\cdots\oplus\mathcal{Z}_{s}}$
is an irreducible component. Let $\tilde{\mathcal{U}_{i}}$ be a non-empty open
(dense) subset of $\mathcal{Z}_{i}$, for each $1\leq i\leq s$. Then there
exists a non-empty open (dense) subset $\tilde{\mathcal{U}}$ of $\mathcal{Z}$
such that
$\tilde{\mathcal{U}}\subseteq\tilde{\mathcal{U}}_{1}\oplus\tilde{\mathcal{U}_{2}}\oplus\cdots\oplus\tilde{\mathcal{U}_{s}}$.Therefore,
a general module in $\mathcal{Z}$ can be written as a direct sum of general
modules in $\mathcal{Z}_{i}$ for $1\leq i\leq s$.

###### Proof.

As
$\operatorname{GL}_{d}(k)\times\tilde{\mathcal{U}_{1}}\times\cdots\times\tilde{\mathcal{U}_{s}}$
is open and dense in
$\operatorname{GL}_{d}(k)\times\mathcal{Z}_{1}\times\cdots\times\mathcal{Z}_{s}$,
by Remark 1.15, the closure of its image is $\mathcal{Z}$. Since
$\tilde{\mathcal{U}_{1}}\oplus\cdots\oplus\tilde{\mathcal{U}_{s}}$ is
constructible and its closure is irreducible, by Theorem 1.18, there exists a
non-empty open (dense) subset $\tilde{\mathcal{U}}$ of $\mathcal{Z}$ such that
$\tilde{\mathcal{U}}\subseteq\tilde{\mathcal{U}_{1}}\oplus\cdots\oplus\tilde{\mathcal{U}_{s}}$
∎

######  Lemma 3.4.

[Pla13, Corollary 2.2] Consider a dimension vector $d$ and an irreducible
component $\mathcal{Z}\subseteq\operatorname{rep}_{d}(\Lambda)$. There exist a
non-empty open (dense) subset $\tilde{\mathcal{U}}$ of $\mathcal{Z}$ and
$P^{-1},P^{0}\in\operatorname{proj}\Lambda$ such that every
$M\in\tilde{\mathcal{U}}$ has a minimal projective presentation of the form

$P^{-1}\rightarrow P^{0}\rightarrow M\rightarrow 0.$

By the above lemma, one can assign a g-vector to each irreducible component as
follows.

######  Definition 3.5.

Let $\mathcal{Z}$ be an irreducible component of
$\operatorname{rep}_{d}(\Lambda)$ for a dimension vector $d$. Then the
g-vector of a general element of $\mathcal{Z}$ is said to be the g-vector of
$\mathcal{Z}$ and denoted by $g^{\mathcal{Z}}$.

######  Lemma 3.6.

[CILFS14, Lemma 5.12] Consider irreducible components $\mathcal{Z}$,
$\mathcal{Z}_{1},\cdots,\mathcal{Z}_{s}$. If
$\mathcal{Z}=\overline{\mathcal{Z}_{1}\oplus\mathcal{Z}_{2}\oplus\cdots\oplus\mathcal{Z}_{s}}$,
then $g^{\mathcal{Z}}=g^{\mathcal{Z}_{1}}+\cdots+g^{\mathcal{Z}_{s}}$.

###### Proof.

By Lemma 3.4, for each $1\leq i\leq s$, there is a non-empty open (dense)
subset $\tilde{\mathcal{U}_{i}}$ of $\mathcal{Z}_{i}$ such that for
$M_{i}\in\tilde{\mathcal{U}_{i}}$, $g^{M_{i}}=g^{\mathcal{Z}_{i}}$. So by
Lemma 3.3 and Lemma 3.4, there exists a non-empty open (dense) subset
$\tilde{\mathcal{U}}$ of $\mathcal{Z}$ such that
$\tilde{\mathcal{U}}\subseteq\tilde{\mathcal{U}}_{1}\oplus\tilde{\mathcal{U}_{2}}\oplus\cdots\oplus\tilde{\mathcal{U}_{s}}$
and for each $M\in\tilde{\mathcal{U}}$, $g^{M}=g^{\mathcal{Z}}$. Therefore,
$g^{\mathcal{Z}}=g^{\mathcal{Z}_{1}}+\cdots+g^{\mathcal{Z}_{s}}$. ∎

By the above assumption, the decomposition of a given irreducible component
does not necessarily provides the generic decomposition of its g-vector.
Later, we will see that

$g^{\mathcal{Z}}=g^{\mathcal{Z}_{1}}\oplus\cdots\oplus g^{\mathcal{Z}_{s}},$

if and only if $\mathcal{Z}$ is a generically $\tau$-reduced component.

######  Remark 3.7.

Let $d$ be a dimension vector and $\mathcal{Z}$ be an irreducible component of
$\operatorname{rep}_{d}(\Lambda)$. It is well known that $\mathcal{Z}$ is
$\operatorname{GL}_{d}(k)$-stable. Additionally, by the Voigt’s Isomorphism
and the Auslander-Reiten Duality, for each $M\in\mathcal{Z}$, we have

$\operatorname{codim}_{\mathcal{Z}}\mathcal{O}_{M}\leq\operatorname{ext}_{\Lambda}^{1}(M,M)\leq\operatorname{hom}_{\Lambda}(M,\tau
M).$

######  Definition 3.8.

Consider a dimension vector $d$. An irreducible component $\mathcal{Z}$ of
$\operatorname{rep}_{d}(\Lambda)$ is said to be generically $\tau$-reduced, if
for a general $M\in\mathcal{Z}$,

$\operatorname{hom}_{\Lambda}(M,\tau
M)=\operatorname{codim}_{\mathcal{Z}}\mathcal{O}_{M}.$

######  Theorem 3.9.

[Pla13, Theorem 1.2] Let $g$ be a g-vector. There is a non-empty open (dense)
subset $\mathcal{U}$ of $\operatorname{Hom}_{\Lambda}(g)$ and a dimension
vector $d(g)$ satisfy in the following conditions.

  1. $(1)$

For any $a\in\mathcal{U}$, $[\operatorname{Coker}(a)]=d(g)$, and the map

$\operatorname{Coker}:\mathcal{U}\rightarrow\operatorname{rep}_{d(g)}(\Lambda)$

is a morphism of varieties.

  2. $(2)$

$\mathcal{Z}_{g}=\overline{\bigcup_{a\in\mathcal{U}}\mathcal{O}_{\operatorname{Coker}(a)}}$
is a generically $\tau$-reduced component of
$\operatorname{rep}_{d(g)}(\Lambda)$.

Moreover, we have a surjective map

$\mathcal{Z}_{-}:K_{0}(\operatorname{proj}\Lambda)\rightarrow\\{\text{generically
$\tau$-reduced components of $\operatorname{rep}(\Lambda)$}\\}$

such that $\mathcal{Z}_{g}=\mathcal{Z}_{h}$ if and only if their generic
decompositions are the same, when one forgets negative direct summands.

######  Corollary 3.10.

Let $g$ be g-vector without negative direct summand. Then the open (dense)
subset $\mathcal{U}$ in Theorem 3.9 can be considered such that it satisfies
in the following extra condition.

  1. $(3)$

All $a\in\mathcal{U}$ are minimal projective presentations.

Moreover, any general $M$ in $\mathcal{Z}_{g}$ is in
$\mathcal{O}_{\operatorname{Coker}(a)}$, for some $a\in\mathcal{U}$.

###### Proof.

Since for all open dense $\mathcal{U}^{\prime}\subseteq\mathcal{U}$ we have

$\overline{\bigcup_{a\in\mathcal{U}^{\prime}}\mathcal{O}_{\operatorname{Coker}(a)}}=\overline{\bigcup_{a\in\mathcal{U}}\mathcal{O}_{\operatorname{Coker}(a)}},$

the statement follows from the next lemma 3.11. For the last part, note that
by Theorem 1.18,
$\bigcup_{a\in\mathcal{U}}\mathcal{O}_{\operatorname{Coker}(a)}$ is
constructible and dense in the irreducible component $\mathcal{Z}_{g}$. So it
is immediately deduced by Lemma 1.17. ∎

######  Lemma 3.11.

[Pla13, Corollary 2.8] Let $g$ be a g-vector. If the generic decomposition of
$g$ has no negative term, then a general element of
$\operatorname{Hom}_{\Lambda}(g)$ is a minimal projective presentation.

###### Proof.

One can check that if a non-zero indecomposable projective presentation is
minimal. Thus by Remark 2.16, a general morphism of
$\operatorname{Hom}_{\Lambda}(g)$ is a minimal projective presentation. ∎

######  Lemma 3.12.

Let $g$ be a g-vector with no negative direct summand. Then $\mathcal{Z}_{g}$
has an open (dense) subset $\tilde{\mathcal{U}}$ such that any
$M\in\tilde{\mathcal{U}}$ admits a minimal projective presentation of the form

${P^{g-}}$${P^{g+}}$${M}$${0.}$

Therefore, by definition, $g^{\mathcal{Z}_{g}}=g$ and so the map

$g^{-}:\\{\text{generically $\tau$-reduced components of
$\operatorname{rep}(\Lambda)$}\\}\rightarrow\mathbb{Z}^{n}$

gives us a bijection between the set of generically $\tau$-reduced components
and g-vectors with non-negative777We say $g$ is non-negative (non–positive,
respectively), if $[P^{g+}]\neq 0$ ($[P^{g-}]\neq 0$, respectively). direct
summands. Moreover, $g$ is a direct summand of any g-vector $h$ with
$\mathcal{Z}_{h}=\mathcal{Z}_{g}$.

###### Proof.

It follows from [Pla13, Corollary 2.17]. ∎

######  Remark 3.13.

In the case that $g$ has no negative direct summand, it was shown in [Pla13,
Section 2.6] that

$\operatorname{Hom}_{\Lambda}(g)_{max}:=\\{a\in\operatorname{Hom}_{\Lambda}(g)\mid\text{$a$
is of maximal rank}\\}$

is equal to

$\operatorname{Hom}_{\Lambda}(g)_{d(g)}:=\\{a\in\operatorname{Hom}_{\Lambda}(g)\mid[\operatorname{Coker}(a)]=d(g)\\}.$

As the first one is an open dense subset of $\operatorname{Hom}_{\Lambda}(g)$,
then so is the second one.

We spend the rest of this section to prove the following conjecture under the
non-decreasing condition. Indeed, we will show that the non-decreasing
condition on a given g-vector $g$ guarantees that $|\mathcal{Z}_{g}|$ is lower
than or equal to $n$, 3.21. Afterwards, we apply results of [Asa21] regarding
the wall and chamber structure of $\Lambda$ to prove the second statement,
when the non-decreasing condition is satisfied.

######  Conjecture 3.14.

[CILFS14, Conjecture 6.3] Let
$\mathcal{Z}=\overline{\mathcal{Z}_{1}\oplus\cdots\oplus\mathcal{Z}_{s}}$ be a
generically $\tau$-reduced component. Then the following statements hold.

  * $(1)$

$|\mathcal{Z}|\leq|\Lambda|$.

  * $(2)$

$|\mathcal{Z}|=|\Lambda|$ if and only if for all $1\leq i\leq s$,

$\min\\{\operatorname{hom}_{\Lambda}(X,\tau X)\mid X\in\mathcal{Z}_{i}\\}=0,$

and $\mathcal{Z}$ is maximal, that is, if $\mathcal{Z}^{\prime}$ and
$\overline{\mathcal{Z}\oplus\mathcal{Z}^{\prime}}$ are generically
$\tau$-reduced components, then
$|\mathcal{Z}|=|\overline{\mathcal{Z}\oplus\mathcal{Z}^{\prime}}|$.

Now, we study the link between generic decompositions of g-vectors and
decompositions of generically $\tau$-reduced components. It is the key to
achieving our goal.

######  Lemma 3.15.

[Pla13, Lemma 2.6] For projective presentations

${P^{-1}}$${P^{0}}$${M}$${0}$$\scriptstyle{a}$

and ${P^{\prime-1}}$${P^{\prime
0}}$${M^{\prime}}$${0,}$$\scriptstyle{a^{\prime}}$

$e(a,a^{\prime})\geq\operatorname{hom}_{\Lambda}(M^{\prime},\tau M)$ and
equality holds if $a$ is minimal.

######  Lemma 3.16.

[GLFS23, Corollary 1.4] For positive integers $d$ and $d^{\prime}$, the
following functions are upper semi-continuous.

$\operatorname{hom}_{\Lambda}(-,\tau?):\operatorname{rep}_{d}(\Lambda)\times\operatorname{rep}_{d^{\prime}}(\Lambda)\rightarrow\mathbb{Z},\hskip
19.91692pt\operatorname{hom}_{\Lambda}(-,\tau-):\operatorname{rep}_{d}(\Lambda)\rightarrow\mathbb{Z}$

The next lemma which plays an important role in the proofs of Theorem 3.18 and
Theorem 4.2 was independently proved by Calvin Pfeifer in a similar way
[Pfe23, Lemma 4.6].

######  Lemma 3.17.

Let $g$ and $h$ be g-vectors with no negative direct summand. Then

$e(g,h)=\min\operatorname{hom}_{\Lambda}(\mathcal{Z}_{h},\tau\mathcal{Z}_{g}).$

###### Proof.

We set
$m_{0}=\min\operatorname{hom}_{\Lambda}(\mathcal{Z}_{h},\tau\mathcal{Z}_{g})$.
By Lemma 3.12, there exist non-empty open (dense) subsets
$\tilde{\mathcal{U}_{g}}\subseteq\mathcal{Z}_{g}$ and
$\tilde{\mathcal{U}_{h}}\subseteq\mathcal{Z}_{h}$ with the property that each
$M\in\tilde{\mathcal{U}_{g}}$ and $N\in\tilde{\mathcal{U}_{h}}$ have minimal
projective presentations of the form

${P^{g-}}$${P^{g+}}$${M}$${0,}$${P^{h-}}$${P^{h+}}$${N}$${0.}$$\scriptstyle{a}$$\scriptstyle{b}$

By Lemma 3.16, the function
$\operatorname{hom}_{\Lambda}(-,\tau?):\mathcal{Z}_{h}\times\mathcal{Z}_{g}\rightarrow\mathbb{Z}$
is upper semi-continuous. Thus there exists a non-empty open (dense) subset
$\tilde{\mathcal{V}}\subseteq\tilde{\mathcal{U}_{h}}\times\tilde{\mathcal{U}_{g}}$
such that for all $(N,M)\in\tilde{\mathcal{V}}$,
$\operatorname{hom}_{\Lambda}(N,\tau M)=m_{0}$. So Lemma 3.15 implies that
$e(a,b)=m_{0}$ and thus $e(g,h)\leq m_{0}$.
On the other hand, as the function
$e(-,?):\operatorname{Hom}_{\Lambda}(g)\times\operatorname{Hom}_{\Lambda}(h)\rightarrow\mathbb{Z}$
is upper semi-continuous,

$\mathcal{V}=\\{(a,b)\in\operatorname{Hom}_{\Lambda}(g)\times\operatorname{Hom}_{\Lambda}(h)\mid
e(a,b)=e(g,h)\\}$

is open and dense. Whereas
$\operatorname{Hom}_{\Lambda}(g)_{d(g)}\subseteq\operatorname{Hom}_{\Lambda}(g)$
and
$\operatorname{Hom}_{\Lambda}(h)_{d(h)}\subseteq\operatorname{Hom}_{\Lambda}(h)$
are non-empty open (dense) subsets, by Corollary 3.10(3),

$\mathcal{V}^{\prime}=\\{(a,b)\in\mathcal{V}\mid\text{$a\in\operatorname{Hom}_{\Lambda}(g)_{d(g)}$
and $b\in\operatorname{Hom}_{\Lambda}(h)_{d(h)}$ are minimal}\\}$

is open (and dense), as well. Hence, by Lemma 3.15, for each
$(a,b)\in\mathcal{V}^{\prime}$, we have

$e(g,h)=e(a,b)=\operatorname{hom}_{\Lambda}(\operatorname{Coker}(b),\tau\operatorname{Coker}(a))\geq
m_{0}.$

∎

######  Theorem 3.18.

Let $\\{\mathcal{Z}_{1},\mathcal{Z}_{2},\cdots,\mathcal{Z}_{s}\\}$ be a set of
irreducible components. Then the following statements are equivalent.

  1. $(1)$

$\mathcal{Z}=\overline{\mathcal{Z}_{1}\oplus\mathcal{Z}_{2}\oplus\cdots\oplus\mathcal{Z}_{s}}$
is a generically $\tau$-reduced component.

  2. $(2)$

Each $\mathcal{Z}_{i}$ is a generically $\tau$-reduced component and

$\min\operatorname{hom}_{\Lambda}(\mathcal{Z}_{i},\tau\mathcal{Z}_{j})=0,$

for all $1\leq i\neq j\leq s$.

  3. $(3)$

Each $\mathcal{Z}_{i}$ is a generically $\tau$-reduced component and

$e(g^{\mathcal{Z}_{i}},g^{\mathcal{Z}_{j}})=0,$

for all $1\leq i\neq j\leq s$.

In this case, $g^{\mathcal{Z}}=g^{\mathcal{Z}_{1}}\oplus\cdots\oplus
g^{\mathcal{Z}_{s}}$ is the generic decomposition of $g^{\mathcal{Z}}$.

###### Proof.

It is an immediate consequence of [CILFS14, Theorem 1.3], and Lemma 3.17
together with Lemma 3.6. ∎

######  Corollary 3.19.

For a g-vector $g$ with no negative direct summand, the following statements
are equivalent.

  * $(1)$

$g=g_{1}\oplus g_{2}\oplus\cdots\oplus g_{s}$ for some $g_{i}\in
K_{0}(\operatorname{proj}\Lambda)$, $1\leq i\leq s$.

  * $(2)$

$\mathcal{Z}_{g}=\overline{\mathcal{Z}_{g_{1}}\oplus\mathcal{Z}_{g_{2}}\oplus\cdots\oplus\mathcal{Z}_{g_{s}}}.$

Moreover, a generically $\tau$-reduced component $\mathcal{Z}$ is generically
indecomposable if and only if $g^{\mathcal{Z}}$ is generically indecomposable.

###### Proof.

It follows from Theorem 3.18 and the uniqueness of generic decompositions. ∎

######  Remark 3.20.

Let $g$ be a g-vector and $g=g_{1}\oplus g_{2}\oplus\cdots\oplus g_{s}$ a
generic decomposition. Then

$d(g)=\sum_{i=1}^{s}d(g_{s}).$

Theorem 3.18 leads us to propose a generalization for [GLFS23, Theorem 2.9] or
[CILFS14, Theorem 6.1]. Note that due to Lemma 3.17, for a generically
$\tau$-reduced component $\mathcal{Z}$,
$\min\operatorname{hom}_{\Lambda}(\mathcal{Z},\tau\mathcal{Z})=0$ if and only
if $g^{\mathcal{Z}}$ is tame.

######  Theorem 3.21.

Let $g$ be an arbitrary g-vector. Then
$|\mathcal{Z}_{g}|\leq|\operatorname{ind}(g)|$ and equality holds if and only
if $g$ has no negative direct summand. In particular, if $g$ satisfies the
non-decreasing condition, then $|\mathcal{Z}_{g}|\leq|\Lambda|$.

###### Proof.

It is a direct consequence of Theorem 3.18 and Theorem 2.30. ∎

According to Remark 2.19, we provide the following consequence which together
with Theorem 3.18 proves Conjecture 3.14($2$), under the non-decreasing
condition.

######  Corollary 3.22.

Let $\mathcal{Z}$ be a generically $\tau$-reduced component. Then the
following statements are equivalent.

  * $(1)$

$g^{\mathcal{Z}}\in\mathcal{C}^{\circ}(\mathrm{P})$ for a $2$-term silting
complex $\mathrm{P}\in K^{b}(\operatorname{proj}\Lambda)$.

  * $(2)$

$|\mathcal{Z}|=|\Lambda|$ and $g^{\mathcal{Z}}$ satisfies the non-decreasing
condition.

##  4\. Numerical criteria for tameness

We use the following lemma to obtain some criteria for tameness of g-vectors.
Then we provide some examples and computations.

######  Lemma 4.1.

[AR85, Theorem 1.4] For modules $M$ and $N$,

$\langle
g^{M},[N]\rangle=\operatorname{hom}_{\Lambda}(M,N)-\operatorname{hom}_{\Lambda}(N,\tau
M)$

######  Theorem 4.2.

Let $g$ and $h$ be g-vectors with no negative direct summand. Then

$e(g,h)=\min\operatorname{hom}_{\Lambda}(\mathcal{Z}_{g},\mathcal{Z}_{h})-\langle
g,d(h)\rangle.$

###### Proof.

It follows from Lemma 3.17 and upper semi-continuity of the functions
$e(-,?)$, $\operatorname{hom}_{\Lambda}(?,\tau-)$ and
$\operatorname{hom}_{\Lambda}(-,?)$. ∎

######  Theorem 4.3.

Consider a g-vector $g$ with no negative direct summand. Then the following
statements hold.

  * $(1)$

$g$ is wild if and only if
$\min\operatorname{hom}_{\Lambda}(\mathcal{Z}_{g},\mathcal{Z}_{g})>\langle
g,d(g)\rangle.$

  * $(2)$

$g$ is tame if and only if $\langle
g,d(g)\rangle=\min\operatorname{hom}_{\Lambda}(\mathcal{Z}_{g},\mathcal{Z}_{g}).$

###### Proof.

It is a direct consequence of Theorem 4.2. ∎

######  Corollary 4.4.

By the above notation, $\langle g,d(g)\rangle\geq 0$, if $g$ is tame. In
particular, if $\Lambda$ is E-tame, then $\langle g,d(g)\rangle\geq 0$, for
all g-vector $g$.

######  Corollary 4.5.

[CILFS14, Corollary 5.14] Let $d$ be a dimension vector and
$\mathcal{Z}\subseteq\operatorname{rep}_{d}(\Lambda)$ be a generically
$\tau$-reduced component. If $\langle g^{\mathcal{Z}},d\rangle<0$, then

$\min\\{\operatorname{hom}_{\Lambda}(M,\tau M)\mid M\in\mathcal{Z}\\}>0.$

###### Proof.

Theorem 4.3 implies that $g^{\mathcal{Z}}$ is wild. Thus
$\min\operatorname{hom}_{\Lambda}(\mathcal{Z},\tau\mathcal{Z})=e(g^{\mathcal{Z}},g^{\mathcal{Z}})>0$.
Now, the statement follows from the following inequality.

$\min\operatorname{hom}_{\Lambda}(\mathcal{Z},\tau\mathcal{Z})\leq\min\\{\operatorname{hom}_{\Lambda}(M,\tau
M)\mid M\in\mathcal{Z}\\}.$

∎

######  Example 4.6.

Consider an $m$-Kronecker’s quiver, for $m\geq 3$.

${1}$${2}$${\vdots}$

We have $S_{(2)}\cong P_{(2)}$ and $[P_{(1)}]=(1,m)$. Consider the g-vector
$g=[P_{(1)}]-[P_{(2)}]=(1,-1)$. It is easy to compute
$d(g)=[P_{(1)}]-[S_{(2)}]=(1,m-1)$. So $\langle g,d(g)\rangle=2-m<0$. Thus by
Corollary 4.4, $(1,-1)$ is a wild g-vector.

A more general statement also holds. Consider projective modules $P_{0}$ and
$P_{1}$ with no non-zero direct summand in common. Assume that there is a
monomorphism in $\operatorname{Hom}_{\Lambda}(P_{1},P_{0})$. Then by upper
semi-continuity of the function

$\operatorname{dim}_{k}\operatorname{Ker}:\operatorname{Hom}_{\Lambda}(P_{1},P_{0})\rightarrow\mathbb{Z},$

the set of monomorphisms from $P_{1}$ to $P_{0}$ are open and dense. So
$g=[P_{0}]-[P_{1}]$ does not admit a negative direct summand. Thus a general
module in $\mathcal{Z}_{g}$ is of projective dimension $1$. Therefore,
$d(g)=[P_{0}]-[P_{1}]\in K_{0}(\operatorname{mod}\Lambda)$.
Now, assume that $P_{0}=P_{(1)}^{a_{1}}\oplus\cdots\oplus P_{(n)}^{a_{n}}$ and
$P_{1}=P_{(1)}^{b_{1}}\oplus\cdots\oplus P_{(n)}^{b_{n}}$. By definition,
$g=(a_{1}-b_{1},\cdots,a_{n}-b_{n})$. Moreover, we know that

$[P_{(i)}]=(\operatorname{dim}_{k}e_{1}\Lambda
e_{i},\cdots,\operatorname{dim}_{k}e_{n}\Lambda e_{i}),$

where $\\{e_{1},\cdots,e_{n}\\}$ is the complete set of primitive orthogonal
idempotents corresponding to $\\{P_{(1)},\cdots,P_{(n)}\\}$. So

$d(g)=(\sum_{i=1}^{n}(a_{i}-b_{i})\operatorname{dim}_{k}e_{1}\Lambda
e_{i},\cdots,\sum_{i=1}^{n}(a_{i}-b_{i})\operatorname{dim}_{k}e_{n}\Lambda
e_{i}).$

Therefore,

$\langle g,d(g)\rangle=\sum_{1\leq i,j\leq
n}(a_{i}-b_{i})(a_{j}-b_{j})\operatorname{dim}_{k}e_{j}\Lambda e_{i}.$

Note that we had assumed that $P_{1}$ and $P_{0}$ do not share any non-zero
direct summands. So in the above formula, if $a_{i}\neq 0$, then $b_{i}=0$.
Consider $A=\\{1\leq i\leq n\mid a_{i}\neq 0\\}$ and $B=\\{1\leq j\leq n\mid
b_{j}\neq 0\\}$. Thus

$\displaystyle\langle g,d(g)\rangle=\sum_{i,i^{\prime}\in
A}a_{i}a_{i^{\prime}}\operatorname{dim}_{k}e_{i}\Lambda
e_{i^{\prime}}+\sum_{j,j^{\prime}\in
B}b_{j}b_{j^{\prime}}\operatorname{dim}_{k}e_{j}\Lambda
e_{j^{\prime}}-\sum_{i\in A,j\in
B}a_{i}b_{j}\operatorname{dim}_{k}(e_{i}\Lambda e_{j}+e_{j}\Lambda e_{i}).$

Moreover, in this case, $d(tg)=td(g)$, for all $t\in\mathbb{N}$. Therefore, if
$\langle g,d(g)\rangle$ is negative, then $tg$ is a wild g-vector, for all
$t\in\mathbb{N}$.

######  Remark 4.7.

Consider a set $A$ consisting of some vertices of $Q$ and a set $B$ consisting
of some sink vertices. For $j\in B$, let

$A_{j}=\\{i\in A\mid\textit{there is a path from $i$ to $j$}\\}.$

Assume that for all $j\in B$, $A_{j}\neq\emptyset$ and

$|\bigcup_{j\in B}A_{j}|>|B|.$

Then there is a monomorphism from $\bigoplus_{j\in B}P_{(j)}$ to
$\bigoplus_{i\in A}P_{(i)}$. Hence, for $g=\sum_{i\in A}[P_{(i)}]-\sum_{j\in
B}[P_{(j)}]$,

$\langle g,d(g)\rangle=\sum_{i,i^{\prime}\in
A}\operatorname{dim}_{k}e_{i}\Lambda e_{i^{\prime}}-\sum_{i\in A,j\in
B}\operatorname{dim}_{k}e_{j}\Lambda e_{i}+|B|.$

######  Example 4.8.

For each of the following quivers, let $g=[P_{(i)}]-[P_{(j)}]$. Then $\langle
g,d(g)\rangle<0$ and thus it is a wild g-vectors.

${\bullet}$${\bullet}$${j}$${i}$${j}$${i}$${j}$${\bullet}$${i}$${\circ}$

The same also holds for each of the following bound quivers.

${\bullet}$${j}$${i}$$\displaystyle
I:(\alpha\gamma_{1}\beta)^{2}=(\alpha\gamma_{2}\beta)^{2}=0,\gamma_{1}\beta\alpha\gamma_{2}=\gamma_{1}\beta\alpha\gamma_{1}=\gamma_{2}\beta\alpha\gamma_{1}$${\circ}$${\bullet}$${j}$${i}$${{I:(\lambda\alpha\delta)^{3}=0}}$${\circ}$$\scriptstyle{\lambda}$$\scriptstyle{\gamma_{1}}$$\scriptstyle{\beta}$$\scriptstyle{\delta}$$\scriptstyle{\alpha}$$\scriptstyle{\gamma_{2}}$$\scriptstyle{\alpha}$$\scriptstyle{\lambda}$$\scriptstyle{\delta}$$\scriptstyle{\gamma}$$\scriptstyle{\beta}$

One may also check the tamness of a g-vector by using the dimension of
corresponding generically $\tau$-reduced component.

######  Lemma 4.9.

Let $g$ be an arbitrary g-vector. Then

$\operatorname{dim}\mathcal{Z}_{g}=\langle d(g),d(g)\rangle-\langle
g,d(g)\rangle.$

###### Proof.

By Lemma 4.1, for a general $M\in\mathcal{Z}_{g}$, we have

$\langle
g,d(g)\rangle=\operatorname{end}_{\Lambda}(M)-\operatorname{hom}_{\Lambda}(M,\tau
M).$

Since $\mathcal{Z}_{g}$ is generically $\tau$-reduced,
$\operatorname{hom}_{\Lambda}(M,\tau
M)=\operatorname{codim}_{\mathcal{Z}_{g}}\mathcal{O}_{M}$. So

$\langle
g,d(g)\rangle=\operatorname{end}_{\Lambda}(M)-\operatorname{codim}_{\mathcal{Z}_{g}}\mathcal{O}_{M}=\operatorname{end}_{\Lambda}(M)-\operatorname{dim}\mathcal{Z}_{g}+\operatorname{dim}\mathcal{O}_{M}=$
$\operatorname{end}_{\Lambda}(M)-\operatorname{dim}\mathcal{Z}_{g}+\operatorname{dim}\operatorname{GL}_{d(g)}(k)-\operatorname{end}_{\Lambda}(M).$

Therefore, $\langle g,d(g)\rangle=\langle
d(g),d(g)\rangle-\operatorname{dim}\mathcal{Z}_{g}$. ∎

######  Remark 4.10.

For an arbitrary g-vector $g$, $\langle d(g),d(g)\rangle\geq\langle
g,d(g)\rangle$.

######  Corollary 4.11.

Let $g$ be an arbitrary g-vector with no negative direct summand. Then

$e(g,g)+\langle
d(g),d(g)\rangle=\operatorname{dim}\mathcal{Z}_{g}+\min\operatorname{hom}_{\Lambda}(\mathcal{Z}_{g},\mathcal{Z}_{g}).$

###### Proof.

It is obtained by combining Lemma 4.9 and Theorem 4.2. ∎

######  Remark 4.12.

Let $g$ be a tame g-vector with no negative direct summand. Then

$\min\operatorname{hom}_{\Lambda}(\mathcal{Z}_{g},\mathcal{Z}_{g})\leq\operatorname{dim}\langle
d(g),d(g)\rangle\geq\operatorname{dim}\mathcal{Z}_{g}.$

## Acknowledgment

We would like to express our gratitude to Pierre-Guy Plamondon for his
contribution in describing some results of [PYK21] and [Pla13] through email.
Additionally, we are also thankful to Sota Asai who helped us to prepare
Corollary 3.22 and informed us that Lemma 3.17 was independently proved by
Calvin Pfeifer. The research of the authors is supported by Iran National
Science Foundation (INSF), Project No. 4003197.

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